diff --git a/_alghierarchy.svg b/_alghierarchy.svg
deleted file mode 100644
index 56a0129122..0000000000
--- a/_alghierarchy.svg
+++ /dev/null
@@ -1,1328 +0,0 @@
-
-
-
diff --git a/_algtophierarchy.svg b/_algtophierarchy.svg
deleted file mode 100644
index d8d026fdca..0000000000
--- a/_algtophierarchy.svg
+++ /dev/null
@@ -1,1092 +0,0 @@
-
-
\ No newline at end of file
diff --git a/_frege_A.svg b/_frege_A.svg
deleted file mode 100644
index 44f801e420..0000000000
--- a/_frege_A.svg
+++ /dev/null
@@ -1,31 +0,0 @@
-
-
diff --git a/_frege_BimA.svg b/_frege_BimA.svg
deleted file mode 100644
index 345b8466ed..0000000000
--- a/_frege_BimA.svg
+++ /dev/null
@@ -1,34 +0,0 @@
-
-
diff --git a/_frege_cfheF.svg b/_frege_cfheF.svg
deleted file mode 100644
index 3e42752c0d..0000000000
--- a/_frege_cfheF.svg
+++ /dev/null
@@ -1,41 +0,0 @@
-
-
diff --git a/_frege_dfAeqB.svg b/_frege_dfAeqB.svg
deleted file mode 100644
index 7444bc98c8..0000000000
--- a/_frege_dfAeqB.svg
+++ /dev/null
@@ -1,28 +0,0 @@
-
-
diff --git a/_frege_fheF.svg b/_frege_fheF.svg
deleted file mode 100644
index 7bb21b83e8..0000000000
--- a/_frege_fheF.svg
+++ /dev/null
@@ -1,41 +0,0 @@
-
-
diff --git a/_frege_funcf.svg b/_frege_funcf.svg
deleted file mode 100644
index bb2d962243..0000000000
--- a/_frege_funcf.svg
+++ /dev/null
@@ -1,24 +0,0 @@
-
-
diff --git a/_frege_funf.svg b/_frege_funf.svg
deleted file mode 100644
index 7bb671eede..0000000000
--- a/_frege_funf.svg
+++ /dev/null
@@ -1,24 +0,0 @@
-
-
diff --git a/_frege_jA.svg b/_frege_jA.svg
deleted file mode 100644
index 36e793ec65..0000000000
--- a/_frege_jA.svg
+++ /dev/null
@@ -1,32 +0,0 @@
-
-
diff --git a/_frege_jABinS.svg b/_frege_jABinS.svg
deleted file mode 100644
index fb526f89b6..0000000000
--- a/_frege_jABinS.svg
+++ /dev/null
@@ -1,32 +0,0 @@
-
-
diff --git a/_frege_jAeqB.svg b/_frege_jAeqB.svg
deleted file mode 100644
index ce4e48db17..0000000000
--- a/_frege_jAeqB.svg
+++ /dev/null
@@ -1,32 +0,0 @@
-
-
diff --git a/_frege_jAinP.svg b/_frege_jAinP.svg
deleted file mode 100644
index 971cfaceb9..0000000000
--- a/_frege_jAinP.svg
+++ /dev/null
@@ -1,32 +0,0 @@
-
-
diff --git a/_frege_jBandA.svg b/_frege_jBandA.svg
deleted file mode 100644
index ee03a1bfc4..0000000000
--- a/_frege_jBandA.svg
+++ /dev/null
@@ -1,37 +0,0 @@
-
-
diff --git a/_frege_jBimA.svg b/_frege_jBimA.svg
deleted file mode 100644
index 372078340b..0000000000
--- a/_frege_jBimA.svg
+++ /dev/null
@@ -1,35 +0,0 @@
-
-
diff --git a/_frege_jBimnA.svg b/_frege_jBimnA.svg
deleted file mode 100644
index effb5e6029..0000000000
--- a/_frege_jBimnA.svg
+++ /dev/null
@@ -1,36 +0,0 @@
-
-
diff --git a/_frege_jBorA.svg b/_frege_jBorA.svg
deleted file mode 100644
index 36f110a744..0000000000
--- a/_frege_jBorA.svg
+++ /dev/null
@@ -1,36 +0,0 @@
-
-
diff --git a/_frege_jBxor1A.svg b/_frege_jBxor1A.svg
deleted file mode 100644
index 9efdc7ce1e..0000000000
--- a/_frege_jBxor1A.svg
+++ /dev/null
@@ -1,45 +0,0 @@
-
-
diff --git a/_frege_jBxor2A.svg b/_frege_jBxor2A.svg
deleted file mode 100644
index dae89db40b..0000000000
--- a/_frege_jBxor2A.svg
+++ /dev/null
@@ -1,45 +0,0 @@
-
-
diff --git a/_frege_jGandBandA.svg b/_frege_jGandBandA.svg
deleted file mode 100644
index 615a0772ca..0000000000
--- a/_frege_jGandBandA.svg
+++ /dev/null
@@ -1,41 +0,0 @@
-
-
diff --git a/_frege_jGandBimA.svg b/_frege_jGandBimA.svg
deleted file mode 100644
index e949e27f63..0000000000
--- a/_frege_jGandBimA.svg
+++ /dev/null
@@ -1,39 +0,0 @@
-
-
diff --git a/_frege_jalAinP.svg b/_frege_jalAinP.svg
deleted file mode 100644
index 8fb097440f..0000000000
--- a/_frege_jalAinP.svg
+++ /dev/null
@@ -1,33 +0,0 @@
-
-
diff --git a/_frege_jexAinP.svg b/_frege_jexAinP.svg
deleted file mode 100644
index 5d71dff665..0000000000
--- a/_frege_jexAinP.svg
+++ /dev/null
@@ -1,35 +0,0 @@
-
-
diff --git a/_frege_jnA.svg b/_frege_jnA.svg
deleted file mode 100644
index 4ec59c59d5..0000000000
--- a/_frege_jnA.svg
+++ /dev/null
@@ -1,33 +0,0 @@
-
-
diff --git a/_frege_jnGandnBimA.svg b/_frege_jnGandnBimA.svg
deleted file mode 100644
index 4016bffc05..0000000000
--- a/_frege_jnGandnBimA.svg
+++ /dev/null
@@ -1,41 +0,0 @@
-
-
diff --git a/_frege_nA.svg b/_frege_nA.svg
deleted file mode 100644
index 719f4d83b3..0000000000
--- a/_frege_nA.svg
+++ /dev/null
@@ -1,32 +0,0 @@
-
-
diff --git a/_frege_njBimA.svg b/_frege_njBimA.svg
deleted file mode 100644
index f79ddea05e..0000000000
--- a/_frege_njBimA.svg
+++ /dev/null
@@ -1,36 +0,0 @@
-
-
diff --git a/_frege_njnAimB.svg b/_frege_njnAimB.svg
deleted file mode 100644
index 786397ec4a..0000000000
--- a/_frege_njnAimB.svg
+++ /dev/null
@@ -1,37 +0,0 @@
-
-
diff --git a/_frege_njnAimnB.svg b/_frege_njnAimnB.svg
deleted file mode 100644
index e7f133ec73..0000000000
--- a/_frege_njnAimnB.svg
+++ /dev/null
@@ -1,38 +0,0 @@
-
-
diff --git a/_frege_yfollowsx.svg b/_frege_yfollowsx.svg
deleted file mode 100644
index 91b1124d56..0000000000
--- a/_frege_yfollowsx.svg
+++ /dev/null
@@ -1,22 +0,0 @@
-
-
diff --git a/_frege_yinfbegx.svg b/_frege_yinfbegx.svg
deleted file mode 100644
index e54bcaaf16..0000000000
--- a/_frege_yinfbegx.svg
+++ /dev/null
@@ -1,22 +0,0 @@
-
-
diff --git a/_frege_yinfendx.svg b/_frege_yinfendx.svg
deleted file mode 100644
index 1f9fa7fe3e..0000000000
--- a/_frege_yinfendx.svg
+++ /dev/null
@@ -1,22 +0,0 @@
-
-
diff --git a/_frege_yprecedsx.svg b/_frege_yprecedsx.svg
deleted file mode 100644
index 40d3cd2d7a..0000000000
--- a/_frege_yprecedsx.svg
+++ /dev/null
@@ -1,22 +0,0 @@
-
-
diff --git a/_relexample.svg b/_relexample.svg
deleted file mode 100644
index dd29b9afc9..0000000000
--- a/_relexample.svg
+++ /dev/null
@@ -1,86 +0,0 @@
-
-
\ No newline at end of file
diff --git a/_tophierarchy.svg b/_tophierarchy.svg
deleted file mode 100644
index d5b2001ae0..0000000000
--- a/_tophierarchy.svg
+++ /dev/null
@@ -1,1162 +0,0 @@
-
-
\ No newline at end of file
diff --git a/changes-set.txt b/changes-set.txt
index d785ab03f3..fdd4558285 100644
--- a/changes-set.txt
+++ b/changes-set.txt
@@ -58,11 +58,7 @@ proposed syl5eqner eqnetrrid compare to eqnetrri or eqnetrrd
proposed syl5ss sstrid compare to sstri or sstrd
proposed syl5eqss eqsstrid compare to eqsstri or eqsstrd
proposed syl5eqssr eqsstrrid compare to eqsstr3i or eqsstr3d
-proposed eqsstr3i eqsstrri
-proposed eqsstr3d eqsstrrd
proposed syl5sseq sseqtrid compare to sseqtri or sseqtrd
-proposed syl5sseqr sseqtrrid
-proposed wl-luk-syl5 wl-luk-imtrid
proposed syl6 imtrdi alternate proposal: syldi
proposed syl6com imtrdicom
proposed syl6d imtrdid
@@ -84,6 +80,8 @@ proposed syl6eleq eleqtrdi compare to eleqtri or eleqtrd
proposed syl6eleqr eleqtrrdi compare to eleqtrri or eleqtrrd
proposed syl6ss sstrdi compare to sstri or sstrd
proposed syl6sseq sseqtrdi compare to sseqtri or sseqtrd
+proposed sseqtr4i sseqtrri
+proposed sseqtr4d sseqtrrd
proposed syl6sseqr sseqtrrdi
proposed syl6eqss eqsstrdi compare to eqsstri or eqsstrd
proposed syl6eqssr eqsstrrdi
@@ -91,12 +89,14 @@ proposed syl6eqbr eqbrtrdi compare to eqbrtri or eqbrtrd
proposed syl6eqbrr eqbrtrrdi compare to eqbrtrri or eqbrtrrd
proposed syl6breq breqtrdi compare to breqtri or breqtrd
proposed syl6breqr breqtrrdi compare to breqtrri or breqtrrd
-proposed wl-luk-syl6 wl-luk-imtrdi
(Please send any comments on these proposals to the mailing list or
make a github issue.)
DONE:
Date Old New Notes
+21-Nov-23 syl5sseqr sseqtrrid
+21-Nov-23 eqsstr3d eqsstrrd
+21-Nov-23 eqsstr3i eqsstrri
18-Nov-23 fun11iun f1iun labeling consistent with fiun
18-Nov-23 inex2ALTV inex2g moved from PM's mathbox to main set.mm
18-Nov-23 bj-rabbid rabbid moved from BJ's mathbox to main set.mm
diff --git a/discouraged b/discouraged
index 9c052cf0d9..4fef7aa6df 100755
--- a/discouraged
+++ b/discouraged
@@ -13015,8 +13015,17 @@
"wl-luk-imim2" is used by "wl-luk-ax2".
"wl-luk-imim2i" is used by "wl-impchain-mp-1".
"wl-luk-imim2i" is used by "wl-impchain-mp-2".
+"wl-luk-imim2i" is used by "wl-luk-imtrdi".
"wl-luk-imim2i" is used by "wl-luk-ja".
-"wl-luk-imim2i" is used by "wl-luk-syl6".
+"wl-luk-imtrdi" is used by "wl-luk-ax3".
+"wl-luk-imtrdi" is used by "wl-luk-pm2.27".
+"wl-luk-imtrid" is used by "wl-luk-ax3".
+"wl-luk-imtrid" is used by "wl-luk-com12".
+"wl-luk-imtrid" is used by "wl-luk-con1i".
+"wl-luk-imtrid" is used by "wl-luk-con4i".
+"wl-luk-imtrid" is used by "wl-luk-ja".
+"wl-luk-imtrid" is used by "wl-luk-mpi".
+"wl-luk-imtrid" is used by "wl-luk-pm2.04".
"wl-luk-ja" is used by "wl-luk-ax2".
"wl-luk-mpi" is used by "wl-luk-imim2i".
"wl-luk-pm2.04" is used by "wl-impchain-com-1.2".
@@ -13033,19 +13042,10 @@
"wl-luk-syl" is used by "wl-luk-a1d".
"wl-luk-syl" is used by "wl-luk-ax1".
"wl-luk-syl" is used by "wl-luk-id".
+"wl-luk-syl" is used by "wl-luk-imtrdi".
+"wl-luk-syl" is used by "wl-luk-imtrid".
"wl-luk-syl" is used by "wl-luk-pm2.18d".
"wl-luk-syl" is used by "wl-luk-pm2.27".
-"wl-luk-syl" is used by "wl-luk-syl5".
-"wl-luk-syl" is used by "wl-luk-syl6".
-"wl-luk-syl5" is used by "wl-luk-ax3".
-"wl-luk-syl5" is used by "wl-luk-com12".
-"wl-luk-syl5" is used by "wl-luk-con1i".
-"wl-luk-syl5" is used by "wl-luk-con4i".
-"wl-luk-syl5" is used by "wl-luk-ja".
-"wl-luk-syl5" is used by "wl-luk-mpi".
-"wl-luk-syl5" is used by "wl-luk-pm2.04".
-"wl-luk-syl6" is used by "wl-luk-ax3".
-"wl-luk-syl6" is used by "wl-luk-pm2.27".
"wl-mps" is used by "wl-syls1".
"wl-sbcom2d-lem1" is used by "wl-sbcom2d".
"wl-sbcom2d-lem2" is used by "wl-sbcom2d".
@@ -15044,7 +15044,6 @@ New usage of "el123" is discouraged (1 uses).
New usage of "el2122old" is discouraged (1 uses).
New usage of "elALT" is discouraged (0 uses).
New usage of "ela" is discouraged (3 uses).
-New usage of "elabgOLD" is discouraged (0 uses).
New usage of "elat2" is discouraged (4 uses).
New usage of "elatcv0" is discouraged (0 uses).
New usage of "elbdop" is discouraged (4 uses).
@@ -15096,7 +15095,6 @@ New usage of "elpqn" is discouraged (24 uses).
New usage of "elprnq" is discouraged (22 uses).
New usage of "elpwgded" is discouraged (2 uses).
New usage of "elpwgdedVD" is discouraged (1 uses).
-New usage of "elrabOLD" is discouraged (0 uses).
New usage of "elreal" is discouraged (7 uses).
New usage of "elreal2" is discouraged (3 uses).
New usage of "elringchomALTV" is discouraged (1 uses).
@@ -16379,6 +16377,7 @@ New usage of "natded" is discouraged (0 uses).
New usage of "negexsr" is discouraged (0 uses).
New usage of "nelne1OLD" is discouraged (0 uses).
New usage of "nelne2OLD" is discouraged (0 uses).
+New usage of "nf5rOLD" is discouraged (0 uses).
New usage of "nf5riOLD" is discouraged (0 uses).
New usage of "nfa1-o" is discouraged (4 uses).
New usage of "nfabd2OLD" is discouraged (0 uses).
@@ -16577,6 +16576,7 @@ New usage of "notnotrALT" is discouraged (0 uses).
New usage of "notnotrALT2" is discouraged (0 uses).
New usage of "notnotrALTVD" is discouraged (0 uses).
New usage of "notnotriALT" is discouraged (0 uses).
+New usage of "notzfausOLD" is discouraged (0 uses).
New usage of "npex" is discouraged (2 uses).
New usage of "npomex" is discouraged (0 uses).
New usage of "nqercl" is discouraged (6 uses).
@@ -16968,6 +16968,7 @@ New usage of "pm2.18dOLD" is discouraged (0 uses).
New usage of "pm2.21ddALT" is discouraged (0 uses).
New usage of "pm2.43bgbi" is discouraged (0 uses).
New usage of "pm2.43cbi" is discouraged (2 uses).
+New usage of "pm2.61iOLD" is discouraged (0 uses).
New usage of "pmap1N" is discouraged (2 uses).
New usage of "pmapglb2N" is discouraged (0 uses).
New usage of "pmapglb2xN" is discouraged (1 uses).
@@ -17761,6 +17762,8 @@ New usage of "wl-luk-con4i" is discouraged (1 uses).
New usage of "wl-luk-id" is discouraged (1 uses).
New usage of "wl-luk-imim2" is discouraged (1 uses).
New usage of "wl-luk-imim2i" is discouraged (4 uses).
+New usage of "wl-luk-imtrdi" is discouraged (2 uses).
+New usage of "wl-luk-imtrid" is discouraged (7 uses).
New usage of "wl-luk-ja" is discouraged (1 uses).
New usage of "wl-luk-mpi" is discouraged (1 uses).
New usage of "wl-luk-notnotr" is discouraged (0 uses).
@@ -17770,8 +17773,6 @@ New usage of "wl-luk-pm2.21" is discouraged (2 uses).
New usage of "wl-luk-pm2.24i" is discouraged (1 uses).
New usage of "wl-luk-pm2.27" is discouraged (1 uses).
New usage of "wl-luk-syl" is discouraged (7 uses).
-New usage of "wl-luk-syl5" is discouraged (7 uses).
-New usage of "wl-luk-syl6" is discouraged (2 uses).
New usage of "wl-mps" is discouraged (1 uses).
New usage of "wl-sbal1" is discouraged (0 uses).
New usage of "wl-sbal2" is discouraged (0 uses).
@@ -18153,6 +18154,8 @@ Proof modification of "bj-nfsab1" is discouraged (12 steps).
Proof modification of "bj-nimn" is discouraged (8 steps).
Proof modification of "bj-nnfa1" is discouraged (25 steps).
Proof modification of "bj-nnfalt" is discouraged (86 steps).
+Proof modification of "bj-nnfand" is discouraged (109 steps).
+Proof modification of "bj-nnfant" is discouraged (109 steps).
Proof modification of "bj-nnfe1" is discouraged (25 steps).
Proof modification of "bj-nnfext" is discouraged (86 steps).
Proof modification of "bj-nnflemaa" is discouraged (25 steps).
@@ -18160,6 +18163,7 @@ Proof modification of "bj-nnflemae" is discouraged (25 steps).
Proof modification of "bj-nnflemea" is discouraged (25 steps).
Proof modification of "bj-nnflemee" is discouraged (25 steps).
Proof modification of "bj-nnfnfTEMP" is discouraged (20 steps).
+Proof modification of "bj-nnfort" is discouraged (106 steps).
Proof modification of "bj-nul" is discouraged (28 steps).
Proof modification of "bj-nuliota" is discouraged (73 steps).
Proof modification of "bj-nuliotaALT" is discouraged (60 steps).
@@ -18481,7 +18485,6 @@ Proof modification of "el12" is discouraged (19 steps).
Proof modification of "el123" is discouraged (26 steps).
Proof modification of "el2122old" is discouraged (25 steps).
Proof modification of "elALT" is discouraged (27 steps).
-Proof modification of "elabgOLD" is discouraged (13 steps).
Proof modification of "eldifsnneqOLD" is discouraged (35 steps).
Proof modification of "eleq2dALT" is discouraged (62 steps).
Proof modification of "elex22VD" is discouraged (111 steps).
@@ -18499,7 +18502,6 @@ Proof modification of "eliminable3b" is discouraged (8 steps).
Proof modification of "elissetOLD" is discouraged (19 steps).
Proof modification of "elpwgded" is discouraged (23 steps).
Proof modification of "elpwgdedVD" is discouraged (23 steps).
-Proof modification of "elrabOLD" is discouraged (16 steps).
Proof modification of "elrefsymrels3" is discouraged (65 steps).
Proof modification of "elunirnALT" is discouraged (38 steps).
Proof modification of "en3lpVD" is discouraged (147 steps).
@@ -18954,6 +18956,7 @@ Proof modification of "n2dvds3OLD" is discouraged (40 steps).
Proof modification of "naecoms-o" is discouraged (19 steps).
Proof modification of "nelne1OLD" is discouraged (26 steps).
Proof modification of "nelne2OLD" is discouraged (26 steps).
+Proof modification of "nf5rOLD" is discouraged (22 steps).
Proof modification of "nf5riOLD" is discouraged (13 steps).
Proof modification of "nfa1-o" is discouraged (8 steps).
Proof modification of "nfabd2OLD" is discouraged (75 steps).
@@ -19013,6 +19016,7 @@ Proof modification of "notnotrALT" is discouraged (12 steps).
Proof modification of "notnotrALT2" is discouraged (2 steps).
Proof modification of "notnotrALTVD" is discouraged (34 steps).
Proof modification of "notnotriALT" is discouraged (7 steps).
+Proof modification of "notzfausOLD" is discouraged (83 steps).
Proof modification of "nsnlpligALT" is discouraged (110 steps).
Proof modification of "nsyl2OLD" is discouraged (12 steps).
Proof modification of "odfvalALT" is discouraged (181 steps).
@@ -19052,6 +19056,7 @@ Proof modification of "pm2.18dOLD" is discouraged (10 steps).
Proof modification of "pm2.21ddALT" is discouraged (10 steps).
Proof modification of "pm2.43bgbi" is discouraged (16 steps).
Proof modification of "pm2.43cbi" is discouraged (34 steps).
+Proof modification of "pm2.61iOLD" is discouraged (13 steps).
Proof modification of "pncan3OLD" is discouraged (49 steps).
Proof modification of "pnfexOLD" is discouraged (4 steps).
Proof modification of "preleqALT" is discouraged (115 steps).
@@ -19454,6 +19459,8 @@ Proof modification of "wl-luk-con4i" is discouraged (14 steps).
Proof modification of "wl-luk-id" is discouraged (12 steps).
Proof modification of "wl-luk-imim2" is discouraged (14 steps).
Proof modification of "wl-luk-imim2i" is discouraged (15 steps).
+Proof modification of "wl-luk-imtrdi" is discouraged (14 steps).
+Proof modification of "wl-luk-imtrid" is discouraged (14 steps).
Proof modification of "wl-luk-ja" is discouraged (20 steps).
Proof modification of "wl-luk-mpi" is discouraged (14 steps).
Proof modification of "wl-luk-notnotr" is discouraged (6 steps).
@@ -19463,8 +19470,6 @@ Proof modification of "wl-luk-pm2.21" is discouraged (8 steps).
Proof modification of "wl-luk-pm2.24i" is discouraged (10 steps).
Proof modification of "wl-luk-pm2.27" is discouraged (27 steps).
Proof modification of "wl-luk-syl" is discouraged (13 steps).
-Proof modification of "wl-luk-syl5" is discouraged (14 steps).
-Proof modification of "wl-luk-syl6" is discouraged (14 steps).
Proof modification of "wl-mps" is discouraged (14 steps).
Proof modification of "wl-sbal1" is discouraged (36 steps).
Proof modification of "wl-sbal2" is discouraged (36 steps).
diff --git a/iset-discouraged b/iset-discouraged
index 95890dc1c5..dba24324d4 100644
--- a/iset-discouraged
+++ b/iset-discouraged
@@ -280,6 +280,7 @@ New usage of "bj-omex2" is discouraged (0 uses).
New usage of "bj-omssonALT" is discouraged (0 uses).
New usage of "bocardo" is discouraged (0 uses).
New usage of "cnm" is discouraged (2 uses).
+New usage of "condcOLD" is discouraged (0 uses).
New usage of "demoivreALT" is discouraged (0 uses).
New usage of "df-div" is discouraged (2 uses).
New usage of "df-ilim" is discouraged (1 uses).
@@ -289,6 +290,7 @@ New usage of "df-iord" is discouraged (1 uses).
New usage of "df-mnf" is discouraged (3 uses).
New usage of "df-pnf" is discouraged (3 uses).
New usage of "df-tru" is discouraged (1 uses).
+New usage of "dftest" is discouraged (0 uses).
New usage of "dftru2" is discouraged (0 uses).
New usage of "difidALT" is discouraged (0 uses).
New usage of "djulclALT" is discouraged (0 uses).
@@ -325,7 +327,7 @@ New usage of "sbieh" is discouraged (7 uses).
New usage of "spimeh" is discouraged (0 uses).
New usage of "spimh" is discouraged (1 uses).
New usage of "spimth" is discouraged (1 uses).
-New usage of "stabtestimpdcOLD" is discouraged (0 uses).
+New usage of "stdcndcOLD" is discouraged (0 uses).
New usage of "stoic2a" is discouraged (0 uses).
New usage of "stoic2b" is discouraged (0 uses).
New usage of "strcollnfALT" is discouraged (0 uses).
@@ -368,7 +370,6 @@ Proof modification of "bj-bdfindes" is discouraged (121 steps).
Proof modification of "bj-bdfindis" is discouraged (84 steps).
Proof modification of "bj-bdfindisg" is discouraged (44 steps).
Proof modification of "bj-d0clsepcl" is discouraged (111 steps).
-Proof modification of "bj-dcbi" is discouraged (31 steps).
Proof modification of "bj-ex" is discouraged (31 steps).
Proof modification of "bj-exlimmp" is discouraged (34 steps).
Proof modification of "bj-exlimmpi" is discouraged (11 steps).
@@ -394,9 +395,6 @@ Proof modification of "bj-nnen2lp" is discouraged (52 steps).
Proof modification of "bj-nnord" is discouraged (53 steps).
Proof modification of "bj-nntrans" is discouraged (244 steps).
Proof modification of "bj-nntrans2" is discouraged (24 steps).
-Proof modification of "bj-notbi" is discouraged (22 steps).
-Proof modification of "bj-notbid" is discouraged (14 steps).
-Proof modification of "bj-notbii" is discouraged (13 steps).
Proof modification of "bj-nvel" is discouraged (11 steps).
Proof modification of "bj-om" is discouraged (101 steps).
Proof modification of "bj-omelon" is discouraged (10 steps).
@@ -425,6 +423,7 @@ Proof modification of "bj-vnex" is discouraged (14 steps).
Proof modification of "bj-vprc" is discouraged (53 steps).
Proof modification of "bj-zfpair2" is discouraged (80 steps).
Proof modification of "brab2ga" is discouraged (79 steps).
+Proof modification of "condcOLD" is discouraged (39 steps).
Proof modification of "demoivreALT" is discouraged (1087 steps).
Proof modification of "difidALT" is discouraged (20 steps).
Proof modification of "djulclALT" is discouraged (53 steps).
@@ -448,7 +447,7 @@ Proof modification of "ruALT" is discouraged (35 steps).
Proof modification of "sbc8g" is discouraged (55 steps).
Proof modification of "sbsbc" is discouraged (21 steps).
Proof modification of "speano5" is discouraged (50 steps).
-Proof modification of "stabtestimpdcOLD" is discouraged (62 steps).
+Proof modification of "stdcndcOLD" is discouraged (62 steps).
Proof modification of "strcollnfALT" is discouraged (79 steps).
Proof modification of "tfri1dALT" is discouraged (247 steps).
Proof modification of "uzind4ALT" is discouraged (16 steps).
diff --git a/iset.mm b/iset.mm
index 84c7c16dd9..fa6ad8cb87 100644
--- a/iset.mm
+++ b/iset.mm
@@ -5110,6 +5110,11 @@ is valid intuitionistically (in the terminology of Section 1.2 of [Bauer]
( wn con3d com12 ) ACEBEABCDFG $.
$}
+ $( Triple negation is equivalent to negation. (Contributed by Jim Kingdon,
+ 28-Jul-2018.) $)
+ notnotnot $p |- ( -. -. -. ph <-> -. ph ) $=
+ ( wn notnot con3i impbii ) ABZBZBFAGACDFCE $.
+
${
con3dimp.1 $e |- ( ph -> ( ps -> ch ) ) $.
$( Variant of ~ con3d with importation. (Contributed by Jonathan Ben-Naim,
@@ -5343,27 +5348,32 @@ is valid intuitionistically (in the terminology of Section 1.2 of [Bauer]
( ex mtod ) ABCDABCEFG $.
$}
+ $( Equivalence property for negation. Closed form. (Contributed by BJ,
+ 27-Jan-2020.) $)
+ notbi $p |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) ) $=
+ ( wb wn bi2 con3d bi1 impbid ) ABCZADBDIBAABEFIABABGFH $.
+
${
notbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
- $( Deduction negating both sides of a logical equivalence. (Contributed by
- NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
+ $( Equivalence property for negation. Deduction form. (Contributed by NM,
+ 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
notbid $p |- ( ph -> ( -. ps <-> -. ch ) ) $=
- ( wn biimprd con3d biimpd impbid ) ABECEACBABCDFGABCABCDHGI $.
+ ( wb wn notbi syl ) ABCEBFCFEDBCGH $.
$}
- $( Contraposition. Bidirectional version of ~ con2 . (Contributed by NM,
- 5-Aug-1993.) $)
- con2b $p |- ( ( ph -> -. ps ) <-> ( ps -> -. ph ) ) $=
- ( wn wi con2 impbii ) ABCDBACDABEBAEF $.
-
${
notbii.1 $e |- ( ph <-> ps ) $.
- $( Negate both sides of a logical equivalence. (Contributed by NM,
+ $( Equivalence property for negation. Inference form. (Contributed by NM,
5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
notbii $p |- ( -. ph <-> -. ps ) $=
- ( wb wn id notbid ax-mp ) ABDZAEBEDCIABIFGH $.
+ ( wb wn notbi ax-mp ) ABDAEBEDCABFG $.
$}
+ $( Contraposition. Bidirectional version of ~ con2 . (Contributed by NM,
+ 5-Aug-1993.) $)
+ con2b $p |- ( ( ph -> -. ps ) <-> ( ps -> -. ph ) ) $=
+ ( wn wi con2 impbii ) ABCDBACDABEBAEF $.
+
${
mtbi.1 $e |- -. ph $.
mtbi.2 $e |- ( ph <-> ps ) $.
@@ -5507,6 +5517,59 @@ is valid intuitionistically (in the terminology of Section 1.2 of [Bauer]
mtt $p |- ( -. ph -> ( -. ps <-> ( ps -> ph ) ) ) $=
( wn wi pm2.21 con3 com12 impbid2 ) ACZBCZBADZBAEKIJBAFGH $.
+ $( Express conjunction in terms of implication. One direction of Theorem
+ *4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable
+ propositions, as can be seen at ~ annimdc . (Contributed by Jim Kingdon,
+ 24-Dec-2017.) $)
+ annimim $p |- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) ) $=
+ ( wn wi pm2.27 con3 syl imp ) ABCZABDZCZAJBDIKDABEJBFGH $.
+
+ $( One direction of Theorem *4.65 of [WhiteheadRussell] p. 120. The converse
+ holds in classical logic. (Contributed by Jim Kingdon, 28-Jul-2018.) $)
+ pm4.65r $p |- ( ( -. ph /\ -. ps ) -> -. ( -. ph -> ps ) ) $=
+ ( wn annimim ) ACBD $.
+
+ $( Express implication in terms of conjunction. The converse only holds
+ given a decidability condition; see ~ imandc . (Contributed by Jim
+ Kingdon, 24-Dec-2017.) $)
+ imanim $p |- ( ( ph -> ps ) -> -. ( ph /\ -. ps ) ) $=
+ ( wn wa wi annimim con2i ) ABCDABEABFG $.
+
+ $( Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM,
+ 3-Jan-2005.) $)
+ pm3.37 $p |- ( ( ( ph /\ ps ) -> ch ) -> ( ( ph /\ -. ch ) -> -. ps ) ) $=
+ ( wa wi wn pm3.3 con3 syl6 impd ) ABDCEZACFZBFZKABCELMEABCGBCHIJ $.
+
+ $( Express implication in terms of conjunction. (Contributed by NM,
+ 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.) $)
+ imnan $p |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) ) $=
+ ( wn wi wa pm3.2im imp con2i pm3.2 con3rr3 impbii ) ABCDZABEZCMLABLCABFGHAB
+ MABIJK $.
+
+ ${
+ imnani.1 $e |- -. ( ph /\ ps ) $.
+ $( Express implication in terms of conjunction. (Contributed by Mario
+ Carneiro, 28-Sep-2015.) $)
+ imnani $p |- ( ph -> -. ps ) $=
+ ( wn wi wa imnan mpbir ) ABDEABFDCABGH $.
+ $}
+
+ $( Theorem to move a conjunct in and out of a negation. (Contributed by NM,
+ 9-Nov-2003.) $)
+ nan $p |- ( ( ph -> -. ( ps /\ ch ) ) <-> ( ( ph /\ ps ) -> -. ch ) ) $=
+ ( wa wn wi impexp imnan imbi2i bitr2i ) ABDCEZFABKFZFABCDEZFABKGLMABCHIJ $.
+
+ $( Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who
+ call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.)
+ (Revised by Mario Carneiro, 2-Feb-2015.) $)
+ pm3.24 $p |- -. ( ph /\ -. ph ) $=
+ ( wn wi wa notnot imnan mpbi ) AABZBCAHDBAEAHFG $.
+
+ $( Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM,
+ 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) $)
+ pm4.15 $p |- ( ( ( ph /\ ps ) -> -. ch ) <-> ( ( ps /\ ch ) -> -. ph ) ) $=
+ ( wa wn wi con2b nan bitr2i ) BCDZAEFAJEFABDCEFJAGABCHI $.
+
$( Two propositions are equivalent if they are both false. Theorem *5.21 of
[WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.) (Revised by
Mario Carneiro, 31-Jan-2015.) $)
@@ -5605,36 +5668,6 @@ is valid intuitionistically (in the terminology of Section 1.2 of [Bauer]
pm4.8 $p |- ( ( ph -> -. ph ) <-> -. ph ) $=
( wn wi pm2.01 ax-1 impbii ) AABZCGADGAEF $.
- $( Express implication in terms of conjunction. (Contributed by NM,
- 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.) $)
- imnan $p |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) ) $=
- ( wn wi wa pm3.2im imp con2i pm3.2 con3rr3 impbii ) ABCDZABEZCMLABLCABFGHAB
- MABIJK $.
-
- ${
- imnani.1 $e |- -. ( ph /\ ps ) $.
- $( Express implication in terms of conjunction. (Contributed by Mario
- Carneiro, 28-Sep-2015.) $)
- imnani $p |- ( ph -> -. ps ) $=
- ( wn wi wa imnan mpbir ) ABDEABFDCABGH $.
- $}
-
- $( Theorem to move a conjunct in and out of a negation. (Contributed by NM,
- 9-Nov-2003.) $)
- nan $p |- ( ( ph -> -. ( ps /\ ch ) ) <-> ( ( ph /\ ps ) -> -. ch ) ) $=
- ( wa wn wi impexp imnan imbi2i bitr2i ) ABDCEZFABKFZFABCDEZFABKGLMABCHIJ $.
-
- $( Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who
- call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.)
- (Revised by Mario Carneiro, 2-Feb-2015.) $)
- pm3.24 $p |- -. ( ph /\ -. ph ) $=
- ( wn wi wa notnot imnan mpbi ) AABZBCAHDBAEAHFG $.
-
- $( Triple negation is equivalent to negation. (Contributed by Jim Kingdon,
- 28-Jul-2018.) $)
- notnotnot $p |- ( -. -. -. ph <-> -. ph ) $=
- ( wn notnot con3i impbii ) ABZBZBFAGACDFCE $.
-
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
@@ -5675,6 +5708,12 @@ is valid intuitionistically (in the terminology of Section 1.2 of [Bauer]
pm2.67-2 $p |- ( ( ( ph \/ ch ) -> ps ) -> ( ph -> ps ) ) $=
( wo orc imim1i ) AACDBACEF $.
+ $( Absorption of disjunction into equivalence. (Contributed by NM,
+ 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) $)
+ oibabs $p |- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) <-> ( ph <-> ps ) ) $=
+ ( wo wb wi pm2.67-2 ibd olc imim1i ibibr sylibr impbid ax-1 impbii ) ABCZAB
+ DZEZPQABQABAPBFGQBPEBAEBOPBAHIBAJKLPOMN $.
+
$( Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM,
3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
pm3.44 $p |- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) ->
@@ -5719,6 +5758,12 @@ is valid intuitionistically (in the terminology of Section 1.2 of [Bauer]
( wa wi adantrd adantld jaoi ) ABEHCIDABCEFJDECBGKL $.
$}
+ $( Implication in terms of disjunction. One direction of theorem *4.6 of
+ [WhiteheadRussell] p. 120. The converse holds for decidable propositions,
+ as seen at ~ imordc . (Contributed by Jim Kingdon, 21-Jul-2018.) $)
+ imorr $p |- ( ( -. ph \/ ps ) -> ( ph -> ps ) ) $=
+ ( wn wi ax-in2 ax-1 jaoi ) ACABDBABEBAFG $.
+
$( Theorem *2.53 of [WhiteheadRussell] p. 107. This holds
intuitionistically, although its converse does not (see ~ pm2.54dc ).
(Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.) $)
@@ -5896,6 +5941,18 @@ inference eliminates the need for a syllogism ( ~ syl ) -type inference
( wn wo wi pm2.21 ax-1 jaoi ax-mp ) ADZBEABFZCKLBABGBAHIJ $.
$}
+ $( One direction of theorem *4.52 of [WhiteheadRussell] p. 120. The converse
+ also holds in classical logic. (Contributed by Jim Kingdon,
+ 27-Jul-2018.) $)
+ pm4.52im $p |- ( ( ph /\ -. ps ) -> -. ( -. ph \/ ps ) ) $=
+ ( wn wa wi wo annimim imorr nsyl ) ABCDABEACBFABGABHI $.
+
+ $( One direction of theorem *4.53 of [WhiteheadRussell] p. 120. The converse
+ also holds in classical logic. (Contributed by Jim Kingdon,
+ 27-Jul-2018.) $)
+ pm4.53r $p |- ( ( -. ph \/ ps ) -> -. ( ph /\ -. ps ) ) $=
+ ( wn wa wo pm4.52im con2i ) ABCDACBEABFG $.
+
$( Negated disjunction in terms of conjunction. This version of DeMorgan's
law is a biconditional for all propositions (not just decidable ones),
unlike ~ oranim , ~ anordc , or ~ ianordc . Compare Theorem *4.56 of
@@ -6065,6 +6122,25 @@ law is a biconditional for all propositions (not just decidable ones),
pm4.44 $p |- ( ph <-> ( ph \/ ( ph /\ ps ) ) ) $=
( wa wo orc id simpl jaoi impbii ) AAABCZDAJEAAJAFABGHI $.
+ $( Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM,
+ 3-Jan-2005.) $)
+ pm4.56 $p |- ( ( -. ph /\ -. ps ) <-> -. ( ph \/ ps ) ) $=
+ ( wo wn wa ioran bicomi ) ABCDADBDEABFG $.
+
+ $( Disjunction in terms of conjunction (DeMorgan's law). One direction of
+ Theorem *4.57 of [WhiteheadRussell] p. 120. The converse does not hold
+ intuitionistically but does hold in classical logic. (Contributed by Jim
+ Kingdon, 25-Jul-2018.) $)
+ oranim $p |- ( ( ph \/ ps ) -> -. ( -. ph /\ -. ps ) ) $=
+ ( wn wa wo pm4.56 biimpi con2i ) ACBCDZABEZIJCABFGH $.
+
+ $( Implication distributes over disjunction. One direction of Theorem *4.78
+ of [WhiteheadRussell] p. 121. The converse holds in classical logic.
+ (Contributed by Jim Kingdon, 15-Jan-2018.) $)
+ pm4.78i $p |- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) ->
+ ( ph -> ( ps \/ ch ) ) ) $=
+ ( wi wo orc imim2i olc jaoi ) ABDABCEZDACDBJABCFGCJACBHGI $.
+
${
mtord.1 $e |- ( ph -> -. ch ) $.
mtord.2 $e |- ( ph -> -. th ) $.
@@ -6351,6 +6427,7 @@ law is a biconditional for all propositions (not just decidable ones),
Stable propositions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
+
$( Declare connective for stability. $)
$c STAB $.
@@ -6361,8 +6438,8 @@ law is a biconditional for all propositions (not just decidable ones),
See Chapter 2 [Moschovakis] p. 2.
Our notation for stability is a connective ` STAB ` which we place before
- the formula in question. For example, ` STAB x = y ` corresponds to "x =
- y is stable".
+ the formula in question. For example, ` STAB x = y ` corresponds
+ to " ` x = y ` is stable".
(Contributed by David A. Wheeler, 13-Aug-2018.) $)
df-stab $a |- ( STAB ph <-> ( -. -. ph -> ph ) ) $.
@@ -6381,19 +6458,13 @@ law is a biconditional for all propositions (not just decidable ones),
stabnot $p |- STAB -. ph $=
( wn wstab wi notnotnot biimpi df-stab mpbir ) ABZCIBBZIDJIAEFIGH $.
- $( Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll]
- p. 176. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf
- Lammen, 30-Oct-2012.) $)
- imanst $p |- ( STAB ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) ) $=
- ( wstab wi wn wa notnot df-stab biimpi impbid2 imbi2d imnan syl6bb ) BCZABD
- ABEZEZDAOFENBPANBPBGNPBDBHIJKAOLM $.
-
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Decidable propositions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
+
$( Declare connective for decidability. $)
$c DECID $.
@@ -6409,7 +6480,7 @@ particular kinds of propositions may be decidable (for example, the
Our notation for decidability is a connective ` DECID ` which we place
before the formula in question. For example, ` DECID x = y ` corresponds
- to "x = y is decidable".
+ to " ` x = y ` is decidable".
We could transform intuitionistic logic to classical logic by adding
unconditional forms of ~ condc , ~ exmiddc , ~ peircedc , or ~ notnotrdc ,
@@ -6436,46 +6507,120 @@ particular kinds of propositions may be decidable (for example, the
pm2.1dc $p |- ( DECID ph -> ( -. ph \/ ph ) ) $=
( wdc wn wo df-dc orcom bitri biimpi ) ABZACZADZIAJDKAEAJFGH $.
- $( A decidable proposition is decidable when negated. (Contributed by Jim
- Kingdon, 25-Mar-2018.) $)
- dcn $p |- ( DECID ph -> DECID -. ph ) $=
- ( wn wo wdc notnot orim2i orcoms df-dc 3imtr4i ) AABZCJJBZCZADJDJALAKJAEFGA
- HJHI $.
-
- ${
- dcbii.1 $e |- ( ph <-> ps ) $.
- $( The equivalent of a decidable proposition is decidable. (Contributed by
- Jim Kingdon, 28-Mar-2018.) $)
- dcbii $p |- ( DECID ph <-> DECID ps ) $=
- ( wn wo wdc notbii orbi12i df-dc 3bitr4i ) AADZEBBDZEAFBFABKLCABCGHAIBIJ
- $.
- $}
-
${
dcbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
- $( The equivalent of a decidable proposition is decidable. (Contributed by
+ $( Equivalence property for decidability. Deduction form. (Contributed by
Jim Kingdon, 7-Sep-2019.) $)
dcbid $p |- ( ph -> ( DECID ps <-> DECID ch ) ) $=
( wn wo wdc notbid orbi12d df-dc 3bitr4g ) ABBEZFCCEZFBGCGABCLMDABCDHIBJC
JK $.
$}
+ $( Equivalence property for decidability. Closed form. (Contributed by BJ,
+ 27-Jan-2020.) $)
+ dcbiit $p |- ( ( ph <-> ps ) -> ( DECID ph <-> DECID ps ) ) $=
+ ( wb id dcbid ) ABCZABFDE $.
+
+ ${
+ dcbii.1 $e |- ( ph <-> ps ) $.
+ $( Equivalence property for decidability. Inference form. (Contributed by
+ Jim Kingdon, 28-Mar-2018.) $)
+ dcbii $p |- ( DECID ph <-> DECID ps ) $=
+ ( wb wdc dcbiit ax-mp ) ABDAEBEDCABFG $.
+ $}
+
+ $( An implication between two decidable propositions is decidable.
+ (Contributed by Jim Kingdon, 28-Mar-2018.) $)
+ dcim $p |- ( DECID ph -> ( DECID ps -> DECID ( ph -> ps ) ) ) $=
+ ( wn wo wi df-dc wa anbi2i andi bitri pm3.4 annimim orim12i sylbi sylibr ex
+ wdc ax-in2 a1d orc syl6 jaoi ) AQAACZDBQZABEZQZEZAFAUGUCAUDUFAUDGZUEUECZDZU
+ FUHABGZABCZGZDZUJUHABULDZGUNUDUOABFHABULIJUKUEUMUIABKABLMNUEFZOPUCUDUEUFUCU
+ EUDABRSUEUJUFUEUITUPOUAUBN $.
+
+ $( The negation of a decidable proposition is decidable. The converse need
+ not hold, but does hold for negated propositions, see ~ dcnn .
+ (Contributed by Jim Kingdon, 25-Mar-2018.) $)
+ dcn $p |- ( DECID ph -> DECID -. ph ) $=
+ ( wn wo wdc notnot orim2i orcoms df-dc 3imtr4i ) AABZCJJBZCZADJDJALAKJAEFGA
+ HJHI $.
+
+ $( Decidability of the negation of a proposition is equivalent to
+ decidability of its double negation. See also ~ dcn . The relation
+ between ~ dcn and ~ dcnn is analogous to that between ~ notnot and
+ ~ notnotnot (and directly stems from it). Using the notion of "testable
+ proposition" (proposition whose negation is decidable), ~ dcnn means that
+ a proposition is testable if and only if its negation is testable, and
+ ~ dcn means that decidability implies testability. (Contributed by David
+ A. Wheeler, 6-Dec-2018.) $)
+ dcnn $p |- ( DECID -. ph <-> DECID -. -. ph ) $=
+ ( wn wo wdc notnotnot orbi2i orcom bitri df-dc 3bitr4ri ) ABZBZLBZCZKLCZLDK
+ DNLKCOMKLAEFLKGHLIKIJ $.
+
+ $( Double negation elimination for a decidable proposition. The converse,
+ ~ notnot , holds for all propositions, not just decidable ones. This is
+ Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability
+ condition added. (Contributed by Jim Kingdon, 11-Mar-2018.) $)
+ notnotrdc $p |- ( DECID ph -> ( -. -. ph -> ph ) ) $=
+ ( wdc wn wo wi df-dc orcom bitri pm2.53 sylbi ) ABZACZADZLCAEKALDMAFALGHLAI
+ J $.
+
+ $( Decidability implies stability. The converse need not hold. (Contributed
+ by David A. Wheeler, 13-Aug-2018.) $)
+ dcstab $p |- ( DECID ph -> STAB ph ) $=
+ ( wdc wn wi wstab notnotrdc df-stab sylibr ) ABACCADAEAFAGH $.
+
+ $( A formula is decidable if and only if its negation is decidable and it is
+ stable (that is, it is testable and stable). (Contributed by David A.
+ Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.) $)
+ stdcndc $p |- ( ( STAB ph /\ DECID -. ph ) <-> DECID ph ) $=
+ ( wstab wn wdc wa wo df-stab df-dc pm2.36 imp syl2anb sylibr dcstab dcn jca
+ wi impbii ) ABZACZDZEZADZUAASFZUBRSCZAPZSUDFZUCTAGSHUEUFUCSUDAIJKAHLUBRTAMA
+ NOQ $.
+
+ $( Obsolete version of ~ stdcndc as of 28-Oct-2023. (Contributed by David A.
+ Wheeler, 13-Aug-2018.) (Proof modification is discouraged.)
+ (New usage is discouraged.) $)
+ stdcndcOLD $p |- ( ( STAB ph /\ DECID -. ph ) <-> DECID ph ) $=
+ ( wstab wn wdc wa wo exmiddc adantl df-stab biimpi orim2d adantr mpd orcomd
+ wi df-dc sylibr dcstab dcn jca impbii ) ABZACZDZEZADZUEAUCFUFUEUCAUEUCUCCZF
+ ZUCAFZUDUHUBUCGHUBUHUIOUDUBUGAUCUBUGAOAIJKLMNAPQUFUBUDARASTUA $.
+
+ $( A formula is stable if and only if the decidability of its negation
+ implies its decidability. Note that the right-hand side of this
+ biconditional is the converse of ~ dcn . (Contributed by BJ,
+ 18-Nov-2023.) $)
+ stdcn $p |- ( STAB ph <-> ( DECID -. ph -> DECID ph ) ) $=
+ ( wstab wn wdc wi wa stdcndc biimpi ex wo imim1i orel2 sylcom df-dc imbi12i
+ olc df-stab 3imtr4i impbii ) ABZACZDZADZEZTUBUCTUBFUCAGHIUAUACZJZAUAJZEZUEA
+ EUDTUHUEUGAUEUFUGUEUAPKUAALMUBUFUCUGUANANOAQRS $.
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Theorems of decidable propositions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
- Many theorems of logic hold in intuitionistic logic just as they do in
- classical (non-inuitionistic) logic, for all propositions. Other theorems
- only hold for decidable propositions, such as the law of the excluded
- middle ( ~ df-dc ), double negation elimination ( ~ notnotrdc ), or
- contraposition ( ~ condc ). Our goal is to prove all well-known
- or important classical theorems, but with suitable decidability
- conditions so that the proofs follow from intuitionistic axioms.
- This section is focused on such proofs, given decidability conditions.
+ Many theorems of logic hold in intuitionistic logic just as they do in
+ classical (non-inuitionistic) logic, for all propositions. Other theorems
+ only hold for decidable propositions, such as the law of the excluded middle
+ ( ~ df-dc ), double negation elimination ( ~ notnotrdc ), or contraposition
+ ( ~ condc ). Our goal is to prove all well-known or important classical
+ theorems, but with suitable decidability conditions so that the proofs follow
+ from intuitionistic axioms. This section is focused on such proofs, given
+ decidability conditions.
+
+ Many theorems of this section actually hold for stable propositions (see
+ ~ df-stab ). Decidable propositions are stable ( ~ dcstab ), but the
+ converse need not hold.
+
$)
+ $( Contraposition of a stable proposition. See comment of ~ condc .
+ (Contributed by BJ, 18-Nov-2023.) $)
+ const $p |- ( STAB ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) ) $=
+ ( wstab wn wi df-stab con3 notnot imim2 syl7 syl5 sylbi ) ACADZDZAEZMBDZEZB
+ AEZEAFQPDZNEZORMPGBSOTABHNASIJKL $.
+
$( Contraposition of a decidable proposition.
This theorem swaps or "transposes" the order of the consequents when
@@ -6489,8 +6634,15 @@ particular kinds of propositions may be decidable (for example, the
is advised that in the field of philosophical logic, "contraposition" has
a different technical meaning.
- (Contributed by Jim Kingdon, 13-Mar-2018.) $)
+ (Contributed by Jim Kingdon, 13-Mar-2018.) (Proof shortened by BJ,
+ 18-Nov-2023.) $)
condc $p |- ( DECID ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) ) $=
+ ( wdc wstab wn wi dcstab const syl ) ACADAEBEFBAFFAGABHI $.
+
+ $( Obsolete proof of ~ condc as of 18-Nov-2023. (Contributed by Jim Kingdon,
+ 13-Mar-2018.) (Proof modification is discouraged.)
+ (New usage is discouraged.) $)
+ condcOLD $p |- ( DECID ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) ) $=
( wdc wn wo wi df-dc ax-1 a1d pm2.27 ax-in2 syl6 jaoi sylbi ) ACAADZEOBDZFZ
BAFZFZAGASOARQABHIOQPROPJBAKLMN $.
@@ -6502,19 +6654,6 @@ particular kinds of propositions may be decidable (for example, the
( wdc wn wi pm2.21 a2i condc syl5 pm2.43d ) ABZACZADZALKLCZDJLADKAMAMEFALGH
I $.
- $( Double negation elimination for a decidable proposition. The converse,
- ~ notnot , holds for all propositions, not just decidable ones. This is
- Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability
- condition added. (Contributed by Jim Kingdon, 11-Mar-2018.) $)
- notnotrdc $p |- ( DECID ph -> ( -. -. ph -> ph ) ) $=
- ( wdc wn wo wi df-dc orcom bitri pm2.53 sylbi ) ABZACZADZLCAEKALDMAFALGHLAI
- J $.
-
- $( Decidability implies stability. The converse is not necessarily true.
- (Contributed by David A. Wheeler, 13-Aug-2018.) $)
- dcimpstab $p |- ( DECID ph -> STAB ph ) $=
- ( wdc wn wi wstab notnotrdc df-stab sylibr ) ABACCADAEAFAGH $.
-
$( Contraposition for a decidable proposition. Based on theorem *2.15 of
[WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.) $)
con1dc $p |- ( DECID ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) ) $=
@@ -6761,39 +6900,19 @@ particular kinds of propositions may be decidable (for example, the
( -. ( -. ph -> -. ps ) <-> ( -. ph /\ ps ) ) ) ) $=
( wdc wn wi wa wb dcn pm4.63dc syl ) ACADZCBCKBDEDKBFGEAHKBIJ $.
- $( Express conjunction in terms of implication. One direction of Theorem
- *4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable
- propositions, as can be seen at ~ annimdc . (Contributed by Jim Kingdon,
- 24-Dec-2017.) $)
- annimim $p |- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) ) $=
- ( wn wi pm2.27 con3 syl imp ) ABCZABDZCZAJBDIKDABEJBFGH $.
-
- $( One direction of Theorem *4.65 of [WhiteheadRussell] p. 120. The converse
- holds in classical logic. (Contributed by Jim Kingdon, 28-Jul-2018.) $)
- pm4.65r $p |- ( ( -. ph /\ -. ps ) -> -. ( -. ph -> ps ) ) $=
- ( wn annimim ) ACBD $.
-
- $( An implication between two decidable propositions is decidable.
- (Contributed by Jim Kingdon, 28-Mar-2018.) $)
- dcim $p |- ( DECID ph -> ( DECID ps -> DECID ( ph -> ps ) ) ) $=
- ( wn wo wi df-dc wa anbi2i andi bitri pm3.4 annimim orim12i sylbi sylibr ex
- wdc ax-in2 a1d orc syl6 jaoi ) AQAACZDBQZABEZQZEZAFAUGUCAUDUFAUDGZUEUECZDZU
- FUHABGZABCZGZDZUJUHABULDZGUNUDUOABFHABULIJUKUEUMUIABKABLMNUEFZOPUCUDUEUFUCU
- EUDABRSUEUJUFUEUITUPOUAUBN $.
-
- $( Express implication in terms of conjunction. The converse only holds
- given a decidability condition; see ~ imandc . (Contributed by Jim
- Kingdon, 24-Dec-2017.) $)
- imanim $p |- ( ( ph -> ps ) -> -. ( ph /\ -. ps ) ) $=
- ( wn wa wi annimim con2i ) ABCDABEABFG $.
+ $( Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll]
+ p. 176. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf
+ Lammen, 30-Oct-2012.) $)
+ imanst $p |- ( STAB ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) ) $=
+ ( wstab wi wn wa notnot df-stab biimpi impbid2 imbi2d imnan syl6bb ) BCZABD
+ ABEZEZDAOFENBPANBPBGNPBDBHIJKAOLM $.
$( Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll]
p. 176, with an added decidability condition. The forward direction,
~ imanim , holds for all propositions, not just decidable ones.
(Contributed by Jim Kingdon, 25-Apr-2018.) $)
imandc $p |- ( DECID ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) ) $=
- ( wdc wi wn wa notnotbdc imbi2d imnan syl6bb ) BCZABDABEZEZDALFEKBMABGHALIJ
- $.
+ ( wdc wstab wi wn wa wb dcstab imanst syl ) BCBDABEABFGFHBIABJK $.
$( Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability
condition. (Contributed by Jim Kingdon, 24-Apr-2018.) $)
@@ -6802,16 +6921,6 @@ particular kinds of propositions may be decidable (for example, the
( wdc wi wn wa con34bdc imbi2d impexp 3bitr4g ) CDZABCEZEACFZBFZEZEABGCEANG
OELMPABCHIABCJANOJK $.
- $( Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM,
- 3-Jan-2005.) $)
- pm3.37 $p |- ( ( ( ph /\ ps ) -> ch ) -> ( ( ph /\ -. ch ) -> -. ps ) ) $=
- ( wa wi wn pm3.3 con3 syl6 impd ) ABDCEZACFZBFZKABCELMEABCGBCHIJ $.
-
- $( Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM,
- 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) $)
- pm4.15 $p |- ( ( ( ph /\ ps ) -> -. ch ) <-> ( ( ps /\ ch ) -> -. ph ) ) $=
- ( wa wn wi con2b nan bitr2i ) BCDZAEFAJEFABDCEFJAGABCHI $.
-
$( Deriving disjunction from implication for a decidable proposition. Based
on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, ~ pm2.53 ,
holds whether the proposition is decidable or not. (Contributed by Jim
@@ -6860,12 +6969,6 @@ particular kinds of propositions may be decidable (for example, the
( wdc wi wn wo notnotbdc imbi1d wb dcn dfordc syl bitr4d ) ACZABDAEZEZBDZOB
FZNAPBAGHNOCRQIAJOBKLM $.
- $( Implication in terms of disjunction. One direction of theorem *4.6 of
- [WhiteheadRussell] p. 120. The converse holds for decidable propositions,
- as seen at ~ imordc . (Contributed by Jim Kingdon, 21-Jul-2018.) $)
- imorr $p |- ( ( -. ph \/ ps ) -> ( ph -> ps ) ) $=
- ( wn wi ax-in2 ax-1 jaoi ) ACABDBABEBAFG $.
-
$( Implication in terms of disjunction. Like Theorem *4.62 of
[WhiteheadRussell] p. 120, but for a decidable antecedent. (Contributed
by Jim Kingdon, 21-Apr-2018.) $)
@@ -6880,12 +6983,6 @@ particular kinds of propositions may be decidable (for example, the
ianordc $p |- ( DECID ph -> ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) ) ) $=
( wa wn wi wdc wo imnan pm4.62dc syl5bbr ) ABCDABDZEAFADKGABHABIJ $.
- $( Absorption of disjunction into equivalence. (Contributed by NM,
- 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) $)
- oibabs $p |- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) <-> ( ph <-> ps ) ) $=
- ( wo wb wi pm2.67-2 ibd olc imim1i ibibr sylibr impbid ax-1 impbii ) ABCZAB
- DZEZPQABQABAPBFGQBPEBAEBOPBAHIBAJKLPOMN $.
-
$( Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability
condition. The reverse direction, ~ pm2.53 , holds for all propositions.
(Contributed by Jim Kingdon, 2-May-2018.) $)
@@ -6897,18 +6994,6 @@ particular kinds of propositions may be decidable (for example, the
pm4.66dc $p |- ( DECID ph -> ( ( -. ph -> -. ps ) <-> ( ph \/ -. ps ) ) ) $=
( wn pm4.64dc ) ABCD $.
- $( One direction of theorem *4.52 of [WhiteheadRussell] p. 120. The converse
- also holds in classical logic. (Contributed by Jim Kingdon,
- 27-Jul-2018.) $)
- pm4.52im $p |- ( ( ph /\ -. ps ) -> -. ( -. ph \/ ps ) ) $=
- ( wn wa wi wo annimim imorr nsyl ) ABCDABEACBFABGABHI $.
-
- $( One direction of theorem *4.53 of [WhiteheadRussell] p. 120. The converse
- also holds in classical logic. (Contributed by Jim Kingdon,
- 27-Jul-2018.) $)
- pm4.53r $p |- ( ( -. ph \/ ps ) -> -. ( ph /\ -. ps ) ) $=
- ( wn wa wo pm4.52im con2i ) ABCDACBEABFG $.
-
$( Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions.
One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.) $)
pm4.54dc $p |- ( DECID ph -> ( DECID ps ->
@@ -6917,25 +7002,6 @@ particular kinds of propositions may be decidable (for example, the
CZBCZADZBEZABDZFZDZGRSEZUATUBHZDZUDRSUAUGGZRTCSUHHAITBJKLUEUFUCRUFUCGSABMNO
PQ $.
- $( Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM,
- 3-Jan-2005.) $)
- pm4.56 $p |- ( ( -. ph /\ -. ps ) <-> -. ( ph \/ ps ) ) $=
- ( wo wn wa ioran bicomi ) ABCDADBDEABFG $.
-
- $( Disjunction in terms of conjunction (DeMorgan's law). One direction of
- Theorem *4.57 of [WhiteheadRussell] p. 120. The converse does not hold
- intuitionistically but does hold in classical logic. (Contributed by Jim
- Kingdon, 25-Jul-2018.) $)
- oranim $p |- ( ( ph \/ ps ) -> -. ( -. ph /\ -. ps ) ) $=
- ( wn wa wo pm4.56 biimpi con2i ) ACBCDZABEZIJCABFGH $.
-
- $( Implication distributes over disjunction. One direction of Theorem *4.78
- of [WhiteheadRussell] p. 121. The converse holds in classical logic.
- (Contributed by Jim Kingdon, 15-Jan-2018.) $)
- pm4.78i $p |- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) ->
- ( ph -> ( ps \/ ch ) ) ) $=
- ( wi wo orc imim2i olc jaoi ) ABDABCEZDACDBJABCFGCJACBHGI $.
-
$( Equivalence between a disjunction of two implications, and a conjunction
and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121
but with additional decidability antecedents. (Contributed by Jim
@@ -7038,45 +7104,6 @@ particular kinds of propositions may be decidable (for example, the
( wi wdc imim1 peircedc syl9r ) ABCZBCBACHACADAHBAEABFG $.
-$(
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
- Testable propositions
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
-$)
-
- $( A proposition is testable iff its negative or double-negative is true.
- See Chapter 2 [Moschovakis] p. 2.
-
- Our notation for testability is ` DECID -. ` before the formula in
- question. For example, ` DECID -. x = y ` corresponds to " ` x = y ` is
- testable". (Contributed by David A. Wheeler, 13-Aug-2018.) $)
- dftest $p |- ( DECID -. ph <-> ( -. ph \/ -. -. ph ) ) $=
- ( wn df-dc ) ABC $.
-
- $( A proposition is testable iff its negation is testable. See also ~ dcn
- (which could be read as "Decidability implies testability"). (Contributed
- by David A. Wheeler, 6-Dec-2018.) $)
- testbitestn $p |- ( DECID -. ph <-> DECID -. -. ph ) $=
- ( wn wo wdc notnotnot orbi2i orcom bitri df-dc 3bitr4ri ) ABZBZLBZCZKLCZLDK
- DNLKCOMKLAEFLKGHLIKIJ $.
-
- $( A formula is decidable if and only if its negation is decidable and it is
- stable (that is, it is testable and stable). (Contributed by David A.
- Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.) $)
- stabtestimpdc $p |- ( ( STAB ph /\ DECID -. ph ) <-> DECID ph ) $=
- ( wstab wn wdc wa wo wi df-stab pm2.36 imp syl2anb sylibr dcimpstab dcn jca
- df-dc impbii ) ABZACZDZEZADZUAASFZUBRSCZAGZSUDFZUCTAHSPUEUFUCSUDAIJKAPLUBRT
- AMANOQ $.
-
- $( Obsolete version of ~ stabtestimpdc as of 28-Oct-2023. (Contributed by
- David A. Wheeler, 13-Aug-2018.) (Proof modification is discouraged.)
- (New usage is discouraged.) $)
- stabtestimpdcOLD $p |- ( ( STAB ph /\ DECID -. ph ) <-> DECID ph ) $=
- ( wstab wn wdc wa wo exmiddc adantl df-stab biimpi orim2d adantr mpd orcomd
- wi df-dc sylibr dcimpstab dcn jca impbii ) ABZACZDZEZADZUEAUCFUFUEUCAUEUCUC
- CZFZUCAFZUDUHUBUCGHUBUHUIOUDUBUGAUCUBUGAOAIJKLMNAPQUFUBUDARASTUA $.
-
-
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Miscellaneous theorems of propositional calculus
@@ -27255,7 +27282,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use
eqsstr3.2 $e |- B C_ C $.
$( Substitution of equality into a subclass relationship. (Contributed by
NM, 19-Oct-1999.) $)
- eqsstr3i $p |- A C_ C $=
+ eqsstrri $p |- A C_ C $=
( eqcomi eqsstri ) ABCBADFEG $.
$}
@@ -27287,11 +27314,11 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use
$}
${
- eqsstr3d.1 $e |- ( ph -> B = A ) $.
- eqsstr3d.2 $e |- ( ph -> B C_ C ) $.
+ eqsstrrd.1 $e |- ( ph -> B = A ) $.
+ eqsstrrd.2 $e |- ( ph -> B C_ C ) $.
$( Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.) $)
- eqsstr3d $p |- ( ph -> A C_ C ) $=
+ eqsstrrd $p |- ( ph -> A C_ C ) $=
( eqcomd eqsstrd ) ABCDACBEGFH $.
$}
@@ -27423,11 +27450,11 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use
$}
${
- syl5sseqr.1 $e |- B C_ A $.
- syl5sseqr.2 $e |- ( ph -> C = A ) $.
+ sseqtrrid.1 $e |- B C_ A $.
+ sseqtrrid.2 $e |- ( ph -> C = A ) $.
$( Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.) $)
- syl5sseqr $p |- ( ph -> B C_ C ) $=
+ sseqtrrid $p |- ( ph -> B C_ C ) $=
( wss a1i sseqtr4d ) ACBDCBGAEHFI $.
$}
@@ -27895,7 +27922,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use
ddifnel.1 $e |- ( -. x e. ( _V \ A ) -> x e. A ) $.
$( Double complement under universal class. The hypothesis corresponds to
stability of membership in ` A ` , which is weaker than decidability
- (see ~ dcimpstab ). Actually, the conclusion is a characterization of
+ (see ~ dcstab ). Actually, the conclusion is a characterization of
stability of membership in a class (see ~ ddifstab ) . Exercise 4.10(s)
of [Mendelson] p. 231, but with an additional hypothesis. For a version
without a hypothesis, but which only states that ` A ` is a subset of
@@ -28553,7 +28580,7 @@ stability of membership in a class (see ~ ddifstab ) . Exercise 4.10(s)
Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM,
27-Apr-1994.) $)
inss2 $p |- ( A i^i B ) C_ B $=
- ( cin incom inss1 eqsstr3i ) ABCBACBBADBAEF $.
+ ( cin incom inss1 eqsstrri ) ABCBACBBADBAEF $.
${
$d x A $. $d x B $. $d x C $.
@@ -30151,9 +30178,9 @@ intersection or the difference (that is, this theorem would be equality
$( Equivalence theorem for conditional operators. (Contributed by Raph
Levien, 15-Jan-2004.) $)
ifbi $p |- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) ) $=
- ( vx wb cv wcel wa wn wo cab cif anbi2 notbid anbi2d orbi12d abbidv df-if
- id 3eqtr4g ) ABFZEGZCHZAIZUCDHZAJZIZKZELUDBIZUFBJZIZKZELACDMBCDMUBUIUMEUB
- UEUJUHULABUDNUBUGUKUFUBABUBTOPQRAECDSBECDSUA $.
+ ( vx wb cv wcel wa wn wo cab cif anbi2 notbi anbi2d orbi12d df-if 3eqtr4g
+ abbidv ) ABFZEGZCHZAIZUBDHZAJZIZKZELUCBIZUEBJZIZKZELACDMBCDMUAUHULEUAUDUI
+ UGUKABUCNUAUFUJUEABOPQTAECDRBECDRS $.
$}
${
@@ -35837,6 +35864,32 @@ This definition and how we use it is easiest to understand (and most
USVOVKFVIIFUTVAVHVIFQVDRVBTTVEVFVG $.
$}
+ ${
+ $d A x $.
+ $( A slight strengthening of ~ pwtrufal . (Contributed by Mario Carneiro
+ and Jim Kingdon, 12-Sep-2023.) $)
+ pwntru $p |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) ) $=
+ ( vx c0 csn wss wne wa cv wcel wex simpr neneqd simpll simpl sselda elsni
+ wn wceq syl eqeltrrd snssd eqssd ex exlimdv mtod notm0 sylib ) ACDZEZAUHF
+ ZGZBHZAIZBJZQACRUKUNAUHRZUKAUHUIUJKLUKUMUOBUKUMUOUKUMGZAUHUIUJUMMUPCAUPUL
+ CAUPULUHIULCRUKAUHULUIUJNOULCPSUKUMKTUAUBUCUDUEBAUFUG $.
+ $}
+
+ ${
+ $d ph x $.
+ exmid1dc.x $e |- ( ( ph /\ x C_ { (/) } ) -> DECID x = { (/) } ) $.
+ $( A convenience theorem for proving that something implies ` EXMID ` .
+ Think of this as an alternative to using a proposition, as in proofs
+ like ~ undifexmid or ~ ordtriexmid . In this context ` x = { (/) } `
+ can be thought of as "x is true". (Contributed by Jim Kingdon,
+ 21-Nov-2023.) $)
+ exmid1dc $p |- ( ph -> EXMID ) $=
+ ( cv c0 csn wss wceq wo wi wal wem wa wn wdc exmiddc syl wne df-ne ex mpd
+ pwntru syl5bir orim2d adantl orcomd alrimiv exmid01 sylibr ) ABDZEFZGZUJE
+ HZUJUKHZIZJZBKLAUPBAULUOAULMZUNUMUQUNUNNZIZUNUMIZUQUNOUSCUNPQULUSUTJAULUR
+ UMUNURUJUKRZULUMUJUKSULVAUMUJUBTUCUDUEUAUFTUGBUHUI $.
+ $}
+
${
$d x y $.
$( Excluded middle is equivalent to any set being empty or inhabited.
@@ -39749,10 +39802,10 @@ can deduce (outside the formal system, since we cannot quantify over
dcextest $p |- DECID -. ph $=
( vy wn wdc wo cab cvv wcel exmiddc ax-mp vprc weq df-v wb pm5.1im notbii
wi equid abbidv syl5req eleq1d mtbiri con2i c0 cv vex elab biimpri eq0rdv
- biidd 0ex syl6eqel impbii orbi12i mpbi dftest mpbir ) AEZFUTUTEZGZABHZIJZ
- VDEZGZVBVDFVFCVDKLVDUTVEVAVDUTAVDAVDIIJMAVCIIAIBBNZBHVCBOAVGABVGAVGAPSBTV
- GAQLUAUBUCUDUEUTVCUFIUTDVCDUGZVCJZEUTVIAAABVHDUHBDNAULUIRUJUKUMUNUOZVDUTV
- JRUPUQAURUS $.
+ biidd 0ex syl6eqel impbii orbi12i mpbi df-dc mpbir ) AEZFUTUTEZGZABHZIJZV
+ DEZGZVBVDFVFCVDKLVDUTVEVAVDUTAVDAVDIIJMAVCIIAIBBNZBHVCBOAVGABVGAVGAPSBTVG
+ AQLUAUBUCUDUEUTVCUFIUTDVCDUGZVCJZEUTVIAAABVHDUHBDNAULUIRUJUKUMUNUOZVDUTVJ
+ RUPUQUTURUS $.
$}
@@ -42333,7 +42386,7 @@ is most interesting when the natural number is a successor (as seen in
$( The domain of the universe is the universe. (Contributed by NM,
8-Aug-2003.) $)
dmv $p |- dom _V = _V $=
- ( cvv cdm ssv cid dmi wss dmss ax-mp eqsstr3i eqssi ) ABZAKCADBZKEDAFLKFD
+ ( cvv cdm ssv cid dmi wss dmss ax-mp eqsstrri eqssi ) ABZAKCADBZKEDAFLKFD
CDAGHIJ $.
$}
@@ -44079,7 +44132,7 @@ is most interesting when the natural number is a successor (as seen in
for its components. (Contributed by Jim Kingdon, 12-Dec-2018.) $)
ssxpbm $p |- ( E. x x e. ( A X. B ) -> ( ( A X. B ) C_ ( C X. D ) <->
( A C_ C /\ B C_ D ) ) ) $=
- ( va vb cv cxp wcel wex wss cdm wceq adantl sylbir adantr eqsstr3d syl6ss
+ ( va vb cv cxp wcel wex wss cdm wceq adantl sylbir adantr eqsstrrd syl6ss
wa xpm dmxpm dmss dmxpss crn rnxpm rnss rnxpss jca ex xpss12 impbid1 ) AH
BCIZJAKZUMDEIZLZBDLZCELZTZUNUPUSUNUPTZUQURUTBUOMZDUTBUMMZVAUNVBBNZUPUNFHB
JFKZGHCJGKZTZVCFGABCUAZVEVCVDGBCUBOPQUPVBVALUNUMUOUCORDEUDSUTCUOUEZEUTCUM
@@ -44092,7 +44145,7 @@ is most interesting when the natural number is a successor (as seen in
14-Dec-2018.) $)
ssxp1 $p |- ( E. x x e. C ->
( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) ) $=
- ( cv wcel wex cxp wss cdm wceq dmxpm adantr adantl eqsstr3d dmxpss syl6ss
+ ( cv wcel wex cxp wss cdm wceq dmxpm adantr adantl eqsstrrd dmxpss syl6ss
wa dmss ex xpss1 impbid1 ) AEDFAGZBDHZCDHZIZBCIZUCUFUGUCUFRZBUEJZCUHBUDJZ
UIUCUJBKUFABDLMUFUJUIIUCUDUESNOCDPQTBCDUAUB $.
$}
@@ -44103,7 +44156,7 @@ is most interesting when the natural number is a successor (as seen in
14-Dec-2018.) $)
ssxp2 $p |- ( E. x x e. C ->
( ( C X. A ) C_ ( C X. B ) <-> A C_ B ) ) $=
- ( cv wcel wex cxp wss crn wceq rnxpm adantr adantl eqsstr3d rnxpss syl6ss
+ ( cv wcel wex cxp wss crn wceq rnxpm adantr adantl eqsstrrd rnxpss syl6ss
wa rnss ex xpss2 impbid1 ) AEDFAGZDBHZDCHZIZBCIZUCUFUGUCUFRZBUEJZCUHBUDJZ
UIUCUJBKUFADBLMUFUJUIIUCUDUESNODCPQTBCDUAUB $.
$}
@@ -47097,7 +47150,7 @@ We use their notation ("onto" under the arrow). (Contributed by NM,
( ( F |` ( A \ B ) ) u. ( G |` ( B \ A ) ) ) )
C_ ( F u. G )
) $=
- ( wfn cin cres wceq w3a cdif cun uneq1i wss inundifss ssres2 ax-mp eqsstr3i
+ ( wfn cin cres wceq w3a cdif cun uneq1i wss inundifss ssres2 ax-mp eqsstrri
resundi syl5eqssr fnresdm unidm un4 simp3 uneq1d uneq2d incom eqsstri mp2an
unss12 syl6eqss uneq12 syl2an 3adant3 sseqtrd ) CAEZDBEZCABFZGZDUQGZHZIZURC
ABJZGZDBAJZGZKZKZCAGZDBGZKZCDKZVAVGURURKZVFKZVJVLURVFURUALVAVMURVCKZURVEKZK
@@ -47753,7 +47806,7 @@ We use their notation ("onto" under the arrow). (Contributed by NM,
( F |` B ) : B -onto-> D ) ->
( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) ) $=
( ccnv wfun cres wfo cdif cima wf1o wceq syl wb ax-mp crn df-ima wa forn wf
- w3a cdm wss fofun difss fof fdm syl5sseqr fores syl2anc indif reseq2i eqtri
+ w3a cdm wss fofun difss fof fdm sseqtrrid fores syl2anc indif reseq2i eqtri
cin resres foeq1 rneqi 3eqtr4i foeq3 bitri funres11 biimpri syl2anr 3adant3
sylib dff1o3 syl5eq anim12i imadif difeq12 sylan9eq sylan2 3impb f1oeq3
mpbid ) EFGZACEAHZIZBDEBHZIZUBZABJZEWCKZEWCHZLZWCCDJZWELZVQVSWFWAVSWCWDWEIZ
@@ -49375,7 +49428,7 @@ We use their notation ("onto" under the arrow). (Contributed by NM,
(Contributed by FL, 25-Jan-2007.) $)
fimacnv $p |- ( F : A --> B -> ( `' F " B ) = A ) $=
( wf ccnv cima crn imassrn cdm dfdm4 fdm ssid syl6eqss syl5eqssr syl5ss wss
- frn wfun wb ffun syl5sseqr funimass3 syl2anc mpbid eqssd ) ABCDZCEZBFZAUFUH
+ frn wfun wb ffun sseqtrrid funimass3 syl2anc mpbid eqssd ) ABCDZCEZBFZAUFUH
UGGZAUGBHUFUICIZACJUFUJAAABCKZALZMNOUFCAFZBPZAUHPZUFUMCGBCAHABCQOUFCRAUJPUN
UOSABCTUFAAUJULUKUAABCUBUCUDUE $.
@@ -52129,7 +52182,8 @@ the framework of set theory (although it already appeared as an exercise
in Errett Bishop's book _Foundations of Constructive Analysis_ from
1967).
- (Contributed by Jim Kingdon, 4-Aug-2019.) $)
+ For this theorem stated using the ~ df-ac and ~ df-exmid syntaxes, see
+ ~ exmidac . (Contributed by Jim Kingdon, 4-Aug-2019.) $)
acexmid $p |- ( ph \/ -. ph ) $=
( va vb vc vd vf ve wel wa cv wral weq wb wrex wreu wex wal weu wsb sb8eu
nfv w3a eleq12 ancoms 3adant3 3ad2antl1 3ad2antl2 simpl3 cbvrexdva2 3expa
@@ -54712,7 +54766,7 @@ associative structure (such as a group). (Contributed by NM,
ofrfval $p |- ( ph -> ( F oR R G <-> A. x e. S C R D ) ) $=
( wcel vf vg cofr wbr cv cfv cdm cin wral cvv wb fnex syl2anc wceq dmeq
wfn wa ineqan12d fveq1 breqan12d raleqbidv df-ofr brabga ineq12d syl6eq
- fndm raleqdv inss1 eqsstr3i sseli sylan2 inss2 breq12d ralbidva 3bitrd
+ fndm raleqdv inss1 eqsstrri sseli sylan2 inss2 breq12d ralbidva 3bitrd
syl ) AIJGUCZUDZBUEZIUFZVSJUFZGUDZBIUGZJUGZUHZUIZWBBHUIEFGUDZBHUIAIUJTZ
JUJTZVRWFUKAICUPZCKTWHMOCKIULUMAJDUPZDLTWINPDLJULUMVSUAUEZUFZVSUBUEZUFZ
GUDZBWLUGZWNUGZUHZUIWFUAUBIJVQUJUJWLIUNZWNJUNZUQWPWBBWSWEWTXAWQWCWRWDWL
@@ -54731,7 +54785,7 @@ associative structure (such as a group). (Contributed by NM,
Carneiro, 20-Jul-2014.) $)
fnofval $p |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) ) $=
( vx wcel wa cof co cfv cv cmpt wceq eqidd offval fveq1d adantr cvv cxp
- simpr wfn inss1 eqsstr3i sseli sylan2 eleq1d mpbird inss2 syl3anc fveq2
+ simpr wfn inss1 eqsstrri sseli sylan2 eleq1d mpbird inss2 syl3anc fveq2
cin fnovex oveq12d eqid fvmptg syl2anc 3eqtrd ) AMGUEZUFZMIJFUGUHZUIZMU
DGUDUJZIUIZWAJUIZFUHZUKZUIZMIUIZMJUIZFUHZDEFUHAVTWFULVQAMVSWEAUDBCWBWCF
GIJKLNOPQRAWABUEUFWBUMAWACUEUFWCUMUNUOUPVRVQWIUQUEZWFWIULAVQUSVRFHKURUT
@@ -54748,7 +54802,7 @@ associative structure (such as a group). (Contributed by NM,
Carneiro, 28-Jul-2014.) $)
ofrval $p |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D ) $=
( wcel vx cofr wbr w3a cfv wa cv wral eqidd ofrfval biimpa wceq breq12d
- wi fveq2 rspccv syl 3impia simp1 cin inss1 eqsstr3i simp3 syl2anc inss2
+ wi fveq2 rspccv syl 3impia simp1 cin inss1 eqsstrri simp3 syl2anc inss2
sseldi 3brtr3d ) AHIFUBUCZLGTZUDZLHUEZLIUEZDEFAVHVIVKVLFUCZAVHUFUAUGZHU
EZVNIUEZFUCZUAGUHZVIVMUNAVHVRAUABCVOVPFGHIJKMNOPQAVNBTUFVOUIAVNCTUFVPUI
UJUKVQVMUALGVNLULVOVKVPVLFVNLHUOVNLIUOUMUPUQURVJALBTVKDULAVHVIUSZVJGBLG
@@ -54778,7 +54832,7 @@ associative structure (such as a group). (Contributed by NM,
$( The function operation produces a function. (Contributed by Mario
Carneiro, 20-Jul-2014.) $)
off $p |- ( ph -> ( F oF R G ) : C --> U ) $=
- ( vz cof co wf cv cfv cmpt wcel wa wral cin inss1 eqsstr3i ffvelrn syl2an
+ ( vz cof co wf cv cfv cmpt wcel wa wral cin inss1 eqsstrri ffvelrn syl2an
sseli inss2 ralrimivva adantr wceq oveq1 eleq1d oveq2 syl21anc eqid fmptd
rspc2va wfn ffn syl eqidd offval feq1d mpbird ) AFJKLGUBUCZUDFJUAFUAUEZKU
FZVPLUFZGUCZUGZUDAUAFVSJVTAVPFUHZUIVQHUHZVRIUHZBUEZCUEZGUCZJUHZCIUJBHUJZV
@@ -54799,7 +54853,7 @@ associative structure (such as a group). (Contributed by NM,
of the operation itself. (Contributed by Mario Carneiro,
15-Sep-2014.) $)
ofres $p |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) ) $=
- ( vx co cfv wcel cvv wfn cof cv cmpt cres eqidd offval wss inss1 eqsstr3i
+ ( vx co cfv wcel cvv wfn cof cv cmpt cres eqidd offval wss inss1 eqsstrri
wa cin fnssres sylancl inss2 ssexg sylancr inidm wceq fvres adantl eqtr4d
) AFGEUAZPODOUBZFQZVCGQZEPUCFDUDZGDUDZVBPAOBCVDVEEDFGHIJKLMNAVCBRUJVDUEAV
CCRUJVEUEUFAODDVDVEEDVFVGSSAFBTDBUGZVFDTJDBCUKZBNBCUHUIZBDFULUMAGCTDCUGVG
@@ -54854,7 +54908,7 @@ associative structure (such as a group). (Contributed by NM,
left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) $)
suppssof1 $p |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) C_ L ) $=
( vx ccnv cvv cof co csn cdif cima cv cfv cmpt wf wfn ffn syl inidm eqidd
- wcel wa offval cnveqd imaeq1d feqmptd eqsstr3d funfni ffvelrnda suppssov1
+ wcel wa offval cnveqd imaeq1d feqmptd eqsstrrd funfni ffvelrnda suppssov1
funfvex sylan eqsstrd ) ACDHUAUBZSZTLUCUDZUERERUFZCUGZVKDUGZHUBUHZSZVJUEG
AVIVOVJAVHVNAREEVLVMHECDJJAEICUICEUJZOEICUKULZAEFDUIDEUJPEFDUKULQQEUMAVKE
UOZUPZVLUNVSVMUNUQURUSARBVLVMEFGHTKLAREVLUHZSZTKUCUDZUECSZWBUEGAWCWAWBACV
@@ -54876,7 +54930,7 @@ associative structure (such as a group). (Contributed by NM,
ofco $p |- ( ph ->
( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) ) $=
( cfv vx vy cof co ccom cv cmpt ffvelrnda feqmptd wcel eqidd offval fveq2
- wa oveq12d fmptco wfn wf wss cin inss1 eqsstr3i fss sylancl fnfco syl2anc
+ wa oveq12d fmptco wfn wf wss cin inss1 eqsstrri fss sylancl fnfco syl2anc
wceq inss2 inidm ffn syl fvco2 sylan eqtr4d ) AGHFUCZUDZIUEUAEUAUFZITZGTZ
VRHTZFUDZUGGIUEZHIUEZVOUDAUAUBEDVRUBUFZGTZWDHTZFUDWAIVPAEDVQIOUHAUAEDIOUI
AUBBCWEWFFDGHJKMNPQSAWDBUJUNWEUKAWDCUJUNWFUKULWDVRVGWEVSWFVTFWDVRGUMWDVRH
@@ -56921,7 +56975,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
( F o. ( x e. ( ( _V X. _V ) u. { (/) } ) |-> U. `' { x } ) ) $=
( vy vz vw cvv csn cun cv ccnv cuni wss wceq cop wcel wbr wa wex vex wb
ctpos cxp cmpt ccom cdm df-tpos cres wrel relcnv df-rel mpbi unss1 resmpt
- c0 mp2b resss eqsstr3i coss2 ax-mp eqsstri relco opelco eleq1 sneq cnveqd
+ c0 mp2b resss eqsstrri coss2 ax-mp eqsstri relco opelco eleq1 sneq cnveqd
unieqd eqeq2d anbi12d anbi2d df-mpt brab wi simplr breldm adantl eqeltrrd
eqeq1 opswapg mp2an eleq1i opelcnv bitr4i eleq1d bibi12d exlimivv biimpcd
elvv mpbiri sylbi elun1 syl6 syl elun2 a1i simpll elun sylib mpjaod simpr
@@ -57258,7 +57312,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) $)
smores2 $p |- ( ( Smo F /\ Ord A ) -> Smo ( F |` A ) ) $=
( vy vx wsmo word wa cdm con0 wf cv cfv wcel wral crn wss wfun dfsmo2 syl
- wceq cres wfn simp1bi ffun funres funfn sylib df-ima imassrn eqsstr3i frn
+ wceq cres wfn simp1bi ffun funres funfn sylib df-ima imassrn eqsstrri frn
cima syl5ss df-f sylanbrc adantr smodm cin ordin dmres ordeq ax-mp sylibr
wb ancoms sylan resss dmss simp3bi ssralv mpsyl wel wi ordtr1 inss1 sseli
eqsstri expcomd imp31 fvres ad2antlr eleq12d ralbidva mpbird syl3anbrc
@@ -57517,7 +57571,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
( ( x g u /\ x h v ) -> u = v ) ) $=
( vz va vw cv wcel cfv wceq wral wa con0 wfn cres wrex wi tfrlem3a reeanv
wbr vex w3a cin fveq2 eqeq12d onin 3ad2ant1 wfun cdm wss simp2ll syl fndm
- fnfun inss1 syl5sseqr simp2rl inss2 simp2lr ssralv simp2rr tfrlem1 simp3l
+ fnfun inss1 sseqtrrid simp2rl inss2 simp2lr ssralv simp2rr tfrlem1 simp3l
mpsyl fnbr syl2anc simp3r elin sylanbrc rspcdva funbrfv sylc 3eqtr3d 3exp
jca rexlimdva rexlimiv sylbir syl2anb ) GNZEOWGKNZUAZLNZWGPZWGWJUBIPQZLWH
RZSZKTUCZHNZMNZUAZWJWPPZWPWJUBIPQZLWQRZSZMTUCZANZDNZWGUGZXDCNZWPUGZSZXEXG
@@ -57759,7 +57813,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
tfrlemiubacc $p |- ( ph ->
A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) ) $=
( cfv wceq wcel cv cuni cres wral crecs cdm wfn tfrlemibfn fndm syl wss
- tfrlemibacc unissd recsfval syl6sseqr dmss eqsstr3d sselda tfrlem9 wfun
+ tfrlemibacc unissd recsfval syl6sseqr dmss eqsstrrd sselda tfrlem9 wfun
wa tfrlem7 a1i adantr eleq2d biimpar funssfv syl3anc word eloni ordelss
sylan sseqtr4d fun2ssres fveq2d 3eqtr3d ralrimiva fveq2 eqeq12d cbvralv
con0 reseq2 sylibr ) AEUAZHUBZRZWEWDUCZLRZSZEBUAZUDFUAZWERZWEWKUCZLRZSZ
@@ -58142,7 +58196,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
tfr1onlemubacc $p |- ( ph ->
A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) $=
( cv cuni cfv cres wceq wral wcel wa crecs cdm wfn tfr1onlembfn syl
- fndm wss tfr1onlembacc unissd tfr1onlemssrecs sstrd eqsstr3d sselda
+ fndm wss tfr1onlembacc unissd tfr1onlemssrecs sstrd eqsstrrd sselda
dmss con0 wrex eqid tfrlem9 tfrfun eleq2d biimpar funssfv mp3an2ani
cab wfun word ordelon syl2anc eloni sylan adantr sseqtr4d fun2ssres
ordelss fveq2d 3eqtr3d ralrimiva reseq2 eqeq12d cbvralv sylibr
@@ -58534,7 +58588,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) $=
( vv vt ve cv cuni cfv cres wceq wral wcel wa crecs cdm tfrcllembfn
wf fdm syl wss tfrcllembacc unissd tfrcllemssrecs sstrd dmss sselda
- eqsstr3d wfn con0 wrex cab eqid tfrlem9 wfun tfrfun biimpar funssfv
+ eqsstrrd wfn con0 wrex cab eqid tfrlem9 wfun tfrfun biimpar funssfv
eleq2d mp3an2ani word ordelon syl2anc eloni ordelss adantr sseqtr4d
sylan fun2ssres fveq2d 3eqtr3d fveq2 reseq2 eqeq12d cbvralv sylibr
ralrimiva ) AEUIZHUJZUKZXAWTULZOUKZUMZEIUNFUIZXAUKZXAXFULZOUKZUMZFI
@@ -59981,7 +60035,7 @@ defined for all sets (being defined for all ordinals might be enough if
Exercise 5 of [TakeutiZaring] p. 62. (Contributed by NM, 6-Dec-2004.) $)
oaword1 $p |- ( ( A e. On /\ B e. On ) -> A C_ ( A +o B ) ) $=
( con0 wcel wa c0 coa co wceq oa0 adantr wss wi 0elon oawordi 3com13 mp3an3
- 0ss mpi eqsstr3d ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBAJKUCFBLZUDUELZBRUAUBFCDZUF
+ 0ss mpi eqsstrrd ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBAJKUCFBLZUDUELZBRUAUBFCDZUF
UGMZNUHUBUAUIFBAOPQST $.
${
@@ -64686,7 +64740,7 @@ the first case of his notation (simple exponentiation) and subscript it
vf wa syl2anc cv wf1 wex wb peano2 brdomg biimpa cfv ad2antrr sssucid a1i
simpr simplll f1imaen2g syl22anc difexg word nnord orddif wfn f1fn sucidg
imaeq2d fnsnfv difeq2d ccnv wfun wf simprbi imadif eqtr4d f1f crn imassrn
- df-f1 syl5ss ssdifd eqsstr3d eqsstrd ssdomg sylc endomtr simpllr ffvelrnd
+ df-f1 syl5ss ssdifd eqsstrrd eqsstrd ssdomg sylc endomtr simpllr ffvelrnd
frn phplem3g domentr exlimddv ex ) ACDZBCDZSZAEZBEZFGZABFGZWLWOSZWMWNRUAZ
UBZWPRWLWOWSRUCZWLWNCDZWOWTUDWKXAWJBUEHZWMWNCRUFIUGWQWSSZAWNAWRUHZJZKZFGZ
XFBLGWPXCAWRAMZLGXHXFFGZXGXCXHAXCWSXAAWMNZWJXHALGWQWSULWLXAWOWSXBUIZXJXCA
@@ -65907,7 +65961,7 @@ texts assume excluded middle (in which case the two intersection
wceq vw vy w3a enfii 3adant2 csn cun breq1 eqeq1 albidv en0 eqcomd adantl
ensym sylib ax-gen wn sseq2 breq2 eqeq2 cbvalv cdif simplr difsn ad3antlr
difun2 syl5eq simprl ssdifd simplll vex a1i simpllr unsnfi syl3anc simprr
- eqsstr3d vsnid elun2 sseldd dif1enen eqbrtrrd jca difexg spcv sylc uneq1d
+ eqsstrrd vsnid elun2 sseldd dif1enen eqbrtrrd jca difexg spcv sylc uneq1d
ax-mp ensymd syl2anc fidifsnid eqtrd alrimiv syl5bi findcard2s syl 3simpc
ex spcgv 3ad2ant1 mp2d ) BFGZABHZABIJZUCZACKZHZAXFIJZLZAXFTZMZCNZXCXDLZAB
TZXEAFGZXLXBXDXOXCABUDUEUAKZXFHZXPXFIJZLZXPXFTZMZCNOXFHZOXFIJZLZOXFTZMZCN
@@ -66272,10 +66326,10 @@ texts assume excluded middle (in which case the two intersection
( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) ) $=
( vy wem cv crn wss wa cuni cdif cima wceq sbthlem2 syl sbthlem1 sylibr
jctil eqss difeq2d adantl imassrn sstr2 ax-mp wcel wstab wal exmidexmid
- wi wb wdc dcimpstab alrimiv dfss4st syl5ib imp eqtr2d ) JFKZLZBMZNBDOZP
- ZBBVCCEKVFQPZQZPZPZVIVEVGVKRJVEVFVJBVEVFVJMZVJVFMZNVFVJRVEVMVLABCDEFGHS
- ABCDEFGHUAUCVFVJUDUBUEUFJVEVKVIRZVEVIBMZJVNVIVDMVEVOUNVCVHUGVIVDBUHUIJI
- KVIUJZUKZIULVOVNUOJVQIJVPUPVQVPUMVPUQTURIVIBUSTUTVAVB $.
+ wi wb wdc dcstab alrimiv dfss4st syl5ib imp eqtr2d ) JFKZLZBMZNBDOZPZBB
+ VCCEKVFQPZQZPZPZVIVEVGVKRJVEVFVJBVEVFVJMZVJVFMZNVFVJRVEVMVLABCDEFGHSABC
+ DEFGHUAUCVFVJUDUBUEUFJVEVKVIRZVEVIBMZJVNVIVDMVEVOUNVCVHUGVIVDBUHUIJIKVI
+ UJZUKZIULVOVNUOJVQIJVPUPVQVPUMVPUQTURIVIBUSTUTVAVB $.
$}
$( Lemma for ~ isbth . (Contributed by NM, 27-Mar-1998.) $)
@@ -67886,7 +67940,7 @@ readily usable (e.g., by ~ djudom and ~ djufun ) while the simpler
caseinj $p |- ( ph -> Fun `' case ( R , S ) ) $=
( vy vx cinl ccnv ccom cinr cun wfun cdm c0 funeqi sylibr wss crn cin cvv
cdjucase cv cop df-inl funmpt2 funcnvcnv ax-mp funco sylancr cnvco df-inr
- wceq c1o df-rn rncoss eqsstr3i ss2in mp2an syl5sseq ss0 syl funun df-case
+ wceq c1o df-rn rncoss eqsstrri ss2in mp2an syl5sseq ss0 syl funun df-case
syl21anc cnveqi cnvun eqtri ) ABIJZKZJZCLJZKZJZMZNZBCUCZJZNAVLNZVONZVLOZV
OOZUAZPUNZVQAVJJZBJZKZNZVTAWFNZWGNWIINWJGUBPGUDUEIGUFUGIUHUIDWFWGUJUKVLWH
BVJULQRAVMJZCJZKZNZWAAWKNZWLNWNLNWOHUBUOHUDUELHUMUGLUHUIEWKWLUJUKVOWMCVMU
@@ -68231,7 +68285,7 @@ property of disjoint unions (see ~ updjud in the case of functions).
djuinj $p |- ( ph -> Fun `' ( R |_|d S ) ) $=
( cA cdm cres ccnv ccom cun wfun cin c0 ax-mp funeqi sylibr wss crn cdjud
cinl cinr wceq cdju inlresf1 f1fun funcnvcnv funco sylancr cnvco inrresf1
- wf1 df-rn rncoss eqsstr3i ss2in mp2an syl5sseq ss0 funun syl21anc df-djud
+ wf1 df-rn rncoss eqsstrri ss2in mp2an syl5sseq ss0 funun syl21anc df-djud
syl cnveqi cnvun eqtri ) ABUBBHZIZJZKZJZCUCCHZIZJZKZJZLZMZBCUAZJZMAVLMZVQ
MZVLHZVQHZNZOUDZVSAVJJZBJZKZMZWBAWHMZWIMWKVIMZWLVHVHGUEZVIUMWMVHGUFVHWNVI
UGPVIUHPDWHWIUIUJVLWJBVJUKQRAVOJZCJZKZMZWCAWOMZWPMWRVNMZWSVMGVMUEZVNUMWTG
@@ -68668,12 +68722,12 @@ elements or fails to hold (is equal to ` (/) ` ) for some element.
exmidomniim $p |- ( EXMID -> A. x x e. Omni ) $=
( vf vy wem cv wcel c2o c0 wceq c1o wo wi wa wn wdc exmidexmid syl adantr
wral wb comni wf cfv wrex wal exmiddc orcomd ffvelrn df2o3 syl6eleq elpri
- cpr ord ralimdva con3d adantl dfrex2dc sylibrd orim1d mpd alrimiv cvv vex
- ex isomni ax-mp sylibr ) DAEZUAFZADVHGBEZUBZCEZVJUCZHIZCVHUDZVMJIZCVHSZKZ
- LZBUEZVIDVSBDVKVRDVKMZVQNZVQKZVRDWCVKDVQWBDVQOVQWBKVQPVQUFQUGRWAWBVOVQWAW
- BVNNZCVHSZNZVOVKWBWFLDVKWEVQVKWDVPCVHVKVLVHFMZVNVPWGVMHJULZFVNVPKWGVMGWHV
- HGVLVJUHUIUJVMHJUKQUMUNUOUPDVOWFTZVKDVOOWIVOPVNCVHUQQRURUSUTVDVAVHVBFVIVT
- TAVCCVHBVBVEVFVGVA $.
+ cpr ord ralimdva adantl dfrex2dc sylibrd orim1d mpd ex alrimiv cvv isomni
+ con3d elv sylibr ) DAEZUAFZADVGGBEZUBZCEZVIUCZHIZCVGUDZVLJIZCVGSZKZLZBUEZ
+ VHDVRBDVJVQDVJMZVPNZVPKZVQDWBVJDVPWADVPOVPWAKVPPVPUFQUGRVTWAVNVPVTWAVMNZC
+ VGSZNZVNVJWAWELDVJWDVPVJWCVOCVGVJVKVGFMZVMVOWFVLHJULZFVMVOKWFVLGWGVGGVKVI
+ UHUIUJVLHJUKQUMUNVDUODVNWETZVJDVNOWHVNPVMCVGUPQRUQURUSUTVAVHVSTACVGBVBVCV
+ EVFVA $.
$}
${
@@ -69376,6 +69430,104 @@ elements or fails to hold (is equal to ` (/) ` ) for some element.
$}
+$(
+=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
+ Axiom of Choice equivalents
+=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
+$)
+
+ $c CHOICE $.
+
+ $( Formula for an abbreviation of the axiom of choice. $)
+ wac $a wff CHOICE $.
+
+ ${
+ $d f x $.
+ $( The expression ` CHOICE ` will be used as a readable shorthand for any
+ form of the axiom of choice; all concrete forms are long, cryptic, have
+ dummy variables, or all three, making it useful to have a short name.
+ Similar to the Axiom of Choice (first form) of [Enderton] p. 49.
+
+ There are some decisions about how to write this definition especially
+ around whether ~ ax-setind is needed to show equivalence to other ways
+ of stating choice, and about whether choice functions are available for
+ nonempty sets or inhabited sets. (Contributed by Mario Carneiro,
+ 22-Feb-2015.) $)
+ df-ac $a |- ( CHOICE <-> A. x E. f ( f C_ x /\ f Fn dom x ) ) $.
+ $}
+
+ ${
+ $d A f u v x $. $d A f u v y $. $d f ph u x $. $d u v w x $.
+ acfun.ac $e |- ( ph -> CHOICE ) $.
+ acfun.a $e |- ( ph -> A e. V ) $.
+ acfun.m $e |- ( ph -> A. x e. A E. w w e. x ) $.
+ $( A convenient form of choice. The goal here is to state choice as the
+ existence of a choice function on a set of inhabited sets, while making
+ full use of our notation around functions and function values.
+ (Contributed by Jim Kingdon, 20-Nov-2023.) $)
+ acfun $p |- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) ) $=
+ ( vu vv vy cv wcel wel wa wex wral cvv wceq copab wss cdm wfn cfv wal cab
+ elexd abid2 vex eqeltri a1i opabex3d wac df-ac sylib sseq2 fneq2d anbi12d
+ dmeq exbidv spcgv sylc simprr elequ2 cbvralv elequ1 cbvexv ralbii dmopab3
+ wb bitri adantr mpbid cop simplrl fnopfv sylan sseldd fvex eleq1 opelopab
+ anbi2d simprd ralrimiva jca ex eximdv mpd ) AEMZJMZDNZKJOZPZJKUAZUBZWJWOU
+ CZUDZPZEQZWJDUDZBMZWJUEZXBNZBDRZPZEQAWOSNWJLMZUBZWJXGUCZUDZPZEQZLUFZWTAWM
+ JKDADFHUHWMKUGZSNAWLPXNWKSKWKUIJUJUKULUMAUNXMGLEUOUPXLWTLWOSXGWOTZXKWSEXO
+ XHWPXJWRXGWOWJUQXOXIWQWJXGWOUTURUSVAVBVCAWSXFEAWSXFAWSPZXAXEXPWRXAAWPWRVD
+ AWRXAVKWSAWQDWJAWMKQZJDRZWQDTACBOZCQZBDRZXRIYACJOZCQZJDRXRXTYCBJDXBWKTXSY
+ BCBJCVEVAVFYCXQJDYBWMCKCKJVGVHVIVLUPWMJKDVJUPURVMVNZXPXDBDXPXBDNZPZYEXDYF
+ XBXCVOZWONYEXDPZYFWJWOYGAWPWRYEVPXPXAYEYGWJNYDDXBWJVQVRVSWNYEKBOZPYHJKXBX
+ CBUJZXBWJSSEUJYJVTWKXBTWLYEWMYIWKXBDWAJBKVEUSKMZXCTYIXDYEYKXCXBWAWCWBUPWD
+ WEWFWGWHWI $.
+ $}
+
+ ${
+ $d A x $. $d A z $. $d B x $. $d B z $. $d C f z $. $d f x y $.
+ $d w z $. $d y z $.
+ exmidaclem.a $e |- A = { x e. { (/) , { (/) } }
+ | ( x = (/) \/ y = { (/) } ) } $.
+ exmidaclem.b $e |- B = { x e. { (/) , { (/) } }
+ | ( x = { (/) } \/ y = { (/) } ) } $.
+ exmidaclem.c $e |- C = { A , B } $.
+ $( Lemma for ~ exmidac . The result, with a few hypotheses to break out
+ commonly used expressions. (Contributed by Jim Kingdon,
+ 21-Nov-2023.) $)
+ exmidaclem $p |- ( CHOICE -> EXMID ) $=
+ ( vz vw cv c0 wa wcel wceq cvv wo simpr eqeq1 orbi1d wac csn wss wfn wral
+ cfv wdc simpl cpr crab pp0ex rabex eqeltri prexg mp2an a1i syl6eleq elpri
+ wex 0ex prid1 eqid orci elrab2 mpbir2an eleq2 mpbiri elex2 syl p0ex prid2
+ vf jaoi 3syl ralrimiva acfun wn 0nep0 simplr eqeq12d mtbiri olc ralrimivw
+ neii rabid2 sylibr syl6eqr eqtr3d fveq2d nsyl olcd orcd id eleq12d simprr
+ fveq2 eleqtrri rspcdva sylib simprd adantr mpjaodan exlimddv exmid1dc
+ df-dc ) UABUABKZLUBZUCZMZVLKZEUDZIKZXJUFZXLNZIEUEZMZXFXGOZUGZVLXIIJEVLPUA
+ XHUHEPNXIECDUIZPHCPNDPNXSPNCAKZLOZXQQZALXGUIZUJZPFYBAYCUKULUMZDXTXGOZXQQZ
+ AYCUJZPGYGAYCUKULUMZCDPPUNUOUMUPXIJKXLNJUSZIEXIXLENZMZXLXSNXLCOZXLDOZQYJY
+ LXLEXSXIYKRHUQXLCDURYMYJYNYMLXLNZYJYMYOLCNZYPLYCNLLOZXQQZLXGUTVAYQXQLVBVC
+ YBYRALYCCYAYAYQXQXTLLSTFVDVEXLCLVFVGJLXLVHVIYNXGXLNZYJYNYSXGDNZYTXGYCNXGX
+ GOZXQQZLXGVJVKUUAXQXGVBVCYGUUBAXGYCDYFYFUUAXQXTXGXGSTGVDVEXLDXGVFVGJXGXLV
+ HVIVMVNVOVPXIXPMZCXJUFZLOZXRXQUUCUUEMZXQXQVQZQZXRUUFDXJUFZXGOZUUHXQUUFUUJ
+ MZUUGXQUUKUUDUUIOZXQUUKUULLXGOLXGVRWDUUKUUDLUUIXGUUCUUEUUJVSUUFUUJRVTWAXQ
+ CDXJXQYCCDXQYCYDCXQYBAYCUEYCYDOXQYBAYCXQYAWBWCYBAYCWEWFFWGXQYCYHDXQYGAYCU
+ EYCYHOXQYGAYCXQYFWBWCYGAYCWEWFGWGWHWIWJWKUUFXQMXQUUGUUFXQRWLUUCUUJXQQZUUE
+ UUCUUIYCNZUUMUUCUUIDNZUUNUUMMUUCXNUUOIEDYNXMUUIXLDXLDXJWPYNWMWNXIXKXOWOZD
+ ENUUCDXSECDYIVKHWQUPWRYGUUMAUUIYCDXTUUIOYFUUJXQXTUUIXGSTGVDWSWTXAXBXQXEZW
+ FUUCXQMZUUHXRUURXQUUGUUCXQRWLUUQWFUUCUUDYCNZUUEXQQZUUCUUDCNZUUSUUTMUUCXNU
+ VAIECYMXMUUDXLCXLCXJWPYMWMWNUUPCENUUCCXSECDYEVAHWQUPWRYBUUTAUUDYCCXTUUDOY
+ AUUEXQXTUUDLSTFVDWSWTXBXCXD $.
+ $}
+
+ ${
+ $d u x y $.
+ $( The axiom of choice implies excluded middle. See ~ acexmid for more
+ discussion of this theorem and a way of stating it without using
+ ` CHOICE ` or ` EXMID ` . (Contributed by Jim Kingdon, 21-Nov-2023.) $)
+ exmidac $p |- ( CHOICE -> EXMID ) $=
+ ( vx vy vu cv c0 wceq csn cpr crab eqeq1 orbi1d cbvrabv eqid exmidaclem
+ wo ) ABCDZEFZBDEGZFZOZCERHZIZPRFZSOZCUAIZUBUEHZTADZEFZSOCAUAPUGFZQUHSPUGE
+ JKLUDUGRFZSOCAUAUIUCUJSPUGRJKLUFMN $.
+ $}
+
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Cardinal number arithmetic
@@ -71050,7 +71202,7 @@ divided by the argument (although we don't define full division since we
( vz vw wcel cnpi wa ceq wceq c1o cltq wbr co adantr clti comu 1onn sylan
com c0 cnq cv cop cec wex wrex nqpi cpli 1pi addclpi mpan2 cmi pinn nnacl
coa sylancl nnm1 syl wss elni2 word nnord ordgt0ge1 biimpa sylbi nnaword1
- adantl wb simprbi sseldd nnmword mp3anl1 syl21anc mpbid eqsstr3d 0lt1o wi
+ adantl wb simprbi sseldd nnmword mp3anl1 syl21anc mpbid eqsstrrd 0lt1o wi
nna0 nnaordi mpi eqeltrrd mulclpi ltpiord syldan mulpiord addpiord oveq1d
eqtrd eleq2d bitrd mpbird mulcompig breq2d jctir ordpipqqs mulidpi breq1d
mpan mpdan breq1 opeq1 eceq1d rspcev syl2anc exlimivv ) BUAECUBZFEZDUBZFE
@@ -107344,7 +107496,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_
syl wceq cc0 cbc cuz elfzuz2 nnuz syl6eleqr fznn0sub nn0p1nn elfznn nnm1nn0
cn0 faccl 3syl nnmulcld divmuldivapd elfzel2 zcnd 1cnd fveq2d oveq1d oveq2d
subsubd oveq12d facp1 eqcomd facnn2 mul32d mulassd 3eqtr4d divcanap5d 0p1e1
- mulcomd oveq1i cz wss 0z fzp1ss ax-mp eqsstr3i sseli bcval2 cc ax-1cn npcan
+ mulcomd oveq1i cz wss 0z fzp1ss ax-mp eqsstrri sseli bcval2 cc ax-1cn npcan
3eqtrrd sylancl peano2zm uzid peano2uz 4syl eqeltrrd fzss2 elfzmlbm sseldd
) ACBDEZFZBGHZBAIEZGHZAGHZJEZKEZWPBACIEZIEZGHZXBGHZJEZKEZXCAKEZJEZBAUAEZBXB
UAEZXHJEWOXIWPWQCLEZJEZWTXLJEZKEZXLWPJEZXLWTJEZKEXAWOWPXLGHZXEJEZKEZXLAKEZJ
@@ -107488,7 +107640,7 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_
nnzd mpbid eleqtrd divap0d divmulap3d div23apd 3eqtr3rd 3eqtrrd nn0re ltp1d
mpbird olcd mpd3an23 sylan9eqr bcnn adantr jaodan syldan bcval3 syl3anc imp
ex jaod sylbid adantlr 00id con3i 3expa sylan2 simpll simplr peano2zm nn0zd
- syl2anr fzp1ss eqsstr3i syl6bir sylbird con3dimp 0zd w3a wdc fzdcel exmiddc
+ syl2anr fzp1ss eqsstrri syl6bir sylbird con3dimp 0zd w3a wdc fzdcel exmiddc
id mpjaodan ) BCDZAEDZFZAGBHIJZKJZDZBALJZBAHMJZLJZIJZUVJALJZNZUVLUAZUVGUVLU
VRUVHUVGUVLUVRUVGUVLAGNZAGHIJZUVJKJZDZOZUVRUVGUVJCDZUVLUWDUBZBUCZUWFUVJGUDU
EZCAGUVJUFUGUHPUVGUVTUVRUWCUVGUVRUVTBGLJZBGHMJZLJZIJZUVJGLJZNUVGHGIJHUWLUWM
@@ -116750,7 +116902,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.) $)
fsumsplit $p |- ( ph -> sum_ k e. U C =
( sum_ k e. A C + sum_ k e. B C ) ) $=
- ( vx csu caddc co wcel cc0 wdc wa wn wceq cv cif ssun1 syl5sseqr wo simpr
+ ( vx csu caddc co wcel cc0 wdc wa wn wceq cv cif ssun1 sseqtrrid wo simpr
cun orcd cin c0 disjel ex con2d imp sylan adantlr olcd eleq2d biimpa elun
sylib mpjaodan df-dc sylibr ralrimiva cc sselda syldan cfn cz cuz cfv wss
wral w3a isumss2 ssun2 ad2antrr sylancom orcanai 0cnd eleq1w dcbid adantr
@@ -128647,7 +128799,7 @@ reduced fraction representation (no common factors, denominator
eqtrd syl3anc wf1o crab ssrab2 eqsstri mpbird syl2an cmpt eqtr3d eqeltrrd
eqtri sylib cfn 0z fzofig sylancr syl5eqel wdc gcdcld nn0zd zdceq ssfirab
1zzd dfphi2 fveq2i syl6eqr simprbi dvdsmul2 opelxpi eqeltrd wfun cdm crth
- dvdstr wfn f1ofn fnfun fndm syl5sseqr funimass4 ccnv mp2an sseqtr4i sseli
+ dvdstr wfn f1ofn fnfun fndm sseqtrrid funimass4 ccnv mp2an sseqtr4i sseli
xpss12 f1ocnvfv2 wf f1ocnv f1of ffvelrn opelxp rpmul funfvima2 imp syldan
cbvmptv ex ssrdv eqssd wf1 f1of1 a1i zmulcld f1imaeng eqbrtrrd 1z sylancl
xpfi hashen hashxp rabeqi oveq12d 3eqtr4d ) AKUCUDZFUCUDZJUCUDZUEUFZHIUEU
@@ -129142,7 +129294,7 @@ reduced fraction representation (no common factors, denominator
increases. (Contributed by Jim Kingdon, 15-Jul-2023.) $)
ennnfonelemss $p |- ( ph -> ( H ` P ) C_ ( H ` ( P + 1 ) ) ) $=
( ccnv cfv cima wcel c1 caddc co wss wceq cdm cop csn cun ennnfonelemp1
- wn wa cif adantr simpr iftrued eqtrd eqimss2 syl iffalsed syl5sseqr wdc
+ wn wa cif adantr simpr iftrued eqtrd eqimss2 syl iffalsed sseqtrrid wdc
ssun1 cn0 com wf1o frechashgf1o f1ocnv f1of mp2b ffvelrnd ennnfonelemdc
wo wf a1i exmiddc mpjaodan ) AEMUBZUCZIUCZIWDUDUEZEKUCZEUFUGUHKUCZUIZWF
UPZAWFUQZWHWGUJWIWKWHWFWGWGWGUKWEULUMZUNZURZWGAWHWNUJZWFABCDEFGHIJKLMNO
@@ -132214,7 +132366,7 @@ _Introduction to General Topology_ (1983), p. 114) and it is convenient
istopon $p |- ( J e. ( TopOn ` B ) <-> ( J e. Top /\ B = U. J ) ) $=
( vj vb ctopon cfv wcel cvv ctop cuni wceq wa cdm wrel wfun funtopon crab
cv cpw wss funrel ax-mp relelfvdm elexd uniexg eleq1 syl5ibrcom imp eqeq1
- mpan rabbidv df-topon vpwex pwex wi rabss pwuni pweq syl5sseqr sylibr a1i
+ mpan rabbidv df-topon vpwex pwex wi rabss pwuni pweq sseqtrrid sylibr a1i
selpw mprgbir ssexi fvmpt3i eleq2d unieq eqeq2d elrab syl6bb pm5.21nii )
BAEFZGZAHGZBIGZABJZKZLZVMAEMZENZVMAVSGEOVTPEUAUBBAEUCUJUDVOVQVNVOVNVQVPHG
BIUEAVPHUFUGUHVNVMBACRZJZKZCIQZGVRVNVLWDBDADRZWBKZCIQZWDHEWEAKWFWCCIWEAWB
@@ -132988,7 +133140,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
{ x e. ~P A | ( P e. x -> x = A ) } e. ( TopOn ` A ) ) $=
( vy vz wcel wa cv wceq wss wral adantr syl eleq2 eqeq1 imbi12d elrab cvv
wi cpw crab ctop cuni ctopon cfv wal cin ssrab simprl sspwuni vuniex elpw
- sylib sylibr wrex eluni2 r19.29 simpr elssuni eqsstr3d rexlimiva ad2antll
+ sylib sylibr wrex eluni2 r19.29 simpr elssuni eqsstrrd rexlimiva ad2antll
impr ex syl5bi jctild eqss syl6ibr sylanbrc alrimiv simprll elpwid syl5ss
inss1 vex inex1 elin simprlr simprrr anim12d ineq12 inidm syl6eq syl6 jca
anbi12i ralrimivv wb pwexg rabexg istopg mpbir2and pwidg eqidd a1d ssrab2
@@ -133274,7 +133426,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
$( A subset of a topology's underlying set is included in its closure.
(Contributed by NM, 22-Feb-2007.) $)
sscls $p |- ( ( J e. Top /\ S C_ X ) -> S C_ ( ( cls ` J ) ` S ) ) $=
- ( vx ctop wcel wss wa cv ccld cfv crab cint ccl ssintub clsval syl5sseqr
+ ( vx ctop wcel wss wa cv ccld cfv crab cint ccl ssintub clsval sseqtrrid
) BFGACHIAEJHEBKLZMNAABOLLEASPEABCDQR $.
$( A subset includes its interior. (Contributed by NM, 3-Oct-2007.)
@@ -137783,7 +137935,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric
the base set. (Contributed by NM, 3-Sep-2006.) (Revised by Mario
Carneiro, 12-Nov-2013.) $)
mopnfss $p |- ( D e. ( *Met ` X ) -> J C_ ~P X ) $=
- ( cxmet cfv wcel cuni cpw pwuni mopnuni pweqd syl5sseqr ) ACEFGZBHZIBCIBJ
+ ( cxmet cfv wcel cuni cpw pwuni mopnuni pweqd sseqtrrid ) ACEFGZBHZIBCIBJ
NCOABCDKLM $.
$( The base set of a metric space is open. Part of Theorem T1 of
@@ -138261,7 +138413,7 @@ S C_ ( P ( ball ` D ) T ) ) $=
( A. x e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\
( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> J = K ) ) $=
( cfv wcel wa cv cbl co crp wrex wral wss cle wbr wceq wi simprrr simplll
- cxmet cxr simplr simprlr rpxrd simprll simprrl syl221anc eqsstr3d simpllr
+ cxmet cxr simplr simprlr rpxrd simprll simprrl syl221anc eqsstrrd simpllr
ssbl eqsstrd jca expr anassrs reximdva r19.40 syl6 ralimdva r19.26 syl6ib
metequiv sylibrd ) BFUGKZLZCVJLZMZGNZHNZUAUBZANZVNBOKZPZVQVNCOKZPZUCZMZGQ
RZHQSZAFSWAVQVOVRPZTZGQRZHQSVSVQVOVTPZTZGQRZHQSMZAFSDEUCVMWEWLAFVMVQFLZMZ
@@ -140445,7 +140597,7 @@ S C_ ( P ( ball ` D ) T ) ) $=
${
$d A d e z $. $d B d e z $. $d F d e z $. $d J d e z $. $d K d e z $.
cnplimcim.k $e |- K = ( MetOpen ` ( abs o. - ) ) $.
- cnplimc.j $e |- J = ( K |`t A ) $.
+ cnplimcim.j $e |- J = ( K |`t A ) $.
$( If a function is continuous at ` B ` , its limit at ` B ` equals the
value of the function there. (Contributed by Mario Carneiro,
28-Dec-2016.) (Revised by Jim Kingdon, 14-Jun-2023.) $)
@@ -140474,6 +140626,79 @@ S C_ ( P ( ball ` D ) T ) ) $=
ABXICXQUVCUVAWTXAXBXC $.
$}
+
+ ${
+ cnplimccntop.k $e |- K = ( MetOpen ` ( abs o. - ) ) $.
+ cnplimc.j $e |- J = ( K |`t A ) $.
+
+ ${
+ cnplimclemr.a $e |- ( ph -> A C_ CC ) $.
+ cnplimclemr.f $e |- ( ph -> F : A --> CC ) $.
+ cnplimclemr.b $e |- ( ph -> B e. A ) $.
+ cnplimclemr.l $e |- ( ph -> ( F ` B ) e. ( F limCC B ) ) $.
+
+ ${
+ cnplimclemle.e $e |- ( ph -> E e. RR+ ) $.
+ cnplimclemle.d $e |- ( ph -> D e. RR+ ) $.
+ cnplimclemle.z $e |- ( ph -> Z e. A ) $.
+ cnplimclemle.im $e |- ( ( ph /\ Z =//= B /\ ( abs ` ( Z - B ) ) < D )
+ -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) ) $.
+ cnplimclemle.zd $e |- ( ph -> ( abs ` ( Z - B ) ) < D ) $.
+ $( Lemma for ~ cnplimccntop . Satisfying the epsilon condition for
+ continuity. (Contributed by Mario Carneiro and Jim Kingdon,
+ 17-Nov-2023.) $)
+ cnplimclemle $p |- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E ) $=
+ ( c2 cdiv co cfv cmin cabs clt wbr wa simpr cr ffvelrnd subcld abscld
+ wcel cc adantr rphalfcld rpred cc0 wceq cap wn simpll syl3anc ltnsymd
+ syl pm2.65da wb sseldd syl2an2r mpbird fveq2d subeq0bd abs00bd rpgt0d
+ apti crp eqbrtrd pm2.21dd wo rphalflt wi axltwlin mpd mpjaodan ) AEUA
+ UBUCZIFUDZCFUDZUEUCZUFUDZUGUHZWKEUGUHZWMAWLUIZWLWMAWLUJZWNWKWGAWKUKUO
+ ZWLAWJAWHWIABUPIFMRULZABUPCFMNULUMUNZUQAWGUKUOZWLAWGAEPURZUSZUQWNWKUT
+ WGUGWNWJWNWHWIAWHUPUOWLWQUQWNICFWNICVAZICVBUHZVCZWNXCWLWNWLXCWOUQWNXC
+ UIZWKWGXEAWPAWLXCVDZWRVGXEAWSXFXAVGXEAXCICUEUCUFUDDUGUHZWKWGUGUHXFWNX
+ CUJXEAXGXFTVGSVEVFVHAIUPUOWLCUPUOZXBXDVIABUPILRVJAXHWLABUPCLNVJUQICVQ
+ VKVLVMVNVOWNWGAWGVRUOWLWTUQVPVSVFVTAWMUJAWGEUGUHZWLWMWAZAEVRUOXIPEWBV
+ GAWSEUKUOWPXIXJWCXAAEPUSWRWGEWKWDVEWEWF $.
+ $}
+
+ $d A d e s z $. $d B d e s z $. $d F d e s z $. $d K d e z $.
+ $d d e ph s z $.
+ $( Lemma for ~ cnplimccntop . The reverse direction. (Contributed by
+ Mario Carneiro and Jim Kingdon, 17-Nov-2023.) $)
+ cnplimclemr $p |- ( ph -> F e. ( ( J CnP K ) ` B ) ) $=
+ ( vz vd cfv co wcel cc clt crp ve vs cabs cmin ccom cxp cres cmopn ccnp
+ wf cv wbr wi wral wrex wa cap cdiv breq2 imbi2d rexralbidv climc sseldd
+ c2 wceq ellimc3ap simprd adantr rphalfcl adantl rspcdva ad5antr simpllr
+ mpbid ffvelrnd cnmetdval syl2anc wss simp-5r simp-4r w3a 3simpc simp1lr
+ eqid ovresd eqtrd simpr eqbrtrrd cnplimclemle eqbrtrd ralimdva reximdva
+ mpd exp31 ralrimiva cxmet cnxmet xmetres2 sylancr a1i metcnp2 mpbir2and
+ wb syl3anc crest metrest syl5eq oveq1d fveq1d eleqtrrd ) ADCUCUDUEZBBUF
+ UGZUHOZFUIPZOZCEFUIPZOADXOQZBRDUJZMUKZCXLPZNUKZSULZXSDOZCDOZXKPZUAUKZSU
+ LZUMZMBUNZNTUOZUATUNZJAYJUATAYFTQZUPZXSCUQULZXSCUDPUCOZYASULZUPZYCYDUDP
+ UCOZYFVDURPZSULZUMZMBUNZNTUOZYJYMYQYRUBUKZSULZUMZMBUNNTUOZUUCUBTYSUUDYS
+ VEZUUFUUANMTBUUHUUEYTYQUUDYSYRSUSUTVAAUUGUBTUNZYLAYDRQZUUIAYDDCVBPQZUUJ
+ UUIUPLAUBNMBCYDDJIABRCIKVCZVFVNVGVHYLYSTQAYFVIVJVKYMUUBYINTYMYATQZUPZUU
+ AYHMBUUNXSBQZUPZUUAYBYGUUPUUAUPZYBUPZYEYRYFSUURYCRQUUJYEYRVEUURBRXSDAXR
+ YLUUMUUOUUAYBJVLZUUNUUOUUAYBVMZVOUURBRCDUUSACBQZYLUUMUUOUUAYBKVLZVOYCYD
+ XKXKWDZVPVQUURBCYAYFDEFXSGHABRVRZYLUUMUUOUUAYBIVLZUUSUVBAUUKYLUUMUUOUUA
+ YBLVLAYLUUMUUOUUAYBVSYMUUMUUOUUAYBVTUUTUURYNYPWAYQYTUURYNYPWBUUPUUAYBYN
+ YPWCWMUURXTYOYASUURXTXSCXKPZYOUURXSCXKBUUTUVBWEUURXSRQCRQZUVFYOVEUURBRX
+ SUVEUUTVCAUVGYLUUMUUOUUAYBUULVLXSCXKUVCVPVQWFUUQYBWGWHWIWJWNWKWLWMWOAXL
+ BWPOQZXKRWPOQZUVAXQXRYKUPXCAUVIUVDUVHWQIXKBRWRWSUVIAWQWTKUANMXLXKCDXMFB
+ RXMWDZGXAXDXBACXPXNAEXMFUIAEFBXEPZXMHAUVIUVDUVKXMVEWQIXKXLFXMRBXLWDGUVJ
+ XFWSXGXHXIXJ $.
+ $}
+
+ $( A function is continuous at ` B ` iff its limit at ` B ` equals the
+ value of the function there. (Contributed by Mario Carneiro,
+ 28-Dec-2016.) $)
+ cnplimccntop $p |- ( ( A C_ CC /\ B e. A ) -> ( F e. ( ( J CnP K ) ` B )
+ <-> ( F : A --> CC /\ ( F ` B ) e. ( F limCC B ) ) ) ) $=
+ ( cc wss wcel wa ccnp co cfv wf climc cnplimcim simpll simprl simplr ex
+ simprr cnplimclemr impbid ) AHIZBAJZKZCBDELMNJZAHCOZBCNCBPMJZKZABCDEFGQUG
+ UKUHUGUKKABCDEFGUEUFUKRUGUIUJSUEUFUKTUGUIUJUBUCUAUD $.
+ $}
+
${
$d x A $. $d x F $.
$( If ` F ` is a continuous function, the limit of the function at each
@@ -140490,6 +140715,22 @@ S C_ ( P ( ball ` D ) T ) ) $=
ABVCVGBHTVJVFVMTVMBVGCVIVHVQVRUQVFVMVAURUSUTVB $.
$}
+ ${
+ $d x A $. $d x F $.
+ $( ` F ` is a continuous function iff the limit of the function at each
+ point equals the value of the function. (Contributed by Mario Carneiro,
+ 28-Dec-2016.) $)
+ cnlimc $p |- ( A C_ CC -> ( F e. ( A -cn-> CC ) <->
+ ( F : A --> CC /\ A. x e. A ( F ` x ) e. ( F limCC x ) ) ) ) $=
+ ( cc wss ccncf co wcel cabs cmin ccom cmopn cfv crest wral wa eqid ctopon
+ ccn wb wf ccnp climc wceq cntoptopon toponrestid cncfcncntop mpan2 eleq2d
+ cv ssid resttopon mpan cncnp sylancl cnplimccntop baibd ralbidva pm5.32da
+ an32s 3bitrd ) BDEZCBDFGZHCIJKLMZBNGZVDSGZHZBDCUAZCAUJZVEVDUBGMHZABOZPZVH
+ VICMCVIUCGHZABOZPVBVCVFCVBDDEVCVFUDDUKBDVDVEVDVDQZVEQZVDDVDVOUEZUFUGUHUIV
+ BVEBRMHZVDDRMHZVGVLTVSVBVRVQBVDDULUMVQACVEVDBDUNUOVBVHVKVNVBVHPVJVMABVBVI
+ BHZVHVJVMTVBVTPVJVHVMBVICVEVDVOVPUPUQUTURUSVA $.
+ $}
+
${
$d x A $. $d x B $. $d x F $.
cnlimci.f $e |- ( ph -> F e. ( A -cn-> D ) ) $.
@@ -140538,7 +140779,7 @@ S C_ ( P ( ball ` D ) T ) ) $=
clt wral wrex ctopon ccnp crest wss cntoptopon resttopon sylancr syl5eqel
wf a1i cnpf2 syl3anc ctop cntoptop cnprcl2k ffvelrnd cxp cres cmopn cxmet
wceq cnxmet eqid metrest syl5eq xmetres2 metcnpd mpbid simprd simplr fssd
- simplll cdm fdmd w3a limcrcl syl simp2d eqsstr3d simp3d ellimc3ap syl2anc
+ simplll cdm fdmd w3a limcrcl syl simp2d eqsstrrd simp3d ellimc3ap syl2anc
r19.21bi oveq2 breq1d fveq2 oveq2d imbi12d simpllr ad5antr rspcdva ovresd
simpr simpld sseldd cnmetdval abssubd 3eqtrd fvco3 fveq2d eqeltrd 3eqtr2d
3imtr3d imim2d ralimdva reximdva mpd rexlimdva2 fco mpbir2and ) ADGPZGFUF
@@ -140608,7 +140849,7 @@ S C_ ( P ( ball ` D ) T ) ) $=
cnex cnprcl2k toponuni syl eleqtrrd opelxp simpld simprd eqid cnmetdval
fovrnd simpl abssubd eqtrd cres jca ovresd sseldd 3eqtrd simprl ralrimi
rpred ltletrd r19.21bi sylc eqbrtrd oveq2 breq1d oveq2d imbi12d anbi2d
- cle cmpt cdm ex dmmptg w3a limcrcl simp2d eqsstr3d simp3d subcld abscld
+ cle cmpt cdm ex dmmptg w3a limcrcl simp2d eqsstrrd simp3d subcld abscld
climc mincl simprr min1inf anbi1d oveq1 rspc2v syl3c eqbrtrrd rspceaimv
min2inf exp31 breq2 ) AJKURUSUTVAZVBVCZBVDZDVEVFZUVGDVGVHZVIVNZUVEUTVFZ
VJZGHLVHZEFLVHZVGVHVIVNZIUTVFZVOZBCVKUVHUVJTVDZUTVFZVJZUVPVOBCVKTVBVLAJ
@@ -140653,7 +140894,7 @@ S C_ ( P ( ball ` D ) T ) ) $=
$( The image of a convergent sequence under a continuous map is convergent
to the image of the original point. Binary operation version.
(Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon,
- 9-Nov-2023.) $)
+ 14-Nov-2023.) $)
limccnp2cntop $p |- ( ph ->
( C H D ) e. ( ( x e. A |-> ( R H S ) ) limCC B ) ) $=
( vd ve vr vj vs vf vg co cmpt climc wcel cc cv cap wbr cmin cabs cfv clt
@@ -140661,42 +140902,41 @@ S C_ ( P ( ball ` D ) T ) ) $=
txtopon xpss12 syl2anc resttopon sylancr syl5eqel cnpf2 syl3anc cuni ctop
a1i cntoptop txtop cnex ssexd xpexg resttop toptopon2 sylib cnprcl2k wceq
cvv toponuni syl eleqtrrd opelxp simpld simprd fovrnd ccom cmopn syl22anc
- cres txrest syl5eq cxmet cnxmet eqid metrest oveq12d eqtrd oveq1d xmetres
- fveq1d ax-mp dfss1 fveq2d syl5eleq mpbid r19.21bi simprl adantr limcmpted
- cin sseldd simplrl sylancom ad4antr nfra1 nfan ad2antrr simprr rexlimddv
- nfv eleqtrd txmetcnp syl31anc simpll cdm dmmptd w3a limcrcl simp2d simp3d
- wb eqsstr3d simp-5l simp-4r simplrr limccnp2lem ralrimiva mpbir2and ) AEF
- IUJZBCGHIUJZUKDULUJUMUUSUNUMBUOZDUPUQZUVADURUJUSUTZUCUOVAUQVBUUTUUSURUJUS
- UTUDUOZVAUQVCBCVDUCVEVFZUDVEVDAEFUNLMIAJLMVGZVHUTZUMZKUNVHUTUMZIEFVIZJKVJ
- UJZUTZUMZUVFUNIVKZAJKKVLUJZUVFVMUJZUVGSAUVOUNUNVGZVHUTUMZUVFUVQVNZUVPUVGU
- MUVIUVIUVRKRVOZUVTKKUNUNVQVPALUNVNZMUNVNZUVSPQLUNMUNVRVSUVFUVOUVQVTWAWBZU
- VIAUVTWGUBUVJIJKUVFUNWCWDZAELUMZFMUMZAUVJUVFUMUWEUWFVBZAUVJJWEZUVFAJUWHVH
- UTUMZKWFUMZUVMUVJUWHUMAJWFUMUWIAJUVPWFSAUVOWFUMZUVFWRUMZUVPWFUMAUWJUWJUWK
- KRWHZUWJAUWMWGZKKWIWAALWRUMZMWRUMZUWLALUNWRUNWRUMAWJWGZPWKZAMUNWRUWQQWKZL
- MWRWRWLVSUVFUVOWRWMVSWBJWNWOUWNUBUVJIJKUWHWPWDAUVHUVFUWHWQUWCUVFJWSWTXAEF
- LMXBWOZXCAUWEUWFUWTXDXEAUVEUDVEAUVDVEUMZVBZEUEUOZUSURXFZLLVGXIZUJUFUOZVAU
- QFUGUOZUXDMMVGXIZUJUXFVAUQVBUUSUXCUXGIUJUXDUJUVDVAUQVCUGMVDUELVDZUVEUFVEA
- UXIUFVEVFZUDVEAUVNUXJUDVEVDZAIUVJUXEXGUTZUXHXGUTZVLUJZKVJUJZUTZUMZUVNUXKV
- BZAIUVLUXPUBAUVJUVKUXOAJUXNKVJAJKLVMUJZKMVMUJZVLUJZUXNAJUVPUYASAUWJUWJUWO
- UWPUVPUYAWQUWNUWNUWRUWSLMKKWFWFWRWRXJXHXKAUXSUXLUXTUXMVLAUXDUNXLUTUMZUWAU
- XSUXLWQXMPUXDUXEKUXLUNLUXEXNRUXLXNZXOWAAUYBUWBUXTUXMWQXMQUXDUXHKUXMUNMUXH
- XNRUXMXNZXOWAXPXQXRXTUUAAUXELXLUTZUMUXHMXLUTZUMUYBUWGUXQUXRUUKAUXEUNLYJZX
- LUTZUYEUYBUXEUYHUMXMUXDLUNXSYAAUYGLXLAUWAUYGLWQPLUNYBWOYCYDAUXHUNMYJZXLUT
- ZUYFUYBUXHUYJUMXMUXDMUNXSYAAUYIMXLAUWBUYIMWQQMUNYBWOYCYDUYBAXMWGUWTUDUFUG
- UEEFUXEUXHUXDIUXLUXMKLMUNUYCUYDRUUBUUCYEXDYFUXBUXFVEUMZUXIVBZVBZUVBUVCUHU
- OZVAUQVBGEURUJUSUTUXFVAUQVCZBCVDZUVEUHVEUYMAUYKUYPUHVEVFZAUXAUYLUUDZUXBUY
- KUXIYGZAUYQUFVEAEUNUMZUYQUFVEVDZAEBCGUKZDULUJUMZUYTVUAVBTAUFUHBCDEGACVUBU
- UEZUNABVUBCGLVUBXNNUUFAVUDUNVUBVKZVUDUNVNZDUNUMZAVUCVUEVUFVUGUUGTDEVUBUUH
- WTZUUIUULZAVUEVUFVUGVUHUUJZAUVACUMZVBZLUNGAUWAVUKPYHNYKYIYEXDYFVSUYMUYNVE
- UMZUYPVBZVBZUVBUVCUIUOZVAUQVBHFURUJUSUTUXFVAUQVCZBCVDZUVEUIVEVUOAUYKVURUI
- VEVFZUYMAVUNUYRYHUXBUYKUXIVUNYLAVUSUFVEAFUNUMZVUSUFVEVDZAFBCHUKDULUJUMZVU
- TVVAVBUAAUFUIBCDFHVUIVUJVULMUNHAUWBVUKQYHOYKYIYEXDYFVSVUOVUPVEUMZVURVBZVB
- ZBCDEFGHUVDUYNVUPIJKUXFLMUGUEUCVVEVUKAGLUMAUXAUYLVUNVVDVUKUUMZNYMVVEVUKAH
- MUMVVFOYMAUWAUXAUYLVUNVVDPYNAUWBUXAUYLVUNVVDQYNRSAVUCUXAUYLVUNVVDTYNAVVBU
- XAUYLVUNVVDUAYNAUVMUXAUYLVUNVVDUBYNVUOVVDBUYMVUNBUYMBYTVUMUYPBVUMBYTUYOBC
- YOYPYPVVCVURBVVCBYTVUQBCYOYPYPAUXAUYLVUNVVDUUNUYMUYKVUNVVDUYSYQUYMUXIVUNV
- VDUXBUYKUXIYRYQUYMVUMUYPVVDYLUYMVUMUYPVVDUUOVUOVVCVURYGVUOVVCVURYRUUPYSYS
- YSUUQAUDUCBCDUUSUUTVUIVUJVULGHUNLMIAUVNVUKUWDYHNOXEYIUUR $.
+ cres txrest syl5eq cxmet cnxmet eqid metrest oveq12d eqtrd oveq1d eleqtrd
+ fveq1d wb xmetres2 txmetcnp syl31anc r19.21bi simpll simprl adantr sseldd
+ mpbid limcmpted simplrl sylancom nfv nfra1 nfan ad2antrr simprr rexlimddv
+ ad4antr cdm dmmptd limcrcl simp2d eqsstrrd simp3d simp-5l simp-4r simplrr
+ w3a limccnp2lem ralrimiva mpbir2and ) AEFIUJZBCGHIUJZUKDULUJUMUUNUNUMBUOZ
+ DUPUQZUUPDURUJUSUTZUCUOVAUQVBUUOUUNURUJUSUTUDUOZVAUQVCBCVDUCVEVFZUDVEVDAE
+ FUNLMIAJLMVGZVHUTZUMZKUNVHUTUMZIEFVIZJKVJUJZUTZUMZUVAUNIVKZAJKKVLUJZUVAVM
+ UJZUVBSAUVJUNUNVGZVHUTUMZUVAUVLVNZUVKUVBUMUVDUVDUVMKRVOZUVOKKUNUNVQVPALUN
+ VNZMUNVNZUVNPQLUNMUNVRVSUVAUVJUVLVTWAWBZUVDAUVOWGUBUVEIJKUVAUNWCWDZAELUMZ
+ FMUMZAUVEUVAUMUVTUWAVBZAUVEJWEZUVAAJUWCVHUTUMZKWFUMZUVHUVEUWCUMAJWFUMUWDA
+ JUVKWFSAUVJWFUMZUVAWRUMZUVKWFUMAUWEUWEUWFKRWHZUWEAUWHWGZKKWIWAALWRUMZMWRU
+ MZUWGALUNWRUNWRUMAWJWGZPWKZAMUNWRUWLQWKZLMWRWRWLVSUVAUVJWRWMVSWBJWNWOUWIU
+ BUVEIJKUWCWPWDAUVCUVAUWCWQUVRUVAJWSWTXAEFLMXBWOZXCAUVTUWAUWOXDXEAUUTUDVEA
+ UUSVEUMZVBZEUEUOZUSURXFZLLVGXIZUJUFUOZVAUQFUGUOZUWSMMVGXIZUJUXAVAUQVBUUNU
+ WRUXBIUJUWSUJUUSVAUQVCUGMVDUELVDZUUTUFVEAUXDUFVEVFZUDVEAUVIUXEUDVEVDZAIUV
+ EUWTXGUTZUXCXGUTZVLUJZKVJUJZUTZUMZUVIUXFVBZAIUVGUXKUBAUVEUVFUXJAJUXIKVJAJ
+ KLVMUJZKMVMUJZVLUJZUXIAJUVKUXPSAUWEUWEUWJUWKUVKUXPWQUWIUWIUWMUWNLMKKWFWFW
+ RWRXJXHXKAUXNUXGUXOUXHVLAUWSUNXLUTUMZUVPUXNUXGWQXMPUWSUWTKUXGUNLUWTXNRUXG
+ XNZXOWAAUXQUVQUXOUXHWQXMQUWSUXCKUXHUNMUXCXNRUXHXNZXOWAXPXQXRXTXSAUWTLXLUT
+ UMZUXCMXLUTUMZUXQUWBUXLUXMYAAUXQUVPUXTXMPUWSLUNYBWAAUXQUVQUYAXMQUWSMUNYBW
+ AUXQAXMWGUWOUDUFUGUEEFUWTUXCUWSIUXGUXHKLMUNUXRUXSRYCYDYJXDYEUWQUXAVEUMZUX
+ DVBZVBZUUQUURUHUOZVAUQVBGEURUJUSUTUXAVAUQVCZBCVDZUUTUHVEUYDAUYBUYGUHVEVFZ
+ AUWPUYCYFZUWQUYBUXDYGZAUYHUFVEAEUNUMZUYHUFVEVDZAEBCGUKZDULUJUMZUYKUYLVBTA
+ UFUHBCDEGACUYMUUAZUNABUYMCGLUYMXNNUUBAUYOUNUYMVKZUYOUNVNZDUNUMZAUYNUYPUYQ
+ UYRUUJTDEUYMUUCWTZUUDUUEZAUYPUYQUYRUYSUUFZAUUPCUMZVBZLUNGAUVPVUBPYHNYIYKY
+ JXDYEVSUYDUYEVEUMZUYGVBZVBZUUQUURUIUOZVAUQVBHFURUJUSUTUXAVAUQVCZBCVDZUUTU
+ IVEVUFAUYBVUIUIVEVFZUYDAVUEUYIYHUWQUYBUXDVUEYLAVUJUFVEAFUNUMZVUJUFVEVDZAF
+ BCHUKDULUJUMZVUKVULVBUAAUFUIBCDFHUYTVUAVUCMUNHAUVQVUBQYHOYIYKYJXDYEVSVUFV
+ UGVEUMZVUIVBZVBZBCDEFGHUUSUYEVUGIJKUXALMUGUEUCVUPVUBAGLUMAUWPUYCVUEVUOVUB
+ UUGZNYMVUPVUBAHMUMVUQOYMAUVPUWPUYCVUEVUOPYTAUVQUWPUYCVUEVUOQYTRSAUYNUWPUY
+ CVUEVUOTYTAVUMUWPUYCVUEVUOUAYTAUVHUWPUYCVUEVUOUBYTVUFVUOBUYDVUEBUYDBYNVUD
+ UYGBVUDBYNUYFBCYOYPYPVUNVUIBVUNBYNVUHBCYOYPYPAUWPUYCVUEVUOUUHUYDUYBVUEVUO
+ UYJYQUYDUXDVUEVUOUWQUYBUXDYRYQUYDVUDUYGVUOYLUYDVUDUYGVUOUUIVUFVUNVUIYGVUF
+ VUNVUIYRUUKYSYSYSUULAUDUCBCDUUNUUOUYTVUAVUCGHUNLMIAUVIVUBUVSYHNOXEYKUUM
+ $.
$}
${
@@ -140947,6 +141187,71 @@ S C_ ( P ( ball ` D ) T ) ) $=
VBVGUTRCVBTCUTTUKUMUNUOUPUQUR $.
$}
+ ${
+ $d y z w A $. $d y z w B $. $d y z w F $. $d y z K $. $d y z w S $.
+ $d y z J $.
+ dvcnp.j $e |- J = ( K |`t A ) $.
+ dvcnpcntop.k $e |- K = ( MetOpen ` ( abs o. - ) ) $.
+ $( A function is continuous at each point for which it is differentiable.
+ (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario
+ Carneiro, 28-Dec-2016.) $)
+ dvcnp2cntop $p |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\
+ B e. dom ( S _D F ) ) -> F e. ( ( J CnP K ) ` B ) ) $=
+ ( vz vw cc wss co cfv wcel cmin cmpt climc cc0 caddc vy wf w3a cv cdv wbr
+ wex ccnp cdm simpl3 simpl1 sstrd simpl2 crest cnt ctop cuni cntoptop cnex
+ wa cvv ssexg sylancl resttop sylancr ctopon cntoptopon resttopon toponuni
+ wceq syl sseqtrd eqid ntrss2 syl2anc crab cdiv simp1 simp2 simp3 simprbda
+ cap eldvap sseldd ctx ffvelrnda ffvelrnd adantr subcld ssid a1i cxp mp2an
+ txtopon toponrestid dvlemap ssrab2 syl5ss sselda simplbda limcresi resmpt
+ cmul cres ax-mp oveq1i sseqtri subidd subcncntop ccncf cncfmptid cncfmptc
+ ccn syl3anc cncfmpt2fcntop oveq1 cnmptlimc eqeltrrd sseldi cop mulcncntop
+ dvcl 0cn opelxpi toponunii cncnpi limccnp2cntop simpr breq1 elrab simprbi
+ mul01d adantl subap0d divcanap1d mpteq2dva oveq1d 3eltr3d limcdifap
+ fmpttd syl6eq eleqtrrd eqidd addcncntop addid2d npcand eqtr4d cnplimclemr
+ feqmptd ex exlimdv eldmg ibi impel ) CKLZAKDUBZACLZUCZBUAUDZCDUEMZUFZUAUG
+ ZDBEFUHMNOZBUUTUIZOZUURUVAUVCUAUURUVAUVCUURUVAUTZABDEFHGUVFACKUUOUUPUUQUV
+ AUJZUUOUUPUUQUVAUKZULZUUOUUPUUQUVAUMZUVFAFCUNMZUONNZABUVFUVKUPOZAUVKUQZLU
+ VLALUVFFUPOCVAOZUVMFHURUVFUUOKVAOUVOUVHUSCKVAVBVCCFVAVDVEUVFACUVNUVGUVFUV
+ KCVFNOZCUVNVJUVFFKVFNOZUUOUVPFHVGZUVHCFKVHVECUVKVIVKVLAUVKUVNUVNVMVNVOUUR
+ UVABUVLOZUUSIJUDZBWBUFZJAVPZIUDZDNZBDNZPMZUWCBPMZVQMZQZBRMOZUURIJABUUSCUV
+ KDUWIFUVKVMHUWIVMUUOUUPUUQVRZUUOUUPUUQVSZUUOUUPUUQVTZWCZWAWDZUVFSUWETMIAU
+ WFUWETMZQZBRMUWEDBRMUVFIABSUWEUWFUWETFFWEMZFKKUVFUWCAOZUTZUWDUWEUVFAKUWCD
+ UVJWFZUVFUWEKOZUWSUVFAKBDUVJUWOWGZWHZWIZUXDKKLZUVFKWJZWKZUXHHUWRKKWLZUVQU
+ VQUWRUXIVFNOUVRUVRFFKKWNWMZWOZUVFSIUWBUWFQZBRMZIAUWFQZBRMZUVFUUSSXCMIUWBU
+ WHUWGXCMZQZBRMSUXMUVFIUWBBUUSSUWHUWGXCUWRFKKUVFJUWCBADUVJUVIUWOWPUVFUWCUW
+ BOZUTZUWCBUVFUWBKUWCUVFUWBAKUWAJAWQZUVIWRWSZUVFBKOZUXRUVFAKBUVIUWOWDZWHZW
+ IZUXHUXHHUXKUURUVAUVSUWJUWNWTUVFIAUWGQZBRMZIUWBUWGQZBRMZSUYGUYFUWBXDZBRMU
+ YIBUWBUYFXAUYJUYHBRUWBALZUYJUYHVJUXTIAUWBUWGXBXEXFXGUVFBBPMZSUYGUVFBUYCXH
+ UVFIABKUWGUYLUVFIUWCBPFAHPUWRFXMMZOUVFFHXIWKUVFAKLZUXFIAUWCQAKXJMZOUVIUXG
+ IAKXKVCUVFUYBUYNUXFIABQUYOOUYCUVIUXHIBAKXLXNXOUWOUWCBBPXPXQXRXSUVFXCUYMOU
+ USSXTZUXIOZXCUYPUWRFUHMZNOFHYAUVFUUSKOSKOZUYQUURABUUSCDUWKUWLUWMYBZYCUUSS
+ KKYDVCUYPXCUWRFUXIUXIUWRUXJYEZYFVEYGUVFUUSUYTYLUVFUXQUXLBRUVFIUWBUXPUWFUX
+ SUWFUWGUXSUWDUWEUXSAKUWCDUVFUUPUXRUVJWHUXSUWBAUWCUXTUVFUXRYHXSWGUVFUXBUXR
+ UXCWHWIUYEUXSUWCBUYAUYDUXRUWCBWBUFZUVFUXRUWSVUBUWAVUBJUWCAUVTUWCBWBYIYJYK
+ YMYNYOYPYQYRUVFUXOUXNUWBXDZBRMUXMUVFJABUXNUVFIAUWFKUXEYTUVIYSVUCUXLBRUYKV
+ UCUXLVJUXTIAUWBUWFXBXEXFUUAUUBUVFIABKUWEUWEUVFUXBUYNUXFIAUWEQUYOOUXCUVIUX
+ HIUWEAKXLXNUWOUWCBVJUWEUUCXQUVFTUYMOSUWEXTZUXIOZTVUDUYRNOFHUUDUVFUYSUXBVU
+ EYCUXCSUWEKKYDVEVUDTUWRFUXIVUAYFVEYGUVFUWEUXCUUEUVFUWQDBRUVFUWQIAUWDQDUVF
+ IAUWPUWDUWTUWDUWEUXAUXDUUFYPUVFIAKDUVJUUIUUGYQYRUUHUUJUUKUVEUVBUABUUTUVDU
+ ULUUMUUN $.
+ $}
+
+ ${
+ $d x A $. $d x F $. $d x S $.
+ $( A differentiable function is continuous. (Contributed by Mario
+ Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.) $)
+ dvcn $p |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\
+ dom ( S _D F ) = A ) -> F e. ( A -cn-> CC ) ) $=
+ ( vx cc wss wf w3a cdv co cdm wceq cabs cfv wcel wral eqid ctopon sylancl
+ wa cmin ccom cmopn crest ccncf cv ccnp simpl2 dvcnp2cntop ralrimiva raleq
+ ccn biimpd mpan9 cntoptopon simpl3 simpl1 sstrd resttopon cncnp mpbir2and
+ wb sylancr ssid toponrestid cncfcncntop eleqtrrd ) BEFZAECGZABFZHZBCIJKZA
+ LZTZCMUAUBUCNZAUDJZVOULJZAEUEJZVNCVQOZVICDUFZVPVOUGJNOZDAPZVHVIVJVMUHVKWA
+ DVLPZVMWBVKWADVLAVTBCVPVOVPQZVOQZUIUJVMWCWBWADVLAUKUMUNVNVPARNOZVOERNOZVS
+ VIWBTVBVNWGAEFZWFVOWEUOZVNABEVHVIVJVMUPVHVIVJVMUQURZAVOEUSVCWIDCVPVOAEUTS
+ VAVNWHEEFVRVQLWJEVDAEVOVPVOWEWDVOEWIVEVFSVG $.
+ $}
+
$(
###############################################################################
@@ -142241,6 +142546,9 @@ Norman Megill (2007) section 1.1.3. Megill then states, "A number of
"";
althtmldef "card" as 'card';
latexdef "card" as "\mathrm{card}";
+htmldef "CHOICE" as "CHOICE";
+ althtmldef "CHOICE" as "CHOICE";
+ latexdef "CHOICE" as "\mathrm{CHOICE}";
htmldef "N." as "";
althtmldef "N." as 'N';
latexdef "N." as "\mathcal{N}";
@@ -144589,35 +144897,6 @@ with definitions ( ` B ` is the definiendum that one wants to prove
ax-bj-d0cl $a |- DECID ph $.
$}
- $( Equivalence property for negation. TODO: minimize all theorems using
- ~ notbid and ~ notbii . (Contributed by BJ, 27-Jan-2020.)
- (Proof modification is discouraged.) $)
- bj-notbi $p |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) ) $=
- ( wb wn bi2 con3d bi1 impbid ) ABCZADBDIBAABEFIABABGFH $.
-
- ${
- bj-notbii.1 $e |- ( ph <-> ps ) $.
- $( Inference associated with ~ bj-notbi . (Contributed by BJ,
- 27-Jan-2020.) (Proof modification is discouraged.) $)
- bj-notbii $p |- ( -. ph <-> -. ps ) $=
- ( wb wn bj-notbi ax-mp ) ABDAEBEDCABFG $.
- $}
-
- ${
- bj-notbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
- $( Deduction form of ~ bj-notbi . (Contributed by BJ, 27-Jan-2020.)
- (Proof modification is discouraged.) $)
- bj-notbid $p |- ( ph -> ( -. ps <-> -. ch ) ) $=
- ( wb wn bj-notbi syl ) ABCEBFCFEDBCGH $.
- $}
-
- $( Equivalence property for ` DECID ` . TODO: solve conflict with ~ dcbi ;
- minimize ~ dcbii and ~ dcbid with it, as well as theorems using those.
- (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) $)
- bj-dcbi $p |- ( ( ph <-> ps ) -> ( DECID ph <-> DECID ps ) ) $=
- ( wb wn wo wdc id bj-notbi orbi12d df-dc 3bitr4g ) ABCZAADZEBBDZEAFBFLABMNL
- GABHIAJBJK $.
-
${
$d a x ph $.
$( ` Delta0 ` -classical logic and separation implies classical logic.
@@ -144626,9 +144905,9 @@ with definitions ( ` B ` is the definiendum that one wants to prove
bj-d0clsepcl $p |- DECID ph $=
( va vx wdc wex c0 cv wcel wb csn wel wal 0ex bj-snex zfauscl wceq anbi1d
wa eleq1 eximii bibi12d spcv snid biantrur bicomi bibi2i exbii ax-bj-d0cl
- mpbi bj-bd0el bj-dcbi mpbii bj-ex ax-mp ) ADZBEUOFBGZHZAIZUOBUQFFJZHZARZI
- ZBEURBECBKZCGZUSHZARZIZCLVBBACBUSFMNOVGVBCFMVDFPZVCUQVFVAVDFUPSVHVEUTAVDF
- USSQUAUBTVBURBVAAUQAVAUTAFMUCUDUEUFUGUIURUQDUOUQBUJUHUQAUKULTUOBUMUN $.
+ mpbi bj-bd0el dcbiit mpbii bj-ex ax-mp ) ADZBEUOFBGZHZAIZUOBUQFFJZHZARZIZ
+ BEURBECBKZCGZUSHZARZIZCLVBBACBUSFMNOVGVBCFMVDFPZVCUQVFVAVDFUPSVHVEUTAVDFU
+ SSQUAUBTVBURBVAAUQAVAUTAFMUCUDUEUFUGUIURUQDUOUQBUJUHUQAUKULTUOBUMUN $.
$}
@@ -145643,17 +145922,6 @@ truth value (as seen in theorems like ~ exmidexmid ), then this theorem
TCPQUSVASTUAUBUIUCUJBAUDUEUMUOUQUFUGUNUPUHUK $.
$}
- ${
- $d A x $.
- $( A slight strengthening of ~ pwtrufal . (Contributed by Mario Carneiro
- and Jim Kingdon, 12-Sep-2023.) $)
- pwntru $p |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) ) $=
- ( vx c0 csn wss wne wa cv wcel wex simpr neneqd simpll simpl sselda elsni
- wn wceq syl eqeltrrd snssd eqssd ex exlimdv mtod notm0 sylib ) ACDZEZAUHF
- ZGZBHZAIZBJZQACRUKUNAUHRZUKAUHUIUJKLUKUMUOBUKUMUOUKUMGZAUHUIUJUMMUPCAUPUL
- CAUPULUHIULCRUKAUHULUIUJNOULCPSUKUMKTUAUBUCUDUEBAUFUG $.
- $}
-
${
$d N x $.
pwle2.t $e |- T = U_ x e. N ( { x } X. 1o ) $.
@@ -145753,7 +146021,7 @@ a singleton (where the latter can also be thought of as representing
wn 1on ontrci peano2 syl df-2o syl6eleq trsucss mpsyl sseqtr4d word simpr
iftrue nnord ordtr unisucg mpbid wne neqned nnsucpred syl2anc suceq fveq2
3syl sseq12d fveq1 ralbidv df-nninf elrab2 simprbi cbvralv sylib ad2antrr
- rspcdva eqsstr3d eqsstrd peano3 neneqd ad2antlr iffalsed 3sstr4d mpjaodan
+ rspcdva eqsstrrd eqsstrd peano3 neneqd ad2antlr iffalsed 3sstr4d mpjaodan
weq wo exmiddc eqeq1 unieq ifbieq2d fvmptg ralrimiva sylanbrc fmpti ) CHH
BIBJZTKZLXTUAZCJZMZUBZUCZADYCHNZYFOIUDUEZNZEJZUFZYFMZYJYFMZPZEIUGZYFHNYGI
OYFUHZYIYGBIYEOYFYGXTINZUIZYALYDOLONZYRUOUJYRIOYBYCYGIOYCUHZYQYCUKZQYQYBI
@@ -146825,6 +147093,26 @@ Limited Principle of Omniscience (LPO) implies excluded middle, so we
$)
+$(
+=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
+ Testable propositions
+=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
+$)
+
+ $( A proposition is testable iff its negative or double-negative is true.
+ See Chapter 2 [Moschovakis] p. 2.
+
+ We do not formally define testability with a new token, but instead use
+ ` DECID -. ` before the formula in question. For example,
+ ` DECID -. x = y ` corresponds to " ` x = y ` is testable". (Contributed
+ by David A. Wheeler, 13-Aug-2018.) For statements about testable
+ propositions, search for the keyword "testable" in the comments of
+ statements, for instance using the Metamath command "MM> SEARCH *
+ "testable" / COMMENTS". (New usage is discouraged.) $)
+ dftest $p |- ( DECID -. ph <-> ( -. ph \/ -. -. ph ) ) $=
+ ( wn df-dc ) ABC $.
+
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Allsome quantifier
diff --git a/mmil.raw.html b/mmil.raw.html
index eef4545027..99a5b64da2 100644
--- a/mmil.raw.html
+++ b/mmil.raw.html
@@ -10876,15 +10876,7 @@
cnplimc
-
~ cnplimcim
-
The converse is conjectured to also be provable, but
- would require a more involved proof.
-
-
-
-
cnlimc
-
~ cnlimcim
-
See cnplimc concerning biconditionalizing this
+
~ cnplimccntop
@@ -11004,29 +10996,19 @@
dvcnp2
-
none
-
The set.mm proof relies on limccnp2
- and the converse of ~ cnplimcim
-
-
-
-
dvcn
-
none
-
the set.mm proof relies on dvcnp2
+
~ dvcnp2cntop
dvaddbr
none
-
the set.mm proof relies on restntr , undif1 , limcres , and
- limccnp2
+
the set.mm proof relies on restntr , undif1 , and limcres
dvmulbr
none
-
the set.mm proof relies on restntr , undif1 , limcres , and
- limccnp2
+
the set.mm proof relies on restntr , undif1 , and limcres
diff --git a/ql.mm b/ql.mm
index d54f068efe..ac726a1cd7 100644
--- a/ql.mm
+++ b/ql.mm
@@ -1,4 +1,16 @@
-$( ql.mm - Version of 11-Apr-2012
+$( This is the Metamath database ql.mm. $)
+
+$( Metamath is a formal language and associated computer program for
+ archiving, verifying, and studying mathematical proofs, created by Norman
+ Dwight Megill (1950--2021). For more information, visit
+ https://us.metamath.org and
+ https://github.com/metamath/set.mm, and feel free to ask questions at
+ https://groups.google.com/group/metamath. $)
+
+$( The database ql.mm was created by Norman Megill on 9-Aug-1997. $)
+
+
+$( !
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
Metamath source file for logic, set theory, numbers, and Hilbert space
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
@@ -7,22 +19,25 @@
This work is waived of all rights, including copyright, according to the CC0
Public Domain Dedication. http://creativecommons.org/publicdomain/zero/1.0/
-Norman Megill - email: nm(at)alum(dot)mit(dot)edu - http://metamath.org
+Norman Megill
$)
+
$( placeholder
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
- AUQL - Algebraic Unified Quantum Logic of M. Pavicic
+ AUQL - Algebraic Unified Quantum Logic of Mladen Pavicic
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
+
$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
Ortholattices
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Basic syntax and axioms
@@ -102,7 +117,6 @@
$v a0 a1 a2 b0 b1 b2 c0 c1 c2 p0 p1 p2 $.
-
$(
Specify some variables that we will use to represent terms.
The fact that a variable represents a wff is relevant only to a theorem
@@ -194,7 +208,6 @@
$( If ` a ` and ` b ` are terms, ` a C b ` is a wff. $)
wc $a wff a C b $.
-
$( If ` a ` is a term, so is ` a ' ` . $)
wn $a term a ' $.
$( If ` a ` and ` b ` are terms, so is ` ( a == b ) ` . $)
@@ -278,7 +291,6 @@
( ( a ' ' v b ' ' ) ' v ( a ' v b ' ) ' ) $.
$)
-
${
r1.1 $e |- a = b $.
$( Inference rule for ortholattices. (Contributed by NM, 9-Aug-1997.) $)
@@ -411,7 +423,6 @@
( ax-r2 ) ABCDEF $.
$}
-
${
3tr1.1 $e |- a = b $.
3tr1.2 $e |- c = a $.
@@ -786,7 +797,7 @@
ZDZCZAEZSRCZSCZFZDZABFZAESUGDUBUACZSEZUFUAAGUITSEUFUHTSTUHTHIJRSKLLUGUAAUGB
AFZUAABMZBANLJUGUESUGUJUEUKBUCAUDBHAHOLPQ $.
- $( Lemma in proof of Th. 1 of Pavicic 1987. (Contributed by NM,
+ $( Lemma in proof of Thm. 1 of Pavicic 1987. (Contributed by NM,
12-Aug-1997.) $)
omlem1 $p |- ( ( a v ( a ' ^ ( a v b ) ) ) v ( a v b ) ) =
( a v b ) $=
@@ -794,7 +805,7 @@
AACZABDZEZDZADBDZRADZSDZTRDZRUDRTDUAUCTRFTABGZRASGHUEUCRRQEZDRUBRSUFUBAADZB
DZRUHUBUHARDUBAABGARFIJUGABAKLIQRMNRQOIP $.
- $( Lemma in proof of Th. 1 of Pavicic 1987. (Contributed by NM,
+ $( Lemma in proof of Thm. 1 of Pavicic 1987. (Contributed by NM,
12-Aug-1997.) $)
omlem2 $p |- ( ( a v b ) ' v ( a v ( a ' ^ ( a v b ) ) ) ) = 1 $=
( wo wn wa wt ax-a2 anor2 2or ax-a3 ax-r1 df-t 3tr1 ) ABCZDZACZADNEZCZAOCZS
@@ -807,7 +818,7 @@
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
- $( Define 'less than or equal to' analogue. (Contributed by NM,
+ $( Define "less than or equal to" analogue. (Contributed by NM,
27-Aug-1997.) $)
df-le $a |- ( a =<2 b ) = ( ( a v b ) == b ) $.
@@ -815,39 +826,39 @@
to eliminate the middle = . $)
${
df-le1.1 $e |- ( a v b ) = b $.
- $( Define 'less than or equal to'. See ~ df-le2 for the other direction.
+ $( Define "less than or equal to". See ~ df-le2 for the other direction.
(Contributed by NM, 27-Aug-1997.) $)
df-le1 $a |- a =< b $.
$}
${
df-le2.1 $e |- a =< b $.
- $( Define 'less than or equal to'. See ~ df-le1 for the other direction.
+ $( Define "less than or equal to". See ~ df-le1 for the other direction.
(Contributed by NM, 27-Aug-1997.) $)
df-le2 $a |- ( a v b ) = b $.
$}
${
df-c1.1 $e |- a = ( ( a ^ b ) v ( a ^ b ' ) ) $.
- $( Define 'commutes'. See ~ df-c2 for the other direction. (Contributed
+ $( Define "commutes". See ~ df-c2 for the other direction. (Contributed
by NM, 27-Aug-1997.) $)
df-c1 $a |- a C b $.
$}
${
df-c2.1 $e |- a C b $.
- $( Define 'commutes'. See ~ df-c1 for the other direction. (Contributed
+ $( Define "commutes". See ~ df-c1 for the other direction. (Contributed
by NM, 27-Aug-1997.) $)
df-c2 $a |- a = ( ( a ^ b ) v ( a ^ b ' ) ) $.
$}
- $( Define 'commutator'. (Contributed by NM, 24-Jan-1999.) $)
+ $( Define "commutator". (Contributed by NM, 24-Jan-1999.) $)
df-cmtr $a |- C ( a , b ) = ( ( ( a ^ b ) v ( a ^ b ' ) ) v
( ( a ' ^ b ) v ( a ' ^ b ' ) ) ) $.
${
df2le1.1 $e |- ( a ^ b ) = a $.
- $( Alternate definition of 'less than or equal to'. (Contributed by NM,
+ $( Alternate definition of "less than or equal to". (Contributed by NM,
27-Aug-1997.) $)
df2le1 $p |- a =< b $=
( leao df-le1 ) ABABACDE $.
@@ -855,7 +866,7 @@
${
df2le2.1 $e |- a =< b $.
- $( Alternate definition of 'less than or equal to'. (Contributed by NM,
+ $( Alternate definition of "less than or equal to". (Contributed by NM,
27-Aug-1997.) $)
df2le2 $p |- ( a ^ b ) = a $=
( df-le2 leoa ) ABBABCDE $.
@@ -864,7 +875,8 @@
${
letr.1 $e |- a =< b $.
letr.2 $e |- b =< c $.
- $( Transitive law for l.e. (Contributed by NM, 27-Aug-1997.) $)
+ $( Transitive law for "less than or equal to". (Contributed by NM,
+ 27-Aug-1997.) $)
letr $p |- a =< c $=
( wa wo df-le2 ax-r5 ax-r1 ax-a3 3tr2 lan anabs ax-r2 df2le1 ) ACACFAABCG
ZGZFACRAQABGZCGZCRTQSBCABDHIJBCEHABCKLMAQNOP $.
@@ -908,14 +920,15 @@
${
bile.1 $e |- a = b $.
- $( Biconditional to l.e. (Contributed by NM, 27-Aug-1997.) $)
+ $( Biconditional to "less than or equal to". (Contributed by NM,
+ 27-Aug-1997.) $)
bile $p |- a =< b $=
( wo ax-r5 oridm ax-r2 df-le1 ) ABABDBBDBABBCEBFGH $.
$}
$( An ortholattice inequality, corresponding to a theorem provable in Hilbert
space. Part of Definition 2.1 p. 2092, in M. Pavicic and N. Megill,
- "Quantum and Classical Implicational Algebras with Primitive Implication,"
+ "Quantum and Classical Implicational Algebras with Primitive Implication",
_Int. J. of Theor. Phys_. 37, 2091-2098 (1998). (Contributed by NM,
3-Feb-2002.) $)
qlhoml1a $p |- a =< a ' ' $=
@@ -929,20 +942,22 @@
${
lebi.1 $e |- a =< b $.
lebi.2 $e |- b =< a $.
- $( L.e. to biconditional. (Contributed by NM, 27-Aug-1997.) $)
+ $( "Less than or equal to" to biconditional. (Contributed by NM,
+ 27-Aug-1997.) $)
lebi $p |- a = b $=
( wo df-le2 ax-r1 ax-a2 ax-r2 ) AABEZBABAEZJKABADFGBAHIABCFI $.
$}
- $( Anything is l.e. 1. (Contributed by NM, 30-Aug-1997.) $)
+ $( Anything is less than or equal to 1. (Contributed by NM, 30-Aug-1997.) $)
le1 $p |- a =< 1 $=
( wt or1 df-le1 ) ABACD $.
- $( 0 is l.e. anything. (Contributed by NM, 30-Aug-1997.) $)
+ $( 0 is less than or equal to anything. (Contributed by NM, 30-Aug-1997.) $)
le0 $p |- 0 =< a $=
( wf wo ax-a2 or0 ax-r2 df-le1 ) BABACABCABADAEFG $.
- $( Identity law for less-than-or-equal. (Contributed by NM, 24-Dec-1998.) $)
+ $( Identity law for "less than or equal to". (Contributed by NM,
+ 24-Dec-1998.) $)
leid $p |- a =< a $=
( id bile ) AAABC $.
@@ -1200,8 +1215,8 @@
Weak "orthomodular law" in ortholattices.
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
- All theorems here do not require R3 and
- are true in all ortholattices.
+ All theorems here do not require R3 and are true in all ortholattices.
+
$)
$( Weak A1. (Contributed by NM, 27-Sep-1997.) $)
@@ -1313,7 +1328,7 @@
${
wlem3.1.1 $e |- ( a v b ) = b $.
wlem3.1.2 $e |- ( b ' v a ) = 1 $.
- $( Weak analogue to lemma used in proof of Th. 3.1 of Pavicic 1993.
+ $( Weak analogue to lemma used in proof of Thm. 3.1 of Pavicic 1993.
(Contributed by NM, 2-Sep-1997.) $)
wlem3.1 $p |- ( a == b ) = 1 $=
( tb wn wo wt wa dfb leoa oran ax-r1 ax-r2 con3 2or ax-a2 ) ABEZBFZAGZHRA
@@ -1350,7 +1365,6 @@
( wo wn wa df-c2 oran ax-a2 anor2 ax-r1 con3 2an ax-r4 3tr1 ax-r2 con1 )
AABDZABEZDZFZAEZUBBFZUBSFZDZUAEZUBBCGUDUCDUDEZUCEZFZEUEUFUDUCHUCUDIUAUIRU
GTUHABHTUCUCTEABJKLMNOPQ $.
-
$}
${
@@ -1392,7 +1406,6 @@
BKZABAVTAUFZRDUGSBVBUHONOOVBBUOVCQOVIVGVDVIVBVTCFZGVGUOWBVBAVTCWAUIPVBCCK
AEUJSONOVBCVDTOVFVBLGLVELVBBCVCGGZURVCGZVELWDWCBCVCTRCBVCUKURUAULPVBUMOOU
NO $.
-
$}
${
@@ -1919,7 +1932,6 @@ Kalmbach axioms (soundness proofs) that are true in all ortholattices
EUDUAUDUBUC $.
$}
-
${
wql1.1 $e |- ( a ->1 b ) = 1 $.
wql1.2 $e |- ( ( a v c ) ->1 ( b v c ) ) = 1 $.
@@ -1941,7 +1953,6 @@ Kalmbach axioms (soundness proofs) that are true in all ortholattices
UCUGBCABNDOPQR $.
$}
-
${
womle2a.1 $e |- ( a ^ ( a ->2 b ) ) =<
( ( a ->2 b ) ' v ( a ->1 b ) ) $.
@@ -2288,28 +2299,28 @@ Kalmbach axioms (soundness proofs) that are true in all ortholattices
( wo tb wn wa wle2 wi2 mi df-le df-i2 3tr1 ) ABCBDBAEBEFCABGABHABIABJABKL
$.
- $( Relevance implication is l.e. Sasaki implication. (Contributed by NM,
- 26-Jun-2003.) $)
+ $( Relevance implication is less than or equal to Sasaki implication.
+ (Contributed by NM, 26-Jun-2003.) $)
i5lei1 $p |- ( a ->5 b ) =< ( a ->1 b ) $=
( wa wn wi5 wi1 ax-a3 ax-a2 ax-r2 lea lel2or leror bltr df-i5 df-i1 le3tr1
wo ) ABCZADZBCZQSBDZCZQZSRQZABEABFUCTUBQZRQZUDUCRUEQUFRTUBGRUEHIUESRTSUBSBJ
SUAJKLMABNABOP $.
- $( Relevance implication is l.e. Dishkant implication. (Contributed by NM,
- 26-Jun-2003.) $)
+ $( Relevance implication is less than or equal to Dishkant implication.
+ (Contributed by NM, 26-Jun-2003.) $)
i5lei2 $p |- ( a ->5 b ) =< ( a ->2 b ) $=
( wa wn wo wi5 wi2 lear lel2or leror df-i5 df-i2 le3tr1 ) ABCZADZBCZEZOBDCZ
EBREABFABGQBRNBPABHOBHIJABKABLM $.
- $( Relevance implication is l.e. Kalmbach implication. (Contributed by NM,
- 26-Jun-2003.) $)
+ $( Relevance implication is less than or equal to Kalmbach implication.
+ (Contributed by NM, 26-Jun-2003.) $)
i5lei3 $p |- ( a ->5 b ) =< ( a ->3 b ) $=
( wa wn wo wi5 wi3 leor lelan leror df-i5 ax-a3 ax-r2 df-i3 ax-a2 le3tr1 )
ABCZADZBCZRBDCZEZEZARBEZCZUAEZABFZABGZQUDUABUCABRHIJUFQSETEUBABKQSTLMUGUAUD
EUEABNUAUDOMP $.
- $( Relevance implication is l.e. non-tollens implication. (Contributed by
- NM, 26-Jun-2003.) $)
+ $( Relevance implication is less than or equal to non-tollens implication.
+ (Contributed by NM, 26-Jun-2003.) $)
i5lei4 $p |- ( a ->5 b ) =< ( a ->4 b ) $=
( wa wn wo wi5 wi4 leo leran lelor df-i5 df-i4 le3tr1 ) ABCADZBCEZNBDZCZEON
BEZPCZEABFABGQSONRPNBHIJABKABLM $.
@@ -2609,7 +2620,7 @@ Kalmbach axioms (soundness proofs) that are true in all ortholattices
${
wdf-le1.1 $e |- ( ( a v b ) == b ) = 1 $.
- $( Define 'less than or equal to' analogue for ` == ` analogue of ` = ` .
+ $( Define "less than or equal to" analogue for ` == ` analogue of ` = ` .
(Contributed by NM, 27-Sep-1997.) $)
wdf-le1 $p |- ( a =<2 b ) = 1 $=
( wle2 wo tb wt df-le ax-r2 ) ABDABEBFGABHCI $.
@@ -2617,7 +2628,7 @@ Kalmbach axioms (soundness proofs) that are true in all ortholattices
${
wdf-le2.1 $e |- ( a =<2 b ) = 1 $.
- $( Define 'less than or equal to' analogue for ` == ` analogue of ` = ` .
+ $( Define "less than or equal to" analogue for ` == ` analogue of ` = ` .
(Contributed by NM, 27-Sep-1997.) $)
wdf-le2 $p |- ( ( a v b ) == b ) = 1 $=
( wo tb wle2 wt df-le ax-r1 ax-r2 ) ABDBEZABFZGLKABHICJ $.
@@ -2682,7 +2693,7 @@ Kalmbach axioms (soundness proofs) that are true in all ortholattices
${
wdf2le1.1 $e |- ( ( a ^ b ) == a ) = 1 $.
- $( Alternate definition of 'less than or equal to'. (Contributed by NM,
+ $( Alternate definition of "less than or equal to". (Contributed by NM,
27-Sep-1997.) $)
wdf2le1 $p |- ( a =<2 b ) = 1 $=
( wleao wdf-le1 ) ABABACDE $.
@@ -2690,7 +2701,7 @@ Kalmbach axioms (soundness proofs) that are true in all ortholattices
${
wdf2le2.1 $e |- ( a =<2 b ) = 1 $.
- $( Alternate definition of 'less than or equal to'. (Contributed by NM,
+ $( Alternate definition of "less than or equal to". (Contributed by NM,
27-Sep-1997.) $)
wdf2le2 $p |- ( ( a ^ b ) == a ) = 1 $=
( wdf-le2 wleoa ) ABBABCDE $.
@@ -2704,11 +2715,11 @@ Kalmbach axioms (soundness proofs) that are true in all ortholattices
wlea $p |- ( ( a ^ b ) =<2 a ) = 1 $=
( wa wo wa2 wa5b wr2 wdf-le1 ) ABCZAIADAIDAIAEABFGH $.
- $( Anything is l.e. 1. (Contributed by NM, 27-Sep-1997.) $)
+ $( Anything is less than or equal to 1. (Contributed by NM, 27-Sep-1997.) $)
wle1 $p |- ( a =<2 1 ) = 1 $=
( wt wo or1 bi1 wdf-le1 ) ABABCBADEF $.
- $( 0 is l.e. anything. (Contributed by NM, 11-Oct-1997.) $)
+ $( 0 is less than or equal to anything. (Contributed by NM, 11-Oct-1997.) $)
wle0 $p |- ( 0 =<2 a ) = 1 $=
( wf wle2 wo tb wt df-le ax-a2 or0 ax-r2 bi1 ) BACBADZAEFBAGLALABDABAHAIJKJ
$.
@@ -2741,7 +2752,6 @@ Kalmbach axioms (soundness proofs) that are true in all ortholattices
wlecon $p |- ( b ' =<2 a ' ) = 1 $=
( wn wa wo ax-a2 bi1 oran wdf-le2 w3tr2 wcon3 wdf2le1 ) BDZADZNOEZBBAFZAB
FZPDZBQRBAGHQSBAIHABCJKLM $.
-
$}
${
@@ -2988,7 +2998,7 @@ Kalmbach axioms (soundness proofs) that are true in all ortholattices
IJUKBUJHZGZHZUOUKUSBUJKLURUCUQUBBUBUQUBUQACMLNOPQUNULHZUMHZFZHZUPUNVCULUM
RLVBUFUFVBUDUTUEVAUDUTBARLUEVABCRLSNPQTUA $.
- $( Th. 4.2 Beran p. 49. (Contributed by NM, 10-Nov-1998.) $)
+ $( Thm. 4.2 Beran p. 49. (Contributed by NM, 10-Nov-1998.) $)
wcom2or $p |- C ( a , ( b v c ) ) = 1 $=
( wo wa wn wcomcom wdf-c2 ancom 2or bi1 wr2 w2or or4 wfh1 wcomcom3 wdf-c1
wr1 ) BCFZAUAAUAABGZACGZFZAHZBGZUECGZFZFZUAAGZUAUEGZFZUAUBUFFZUCUGFZFZUIB
@@ -2996,11 +3006,10 @@ Kalmbach axioms (soundness proofs) that are true in all ortholattices
USUCUTUGCAKCUEKLMNOUOUIUBUFUCUGPMNULUIUJUDUKUHUJAUAGZUDUJVBUAAKMABCDEQNUK
UEUAGZUHUKVCUAUEKMUEBCABDRACERQNOTNSI $.
- $( Th. 4.2 Beran p. 49. (Contributed by NM, 10-Nov-1998.) $)
+ $( Thm. 4.2 Beran p. 49. (Contributed by NM, 10-Nov-1998.) $)
wcom2an $p |- C ( a , ( b ^ c ) ) = 1 $=
( wa wn wo wcomcom4 wcom2or df-a con2 ax-r1 bi1 wcbtr wcomcom5 ) ABCFZAGZ
BGZCGZHZQGZRSTABDIACEIJUAUBUBUAQUABCKLMNOP $.
-
$}
$( Negated biconditional (distributive form) (Contributed by NM,
@@ -3138,6 +3147,7 @@ Kalmbach axioms (soundness proofs) that require WOML
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Orthomodular law
@@ -3179,18 +3189,18 @@ Kalmbach axioms (soundness proofs) that require WOML
${
lem3.1.1 $e |- ( a v b ) = b $.
lem3.1.2 $e |- ( b ' v a ) = 1 $.
- $( Lemma used in proof of Th. 3.1 of Pavicic 1993. (Contributed by NM,
+ $( Lemma used in proof of Thm. 3.1 of Pavicic 1993. (Contributed by NM,
12-Aug-1997.) $)
lem3.1 $p |- a = b $=
( tb wt wlem3.1 ax-r1 r3a ) ABABEFABCDGHI $.
- $( Lemma used in proof of Th. 3.1 of Pavicic 1993. (Contributed by NM,
+ $( Lemma used in proof of Thm. 3.1 of Pavicic 1993. (Contributed by NM,
12-Aug-1997.) $)
lem3a.1 $p |- ( a v b ) = a $=
( wo lem3.1 ax-r1 lor oridm ax-r2 ) ABEAAEABAAABABCDFGHAIJ $.
$}
- $( Orthomodular law. Compare Th. 1 of Pavicic 1987. (Contributed by NM,
+ $( Orthomodular law. Compare Thm. 1 of Pavicic 1987. (Contributed by NM,
12-Aug-1997.) $)
oml $p |- ( a v ( a ' ^ ( a v b ) ) ) = ( a v b ) $=
( wn wo wa omlem1 omlem2 lem3.1 ) AACABDZEDIABFABGH $.
@@ -3443,18 +3453,17 @@ Kalmbach axioms (soundness proofs) that require WOML
( wa wo fh4r ancom ax-r5 3tr1 ) ACFZBGABGZCBGZFCAFZBGNMFABCDEHOLBCAIJNMIK
$.
- $( Th. 4.2 Beran p. 49. (Contributed by NM, 7-Nov-1997.) $)
+ $( Thm. 4.2 Beran p. 49. (Contributed by NM, 7-Nov-1997.) $)
com2or $p |- a C ( b v c ) $=
( wo wa wn comcom df-c2 ancom 2or ax-r2 or4 fh1 comcom3 ax-r1 df-c1 ) BCF
ZASASABGZACGZFZAHZBGZUCCGZFZFZSAGZSUCGZFZSTUDFZUAUEFZFUGBUKCULBBAGZBUCGZF
UKBAABDIJUMTUNUDBAKBUCKLMCCAGZCUCGZFULCAACEIJUOUAUPUECAKCUCKLMLTUDUAUENMU
JUGUHUBUIUFUHASGUBSAKABCDEOMUIUCSGUFSUCKUCBCABDPACEPOMLQMRI $.
- $( Th. 4.2 Beran p. 49. (Contributed by NM, 7-Nov-1997.) $)
+ $( Thm. 4.2 Beran p. 49. (Contributed by NM, 7-Nov-1997.) $)
com2an $p |- a C ( b ^ c ) $=
( wa wn wo comcom4 com2or df-a con2 ax-r1 cbtr comcom5 ) ABCFZAGZBGZCGZHZ
PGZQRSABDIACEIJUATPTBCKLMNO $.
-
$}
$( Commutation theorem for Sasaki implication. (Contributed by NM,
@@ -3488,7 +3497,7 @@ Kalmbach axioms (soundness proofs) that require WOML
gsth.1 $e |- a C b $.
gsth.2 $e |- b C c $.
gsth.3 $e |- a C ( b ^ c ) $.
- $( Gudder-Schelp's Theorem. Beran, p. 262, Th. 4.1. (Contributed by NM,
+ $( Gudder-Schelp's Theorem. Beran, p. 262, Thm. 4.1. (Contributed by NM,
20-Sep-1998.) $)
gsth $p |- ( a ^ b ) C c $=
( wa wo wn comcom fh4rc comcom2 lan fh1r ran lea ancom wf ax-r1 3tr lecom
@@ -3505,7 +3514,7 @@ Kalmbach axioms (soundness proofs) that require WOML
${
gsth2.1 $e |- b C c $.
gsth2.2 $e |- a C ( b ^ c ) $.
- $( Stronger version of Gudder-Schelp's Theorem. Beran, p. 263, Th. 4.2.
+ $( Stronger version of Gudder-Schelp's Theorem. Beran, p. 263, Thm. 4.2.
(Contributed by NM, 20-Sep-1998.) $)
gsth2 $p |- ( a ^ b ) C c $=
( wa wn comcom ancom ax-a2 ran ax-r2 comor2 comcom7 comcom2 coman1 com2or
@@ -3769,7 +3778,8 @@ Kalmbach axioms (soundness proofs) that require WOML
${
binr1.1 $e |- ( a ->3 b ) = 1 $.
- $( Pavicic binary logic ax-r1 analog. (Contributed by NM, 7-Nov-1997.) $)
+ $( Pavicic binary logic ~ ax-r1 analog. (Contributed by NM,
+ 7-Nov-1997.) $)
binr1 $p |- ( b ' ->3 a ' ) = 1 $=
( wn i3le lecon lei3 ) BDADABABCEFG $.
$}
@@ -3777,7 +3787,8 @@ Kalmbach axioms (soundness proofs) that require WOML
${
binr2.1 $e |- ( a ->3 b ) = 1 $.
binr2.2 $e |- ( b ->3 c ) = 1 $.
- $( Pavicic binary logic ax-r2 analog. (Contributed by NM, 7-Nov-1997.) $)
+ $( Pavicic binary logic ~ ax-r2 analog. (Contributed by NM,
+ 7-Nov-1997.) $)
binr2 $p |- ( a ->3 c ) = 1 $=
( i3le letr lei3 ) ACABCABDFBCEFGH $.
$}
@@ -3785,7 +3796,8 @@ Kalmbach axioms (soundness proofs) that require WOML
${
binr3.1 $e |- ( a ->3 c ) = 1 $.
binr3.2 $e |- ( b ->3 c ) = 1 $.
- $( Pavicic binary logic axr3 analog. (Contributed by NM, 7-Nov-1997.) $)
+ $( Pavicic binary logic ~ ax-r3 analog. (Contributed by NM,
+ 7-Nov-1997.) $)
binr3 $p |- ( ( a v b ) ->3 c ) = 1 $=
( wo i3le le2or oridm lbtr lei3 ) ABFZCLCCFCACBCACDGBCEGHCIJK $.
$}
@@ -5254,7 +5266,6 @@ Kalmbach axioms (soundness proofs) that require WOML
u5lemc2 $p |- a C ( b ->5 c ) $=
( wa wn wo wi5 com2an comcom2 com2or df-i5 ax-r1 cbtr ) ABCFZBGZCFZHZQCGZ
FZHZBCIZASUAAPRABCDEJAQCABDKZEJLAQTUDACEKJLUCUBBCMNO $.
-
$}
${
@@ -5349,7 +5360,6 @@ Kalmbach axioms (soundness proofs) that require WOML
ax-r5 comcom3 comcom4 fh4 ax-a2 2an ) ABDABEAFZBEGZUEBFZEZGZUEBGZABHUIBUH
GZUJUFBUHUFAUEGZBEZBUMUFABUECAAAIJKLUMBULEZBULBMUNBNEBULNBNULAOLPBQRRRSUK
BUEGZBUGGZEZUJUEBUGABCTABCUAUBUQUJNEUJUOUJUPNBUEUCNUPBOLUDUJQRRRR $.
-
$}
$( Commutation theorem for Sasaki implication. (Contributed by NM,
@@ -5405,7 +5415,6 @@ Kalmbach axioms (soundness proofs) that require WOML
$( L.e. to relevance implication. (Contributed by NM, 11-Jan-1998.) $)
u5lemle1 $p |- ( a ->5 b ) = 1 $=
( wi5 wn wo wt lecom u5lemc4 sklem ax-r2 ) ABDAEBFGABABCHIABCJK $.
-
$}
${
@@ -5580,6 +5589,7 @@ Kalmbach axioms (soundness proofs) that require WOML
LQHR $.
$}
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
More lemmas for unified implication
@@ -5603,7 +5613,6 @@ Kalmbach axioms (soundness proofs) that require WOML
?$.
$)
-
$( Lemma for unified implication study. (Contributed by NM, 14-Dec-1997.) $)
u1lem1 $p |- ( ( a ->1 b ) ->1 a ) = a $=
( wi1 wn wo u1lemc1 comcom u1lemc4 u1lemnoa ax-r2 ) ABCZACKDAEAKAAKABFGHABI
@@ -6118,7 +6127,6 @@ Kalmbach axioms (soundness proofs) that require WOML
VMWAVIVQVLVTVHVPABQOVLVBEZVCEZLZVTVBVCQVTWDVRWBVSWCVRCVNMWBVNCRCATKVSCVOMWC
VOCRCBTKUFPKUGPKVLVDVIVLVDVCVBDZMZVDWFVLVBVCNPVDWEUHSUISUJUKKK $.
-
$( Possible axiom for Kalmbach implication system. (Contributed by NM,
23-Jan-1998.) $)
u3lemax5 $p |- ( ( a ->3 b ) ->3 ( ( a ->3 b )
@@ -6252,7 +6260,6 @@ Kalmbach axioms (soundness proofs) that require WOML
IVDULBUIBUDUHBBAEZBEZGZFZUDBVGPUDVHABQISBBCEZVFGZFZUHBVJPUHVKCBQISRTCULCUJU
KCCVEVIGZFZUJCVLPUJVMACQISCCVFVIGZFZUKCVNPUKVOBCQISRTNMMMMMM $.
-
$( A 3-variable theorem. (Contributed by NM, 18-Oct-1998.) $)
3vth7 $p |- ( ( a ->2 b ) ' ->2 ( b v c ) ) =
( a ->2 ( b v c ) ) $=
@@ -6491,7 +6498,6 @@ Kalmbach axioms (soundness proofs) that require WOML
OPREZHRTOUAOSIDJABCKLOQMN $.
$}
-
${
3vroa.1 $e |- ( ( a ->2 b ) ^
( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) = 1 $.
@@ -6517,7 +6523,6 @@ Kalmbach axioms (soundness proofs) that require WOML
ZDZVMVHVDVFJWEVHVMVHWDUKVHVLTPMQVPVCVIFZVFDVKVNWFVOVFABULBCRUMVIVFVCVIVFVIV
DVFVHVDOVDVETPUOVIVCVIVHVDFVCEVIVHVDVHVDUNVHVDUPUQABURUSUTVANACRVB $.
-
$( Mladen's OML. (Contributed by NM, 4-Nov-1998.) $)
mlaoml $p |- ( ( a == b ) ^ ( b == c ) ) =< ( a == c ) $=
( wi1 wa tb u1lembi ran mlalem bltr ancom an32 3tr le2an an12 id 3tr1 anass
@@ -6554,7 +6559,6 @@ Kalmbach axioms (soundness proofs) that require WOML
BUTCUQCTLUK $.
$}
-
$(
lattice (((-xIy)vy)Iy)=(x2y)
lattice "((xIw)v(yIw))<((((-xIw)^(-yIw))Iw)vw)"
@@ -6573,7 +6577,6 @@ Kalmbach axioms (soundness proofs) that require WOML
BCZBEZCZUGUCUJUBBGZHUGUKUBBIJKUEUJBQZBBUIEZQZBUCUJBULLUMUNBQUOUJUNBUIBMLUNB
NKBUIORSUGBMKUCBPABTUA $.
-
$( Weak DeMorgan's law for attempt at Sasaki algebra. (Contributed by NM,
4-Jan-1999.) $)
sadm3 $p |- ( ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ->1 c ) =<
@@ -6707,7 +6710,6 @@ Kalmbach axioms (soundness proofs) that require WOML
$.
$}
-
$( For 5GO proof of Mladen's conjecture. (Contributed by NM,
20-Jan-2002.) $)
mlaconj $p |- ( ( a == b ) ^ ( ( a == c ) v ( b == c ) ) ) =<
@@ -7649,13 +7651,13 @@ Kalmbach axioms (soundness proofs) that require WOML
WAVFUHWIVFUILVFUJFVFVGVH $.
$}
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
OML Lemmas for studying orthoarguesian laws
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
-
${
oas.1 $e |- ( a ' ^ ( a v b ) ) =< c $.
$( "Strengthening" lemma for studying the orthoarguesian law. (Contributed
@@ -7834,7 +7836,6 @@ Kalmbach axioms (soundness proofs) that require WOML
DGIZUAGPSQTABCDEFKABCDEFLMUAUBNO $.
$}
-
${
$( Substitutions into OA distributive law. $)
distoa.1 $e |- d = ( a ->2 b ) $.
@@ -7864,7 +7865,6 @@ Kalmbach axioms (soundness proofs) that require WOML
AUCNDSTGOFUBUAJUDIUBUAPQR $.
${
-
$( OA distributive law as hypothesis. $)
distoa.4 $e |- ( d ^ ( e v f ) ) = ( ( d ^ e ) v ( d ^ f ) ) $.
$( Derivation in OM of OA, assuming OA distributive law ~ oadistd .
@@ -7880,7 +7880,6 @@ Kalmbach axioms (soundness proofs) that require WOML
$}
$}
-
${
oa3to4lem.1 $e |- a ' =< b $.
oa3to4lem.2 $e |- c ' =< d $.
@@ -8106,7 +8105,6 @@ Kalmbach axioms (soundness proofs) that require WOML
BDGZHZCDGZHDEQDRODPADIBDIJCDIJK $.
$}
-
${
oa4to6lem.1 $e |- a ' =< b $.
oa4to6lem.2 $e |- c ' =< d $.
@@ -8204,7 +8202,6 @@ Kalmbach axioms (soundness proofs) that require WOML
VIWEIUNJUPPQUEVAVQVEWAWLWMUEUFUEUFUEUFUENUGUHUI $.
$}
-
${
oa4btoc.1 $e |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
@@ -8502,7 +8499,6 @@ Kalmbach axioms (soundness proofs) that require WOML
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
-
$( Lemma for 3-OA(2). Equivalence with substitution into 4-OA. (Contributed
by NM, 24-Dec-1998.) $)
oa3-2lema $p |- ( ( a ->1 c ) ^ ( a v ( b ^ ( (
@@ -9056,7 +9052,6 @@ Kalmbach axioms (soundness proofs) that require WOML
VBVBUBUCUDUEUFRABCUGUHUOVEVCVJABUIUTVFVBVIVFUTBCUJPUMVGVAVHABTACTZSUKSWAUL
$.
-
${
oadist.1 $e |- d =< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
$( Distributive law derived from OAL. (Contributed by NM, 20-Nov-1998.) $)
@@ -9170,7 +9165,6 @@ Kalmbach axioms (soundness proofs) that require WOML
( wi2 wn wo wa ud2lem0c lea bltr ax-oal4 id oa4v3v oal42 oa23 ) ABCACBACBAC
DEZABDEZPCEZACFZGRACHRSIJZQBEZABFZGUAABHUAUBIJZPCQBTUCKPLQLMNO $.
-
${
oa3moa3.1 $e |- a =< b ' $.
oa3moa3.2 $e |- c =< d ' $.
@@ -9195,6 +9189,7 @@ Kalmbach axioms (soundness proofs) that require WOML
KXMXKBVDVPULUNVEVKVOUOUPPUHUM $.
$}
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
6-variable orthoarguesian law
@@ -9212,7 +9207,6 @@ Kalmbach axioms (soundness proofs) that require WOML
( ( ( a v e ) ^ ( b v f ) ) v ( ( c v e ) ^ ( d v f ) ) ) ) ) ) ) $.
$}
-
${
oa64v.1 $e |- a =< b ' $.
oa64v.2 $e |- c =< d ' $.
@@ -9223,7 +9217,6 @@ Kalmbach axioms (soundness proofs) that require WOML
( wf wt wn le0 ax-oa6 id oa6v4v ) ABCDGHABCDGHEFHIJKGLHLM $.
$}
-
$( Derivation of 3-variable OA from 6-variable OA. (Contributed by NM,
28-Nov-1998.) $)
oa63v $p |- ( ( a ->2 b ) ^
@@ -9359,35 +9352,50 @@ Kalmbach axioms (soundness proofs) that require WOML
UCLUHTUGTVEVDVAUSVDUJVDVAVCQZVAVDVDIQZVMVNVDVDIOUESGVCIUKTVAVCULUMUNUMV
AUTUOUNGHIUPUQ $.
$}
-
$}
+
$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
Other stronger-than-OML laws
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
New state-related equation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
- $( New equation that holds in Hilbert space, discovered by Pavicic and Megill
- (unpublished).
+ $( This is the simplest known example of an equation implied by the set of
+ Mayet--Godowski equations that is independent from all Godowski equations.
+ It was discovered by Norman Megill and Mladen Pavicic between 1997 and
+ August 2003. This is Equation (54) in
+
+ Mladen Pavicic, Norman D. Megill, _Quantum Logic and Quantum Computation_,
+ in _Handbook of Quantum Logic and Quantum Structures_, Volume _Quantum
+ Structures_, Elsevier, Amsterdam, 2007, pp. 751--787.
+ ~ https://arxiv.org/abs/0812.3072
- [Editor's note: The date this was added is unknown within the range
- 1997 to 9-Aug-2003.] (Contributed by NM, 1-Jan-1998.) $)
+ and Equation (15) in
+
+ Mladen Pavicic, Brendan D. McKay, Norman D. Megill, Kresimir Fresl, _Graph
+ Approach to Quantum Systems_, Journal of Mathematical Physics, Volume 51,
+ Issue 10, October 2010. ~ https://arxiv.org/abs/1004.0776
+
+ (Contributed by NM, 1-Jan-1998.) $)
ax-newstateeq $a |- ( ( ( a ->1 b ) ->1 ( c ->1 b ) ) ^
( ( a ->1 c ) ^ ( b ->1 a ) ) ) =< ( c ->1 a ) $.
+
$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
Contributions of Roy Longton
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Roy's first section
@@ -10059,7 +10067,6 @@ Kalmbach axioms (soundness proofs) that require WOML
BAAAUCUDUKBCUEUFUGUHUNUMABUIKUJ $.
$}
-
$( $t
/* The '$t' token indicates the beginning of the typesetting definition
@@ -10092,12 +10099,10 @@ Kalmbach axioms (soundness proofs) that require WOML
/* Variable color key */
htmlvarcolor 'term';
-
/* GIF and Symbol Font HTML directories */
htmldir "../qlegif/";
althtmldir "../qleuni/";
-
/* Symbol definitions */
htmldef "a" as "";
htmldef "b" as "";
@@ -10194,7 +10199,6 @@ Kalmbach axioms (soundness proofs) that require WOML
htmldef "=" as ' = ';
*/
-
/* Definitions for Unicode version */
althtmldef "a" as 'a';
althtmldef "b" as 'b';
@@ -10290,7 +10294,6 @@ Kalmbach axioms (soundness proofs) that require WOML
*/
/* End of Unicode defintions */
-
latexdef "a" as "a";
latexdef "b" as "b";
latexdef "c" as "c";
@@ -10372,6 +10375,7 @@ Weakly distributive ortholattices (WDOL)
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
WDOL law
@@ -10409,7 +10413,6 @@ Weakly distributive ortholattices (WDOL)
wddi1 $p |- ( ( a ^ ( b v c ) ) == ( ( a ^ b ) v ( a ^ c ) ) ) = 1 $=
( wdcom wfh1 ) ABCABDACDE $.
-
$( The weak distributive law in WDOL. (Contributed by NM, 5-Mar-2006.) $)
wddi2 $p |- ( ( ( a v b ) ^ c ) == ( ( a ^ c ) v ( b ^ c ) ) ) = 1 $=
( wo wa wancom wddi1 w2or wr2 ) ABDZCECJEZACEZBCEZDZJCFKCAEZCBEZDNCABGOLPMC
@@ -10426,7 +10429,6 @@ Weakly distributive ortholattices (WDOL)
( wa wo wa2 wddi3 w2an wr2 ) ABDZCECJEZACEZBCEZDZJCFKCAEZCBEZDNCABGOLPMCAFC
BFHII $.
-
${
wdid0id5.1 $e |- ( a ==0 b ) = 1 $.
$( Show that quantum identity follows from classical identity in a WDOL.
@@ -10533,7 +10535,6 @@ Modular ortholattices (MOL)
mli $p |- ( ( a ^ b ) v c ) = ( a ^ ( b v c ) ) $=
( wa wo ancom ror orcom ml2i 3tr ran ) ABEZCFZCBFZAEZBCFZAEAQENBAEZCFCRFP
MRCABGHRCIABCDJKOQACBILQAGK $.
-
$}
${
@@ -10563,7 +10564,6 @@ Modular ortholattices (MOL)
ml3 $p |- ( a v ( b ^ ( c v a ) ) ) = ( a v ( c ^ ( b v a ) ) ) $=
( wo wa ml3le lebi ) ABCADEDACBADEDABCFACBFG $.
-
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 (Contributed by NM,
15-Mar-2010.) (Revised by NM, 31-Mar-2011.) $)
vneulem1 $p |- ( ( ( x v y ) v u ) ^ w )
@@ -11238,7 +11238,6 @@ Modular ortholattices (MOL)
UDBUBRST $.
$}
-
${
xdp41.c0 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
xdp41.c1 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
@@ -11335,7 +11334,6 @@ Modular ortholattices (MOL)
EBEUQUJVPWLUIVEWLBVPBEUHVLUGUOVFTVAUO $.
$}
-
${
xxdp.1 $e |- p2 =< ( a2 v b2 ) $.
xxdp.c0 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
@@ -11588,7 +11586,6 @@ Modular ortholattices (MOL)
IUOUIUJYFXIGDGVDUQXJYFUOVLYFDXJDGUNVHUMVBWPUPWDVB $.
$}
-
${
xxxdp.c0 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
xxxdp.c1 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
@@ -11726,7 +11723,6 @@ Modular ortholattices (MOL)
GDGVCUPXIYEUNVKYEDXIDGUMVGULVAWOUOWCVA $.
$}
-
${
3dp.c0 $e |- c0 = ( ( a1 v a1 ) ^ ( b1 v b1 ) ) $.
3dp.c1 $e |- c1 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
@@ -11808,7 +11804,6 @@ Modular ortholattices (MOL)
XFFDFVAUNXGYBULVIYBDXGDFUKVEUJUSWMUMWAUS $.
$}
-
${
oadp35lem.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
oadp35lem.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
@@ -11882,7 +11877,6 @@ Modular ortholattices (MOL)
$)
$}
-
${
oadp35.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
oadp35.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
diff --git a/set.mm b/set.mm
index cae1f55ff8..3814e09f3b 100644
--- a/set.mm
+++ b/set.mm
@@ -1854,6 +1854,15 @@ In classical logic (our logic) this is always true. In intuitionistic
( nsyl3 con2i ) CAABCDEFG $.
$}
+ ${
+ nsyl2.1 $e |- ( ph -> -. ps ) $.
+ nsyl2.2 $e |- ( -. ch -> ps ) $.
+ $( A negated syllogism inference. (Contributed by NM, 26-Jun-1994.)
+ (Proof shortened by Wolf Lammen, 14-Nov-2023.) $)
+ nsyl2 $p |- ( ph -> ch ) $=
+ ( wn nsyl3 con4i ) CAABCFDEGH $.
+ $}
+
$( Double negation introduction. Converse of ~ notnotr and one implication
of ~ notnotb . Theorem *2.12 of [WhiteheadRussell] p. 101. This was the
sixth axiom of Frege, specifically Proposition 41 of [Frege1879] p. 47.
@@ -1886,6 +1895,22 @@ In classical logic (our logic) this is always true. In intuitionistic
( wn notnot syl6 con4d ) ABCEZABECIEDCFGH $.
$}
+ $( Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. Its
+ associated inference is ~ con1i . (Contributed by NM, 29-Dec-1992.)
+ (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)
+ con1 $p |- ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) $=
+ ( wn wi id con1d ) ACBDZABGEF $.
+
+ ${
+ con1i.1 $e |- ( -. ph -> ps ) $.
+ $( A contraposition inference. Inference associated with ~ con1 . Its
+ associated inference is ~ mt3 . (Contributed by NM, 3-Jan-1993.)
+ (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (Proof shortened by
+ Wolf Lammen, 19-Jun-2013.) $)
+ con1i $p |- ( -. ps -> ph ) $=
+ ( wn id nsyl2 ) BDZBAGECF $.
+ $}
+
${
mt3d.1 $e |- ( ph -> -. ch ) $.
mt3d.2 $e |- ( ph -> ( -. ps -> ch ) ) $.
@@ -1904,13 +1929,8 @@ In classical logic (our logic) this is always true. In intuitionistic
$}
${
- nsyl2.1 $e |- ( ph -> -. ps ) $.
- nsyl2.2 $e |- ( -. ch -> ps ) $.
- $( A negated syllogism inference. (Contributed by NM, 26-Jun-1994.)
- (Proof shortened by Wolf Lammen, 14-Nov-2023.) $)
- nsyl2 $p |- ( ph -> ch ) $=
- ( wn nsyl3 con4i ) CAABCFDEGH $.
-
+ nsyl2OLD.1 $e |- ( ph -> -. ps ) $.
+ nsyl2OLD.2 $e |- ( -. ch -> ps ) $.
$( Obsolete version of ~ nsyl2 as of 14-Nov-2023. (Contributed by NM,
26-Jun-1994.) (Proof modification is discouraged.)
(New usage is discouraged.) $)
@@ -1918,22 +1938,6 @@ In classical logic (our logic) this is always true. In intuitionistic
( wn wi a1i mt3d ) ACBDCFBGAEHI $.
$}
- $( Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. Its
- associated inference is ~ con1i . (Contributed by NM, 29-Dec-1992.)
- (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)
- con1 $p |- ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) $=
- ( wn wi id con1d ) ACBDZABGEF $.
-
- ${
- con1i.1 $e |- ( -. ph -> ps ) $.
- $( A contraposition inference. Inference associated with ~ con1 . Its
- associated inference is ~ mt3 . (Contributed by NM, 3-Jan-1993.)
- (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (Proof shortened by
- Wolf Lammen, 19-Jun-2013.) $)
- con1i $p |- ( -. ps -> ph ) $=
- ( wn id nsyl2 ) BDZBAGECF $.
- $}
-
${
pm2.24i.1 $e |- ph $.
$( Inference associated with ~ pm2.24 . Its associated inference is
@@ -2137,32 +2141,13 @@ In classical logic (our logic) this is always true. In intuitionistic
( wi a1i pm2.61d ) ABCBCFAEGDH $.
$}
- ${
- ja.1 $e |- ( -. ph -> ch ) $.
- ja.2 $e |- ( ps -> ch ) $.
- $( Inference joining the antecedents of two premises. For partial
- converses, see ~ jarri and ~ jarli . (Contributed by NM, 24-Jan-1993.)
- (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) $)
- ja $p |- ( ( ph -> ps ) -> ch ) $=
- ( wi imim2i pm2.61d1 ) ABFACBCAEGDH $.
- $}
-
- ${
- jad.1 $e |- ( ph -> ( -. ps -> th ) ) $.
- jad.2 $e |- ( ph -> ( ch -> th ) ) $.
- $( Deduction form of ~ ja . (Contributed by Scott Fenton, 13-Dec-2010.)
- (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
- jad $p |- ( ph -> ( ( ps -> ch ) -> th ) ) $=
- ( wi wn com12 ja ) BCGADBCADGABHDEIACDFIJI $.
- $}
-
${
pm2.61i.1 $e |- ( ph -> ps ) $.
pm2.61i.2 $e |- ( -. ph -> ps ) $.
$( Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.)
- (Proof shortened by Wolf Lammen, 12-Sep-2013.) $)
+ (Proof shortened by Wolf Lammen, 19-Nov-2023.) $)
pm2.61i $p |- ps $=
- ( wi id ja ax-mp ) AAEBAFAABDCGH $.
+ ( nsyl4 pm2.18i ) BABBCDEF $.
$}
${
@@ -2197,6 +2182,35 @@ In classical logic (our logic) this is always true. In intuitionistic
( wn wi a1d pm2.61ii pm2.61i ) CDHABCIZDJEADNFKBDNGKLM $.
$}
+ ${
+ ja.1 $e |- ( -. ph -> ch ) $.
+ ja.2 $e |- ( ps -> ch ) $.
+ $( Inference joining the antecedents of two premises. For partial
+ converses, see ~ jarri and ~ jarli . (Contributed by NM, 24-Jan-1993.)
+ (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) $)
+ ja $p |- ( ( ph -> ps ) -> ch ) $=
+ ( wi imim2i pm2.61d1 ) ABFACBCAEGDH $.
+ $}
+
+ ${
+ jad.1 $e |- ( ph -> ( -. ps -> th ) ) $.
+ jad.2 $e |- ( ph -> ( ch -> th ) ) $.
+ $( Deduction form of ~ ja . (Contributed by Scott Fenton, 13-Dec-2010.)
+ (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
+ jad $p |- ( ph -> ( ( ps -> ch ) -> th ) ) $=
+ ( wi wn com12 ja ) BCGADBCADGABHDEIACDFIJI $.
+ $}
+
+ ${
+ pm2.61iOLD.1 $e |- ( ph -> ps ) $.
+ pm2.61iOLD.2 $e |- ( -. ph -> ps ) $.
+ $( Obsolete version of ~ pm2.61i as of 19-Nov-2023. (Contributed by NM,
+ 5-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.)
+ (Proof modification is discouraged.) (New usage is discouraged.) $)
+ pm2.61iOLD $p |- ps $=
+ ( wi id ja ax-mp ) AAEBAFAABDCGH $.
+ $}
+
$( Weak Clavius law. If a formula implies its negation, then it is false. A
form of "reductio ad absurdum", which can be used in proofs by
contradiction. Theorem *2.01 of [WhiteheadRussell] p. 100. Provable in
@@ -2495,8 +2509,8 @@ statements not containing the new symbol (or new combination) should
~ dfbi1ALT for an unusual version proved directly from axioms.
(Contributed by NM, 29-Dec-1992.) $)
dfbi1 $p |- ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) $=
- ( wb wi wn df-bi pm2.21 mt3 impbi impi impbii ) ABCZABDZBADZEDEZLODZPOLDEZD
- ABFPQGHMNLABIJK $.
+ ( wb wi wn df-bi impbi con3rr3 mt3 ) ABCZABDBADEDEZCZJKDZKJDZEDABFMNLJKGHI
+ $.
$( Alternate proof of ~ dfbi1 . This proof, discovered by Gregory Bush on
8-Mar-2004, has several curious properties. First, it has only 17 steps
@@ -3486,7 +3500,7 @@ intermediate steps that are essentially incomprehensible to humans (other
$( A mixed syllogism inference from an implication and a biconditional.
(Contributed by Wolf Lammen, 16-Dec-2013.) $)
sylnib $p |- ( ph -> -. ch ) $=
- ( wb a1i mtbid ) ABCDBCFAEGH $.
+ ( biimpri nsyl ) ABCDBCEFG $.
$}
${
@@ -4304,7 +4318,7 @@ use disjunction (although this is not required since definitions are
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
imp5g $p |- ( ( ph /\ ps ) -> ( ( ( ch /\ th ) /\ ta ) -> et ) ) $=
- ( wa wi imp imp4c ) ABHCDEFABCDEFIIIGJK $.
+ ( wa wi imp4b impd ) ABHCDHEFABCDEFIGJK $.
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
imp55 $p |- ( ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) /\ ta ) -> et ) $=
@@ -4468,7 +4482,7 @@ use disjunction (although this is not required since definitions are
$( Deduction commuting conjunction in antecedent. (Contributed by NM,
12-Dec-2004.) $)
ancomsd $p |- ( ph -> ( ( ch /\ ps ) -> th ) ) $=
- ( wa ancom syl5bi ) CBFBCFADCBGEH $.
+ ( expcomd impd ) ACBDABCDEFG $.
$}
${
@@ -4957,7 +4971,7 @@ use disjunction (although this is not required since definitions are
$( A wff is equivalent to its conjunction with truth. (Contributed by NM,
3-Aug-1994.) $)
biantrur $p |- ( ps <-> ( ph /\ ps ) ) $=
- ( wa wb ibar ax-mp ) ABABDECABFG $.
+ ( biantru biancomi ) BABABCDE $.
$}
${
@@ -5005,7 +5019,7 @@ use disjunction (although this is not required since definitions are
Carneiro, 11-Sep-2015.) (Proof shortened by Wolf Lammen,
19-Jan-2020.) $)
rbaibr $p |- ( ch -> ( ps <-> ph ) ) $=
- ( wa iba syl6bbr ) CBBCEACBFDG $.
+ ( biancomi baibr ) ACBACBDEF $.
$( Move conjunction outside of biconditional. (Contributed by Mario
Carneiro, 11-Sep-2015.) (Proof shortened by Wolf Lammen,
@@ -5024,7 +5038,7 @@ use disjunction (although this is not required since definitions are
$( Move conjunction outside of biconditional. (Contributed by Mario
Carneiro, 11-Sep-2015.) $)
rbaibd $p |- ( ( ph /\ th ) -> ( ps <-> ch ) ) $=
- ( wa iba bicomd sylan9bb ) ABCDFZDCEDCJDCGHI $.
+ ( biancomd baibd ) ABDCABDCEFG $.
$}
${
@@ -5167,7 +5181,7 @@ use disjunction (although this is not required since definitions are
Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct
reversed). (Contributed by NM, 1-Dec-2003.) $)
pm4.71ri $p |- ( ph <-> ( ps /\ ph ) ) $=
- ( wi wa wb pm4.71r mpbi ) ABDABAEFCABGH $.
+ ( pm4.71i biancomi ) ABAABCDE $.
$}
${
@@ -5182,7 +5196,7 @@ use disjunction (although this is not required since definitions are
Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed
by NM, 10-Feb-2005.) $)
pm4.71rd $p |- ( ph -> ( ps <-> ( ch /\ ps ) ) ) $=
- ( wi wa wb pm4.71r sylib ) ABCEBCBFGDBCHI $.
+ ( pm4.71d biancomd ) ABCBABCDEF $.
$}
$( Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM,
@@ -5250,7 +5264,7 @@ use disjunction (although this is not required since definitions are
3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) $)
pm5.3 $p |- ( ( ( ph /\ ps ) -> ch ) <->
( ( ph /\ ps ) -> ( ph /\ ch ) ) ) $=
- ( wa wi impexp imdistan bitri ) ABDZCEABCEEIACDEABCFABCGH $.
+ ( wa simpl biantrurd pm5.74i ) ABDZCACDHACABEFG $.
$( Distribution of implication over biconditional. Theorem *5.32 of
[WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) $)
@@ -5393,7 +5407,7 @@ use disjunction (although this is not required since definitions are
$( Syllogism inference combined with a biconditional. (Contributed by BJ,
25-Apr-2019.) $)
sylanblrc $p |- ( ph -> th ) $=
- ( wa biimpri sylancl ) ABCDEFDBCHGIJ $.
+ ( a1i sylanbrc ) ABCDECAFHGI $.
$}
${
@@ -5629,7 +5643,7 @@ use disjunction (although this is not required since definitions are
$( Deduction joining nested implications to form implication of
conjunctions. (Contributed by NM, 29-Feb-1996.) $)
im2anan9 $p |- ( ( ph /\ th ) -> ( ( ps /\ ta ) -> ( ch /\ et ) ) ) $=
- ( wa wi adantr adantl anim12d ) ADIBCEFABCJDGKDEFJAHLM $.
+ ( wa adantrd adantld anim12ii ) ABEICDFABCEGJDEFBHKL $.
$( Deduction joining nested implications to form implication of
conjunctions. (Contributed by NM, 29-Feb-1996.) $)
@@ -5723,7 +5737,7 @@ use disjunction (although this is not required since definitions are
conjunct on the right-hand side of an equivalence. Deduction form.
(Contributed by Peter Mazsa, 22-May-2021.) $)
anbi1cd $p |- ( ph -> ( ( th /\ ps ) <-> ( ch /\ th ) ) ) $=
- ( wb wa anbi2 biancomd syl ) ABCFZDBGZCDGFEKLCDBCDHIJ $.
+ ( wa anbi2d biancomd ) ADBFCDABCDEGH $.
$}
$( Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM,
@@ -5766,14 +5780,14 @@ use disjunction (although this is not required since definitions are
$( Swap two conjuncts. (Contributed by Peter Mazsa, 18-Sep-2022.) $)
an21 $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
- ( wa ancom anbi1i anass bitri ) ABDZCDBADZCDBACDDIJCABEFBACGH $.
+ ( wa biid bianassc bicomi ) BACDZDABDCDHACBHEFG $.
$( Swap two conjuncts. Note that the first digit (1) in the label refers to
the outer conjunct position, and the next digit (2) to the inner conjunct
position. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Peter
Mazsa, 18-Sep-2022.) $)
an12 $p |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
- ( wa anass an21 bitr3i ) ABCDDABDCDBACDDABCEABCFG $.
+ ( wa ancom bianass biancomi ) ABCDZDBACDHCBABCEFG $.
$( A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof
shortened by Wolf Lammen, 25-Dec-2012.) $)
@@ -5835,8 +5849,7 @@ use disjunction (although this is not required since definitions are
$( Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.) $)
an4 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
( ( ph /\ ch ) /\ ( ps /\ th ) ) ) $=
- ( wa an12 anbi2i anass 3bitr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHAC
- LHI $.
+ ( wa anass an12 bianass bitri ) ABECDEZEABJEZEACEBDEZEABJFKCLABCDGHI $.
$( Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.) $)
an42 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
@@ -6014,7 +6027,7 @@ use disjunction (although this is not required since definitions are
sylanl2.2 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
$( A syllogism inference. (Contributed by NM, 1-Jan-2005.) $)
sylanl2 $p |- ( ( ( ps /\ ph ) /\ th ) -> ta ) $=
- ( wa anim2i sylan ) BAHBCHDEACBFIGJ $.
+ ( adantl syldanl ) BACDEACBFHGI $.
$}
${
@@ -6040,7 +6053,7 @@ use disjunction (although this is not required since definitions are
$( A syllogism deduction combined with conjoining antecedents.
(Contributed by Alan Sare, 28-Oct-2011.) $)
syl6an $p |- ( ph -> ( ch -> ta ) ) $=
- ( wa jctild syl6 ) ACBDIEACDBGFJHK $.
+ ( ex sylsyld ) ABCDEFGBDEHIJ $.
$}
${
@@ -6061,7 +6074,7 @@ use disjunction (although this is not required since definitions are
$( ~ syl2an with antecedents in standard conjunction form. (Contributed by
Alan Sare, 27-Aug-2016.) $)
syl2an2 $p |- ( ( ch /\ ph ) -> ta ) $=
- ( wa syl2an anabss7 ) CAEABDECAIFGHJK $.
+ ( wa adantl syl2anc ) CAIBDEABCFJGHK $.
$}
${
@@ -6267,7 +6280,7 @@ use disjunction (although this is not required since definitions are
$( Detach truth from conjunction in biconditional. Deduction form.
(Contributed by Peter Mazsa, 24-Sep-2022.) $)
mpbiran2d $p |- ( ph -> ( ps <-> ch ) ) $=
- ( wa biantrud bitr4d ) ABCDGCFADCEHI $.
+ ( biancomd mpbirand ) ABDCEABDCFGH $.
$}
${
@@ -6285,7 +6298,7 @@ use disjunction (although this is not required since definitions are
$( Detach truth from conjunction in biconditional. (Contributed by NM,
22-Feb-1996.) $)
mpbiran2 $p |- ( ph <-> ps ) $=
- ( wa biantru bitr4i ) ABCFBECBDGH $.
+ ( biancomi mpbiran ) ACBDACBEFG $.
$}
${
@@ -6402,7 +6415,7 @@ use disjunction (although this is not required since definitions are
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
19-Oct-1999.) $)
ad2antrl $p |- ( ( ch /\ ( ph /\ th ) ) -> ps ) $=
- ( wa adantr adantl ) ADFBCABDEGH $.
+ ( adantl adantrr ) CABDABCEFG $.
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
19-Oct-1999.) $)
@@ -6530,7 +6543,7 @@ use disjunction (although this is not required since definitions are
$( Deduction adding 1 conjunct to antecedent. (Contributed by Alan Sare,
17-Oct-2017.) $)
adantl3r $p |- ( ( ( ( ( ph /\ et ) /\ ps ) /\ ch ) /\ th ) -> ta ) $=
- ( wa wi ex adantllr imp ) AFHBHCHDEABCDEIFABHCHDEGJKL $.
+ ( wa id adantlr sylanl1 ) AFHBHABHZCDEABLFLIJGK $.
$}
${
@@ -6538,7 +6551,7 @@ use disjunction (although this is not required since definitions are
$( Deduction adding conjuncts to antecedent. (Contributed by Alan Sare,
17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
ad4ant13 $p |- ( ( ( ( ph /\ th ) /\ ps ) /\ ta ) -> ch ) $=
- ( wa adantlr adantr ) ADGBGCEABCDFHI $.
+ ( wa adantr adantllr ) ABECDABGCEFHI $.
$( Deduction adding conjuncts to antecedent. (Contributed by Alan Sare,
17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
@@ -6548,12 +6561,12 @@ use disjunction (although this is not required since definitions are
$( Deduction adding conjuncts to antecedent. (Contributed by Alan Sare,
17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
ad4ant23 $p |- ( ( ( ( th /\ ph ) /\ ps ) /\ ta ) -> ch ) $=
- ( wa adantll adantr ) DAGBGCEABCDFHI $.
+ ( wa adantr adantlll ) ABECDABGCEFHI $.
$( Deduction adding conjuncts to antecedent. (Contributed by Alan Sare,
17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
ad4ant24 $p |- ( ( ( ( th /\ ph ) /\ ta ) /\ ps ) -> ch ) $=
- ( wa adantll adantlr ) DAGBCEABCDFHI $.
+ ( adantlr adantlll ) AEBCDABCEFGH $.
$}
${
@@ -7102,7 +7115,7 @@ _praeclarum theorema_ (splendid theorem). (Contributed by NM,
$( Syllogism combined with contraction. (Contributed by Jeff Hankins,
1-Aug-2009.) $)
syl12anc $p |- ( ph -> ta ) $=
- ( wa jca32 syl ) ABCDJJEABCDFGHKIL $.
+ ( wa jca syl2anc ) ABCDJEFACDGHKIL $.
$}
${
@@ -7110,7 +7123,7 @@ _praeclarum theorema_ (splendid theorem). (Contributed by NM,
$( Syllogism combined with contraction. (Contributed by Jeff Hankins,
1-Aug-2009.) $)
syl21anc $p |- ( ph -> ta ) $=
- ( wa jca31 syl ) ABCJDJEABCDFGHKIL $.
+ ( wa jca syl2anc ) ABCJDEABCFGKHIL $.
$}
${
@@ -7134,7 +7147,7 @@ _praeclarum theorema_ (splendid theorem). (Contributed by NM,
unification theorem uses left-nested conjunction. (Contributed by Alan
Sare, 17-Oct-2017.) $)
syl1111anc $p |- ( ph -> et ) $=
- ( wi exp41 syl3c mpd ) AEFJABCDEFLGHIBCDEFKMNO $.
+ ( wa jca syl21anc ) ABCLDEFABCGHMIJKN $.
$}
${
@@ -7145,7 +7158,7 @@ _praeclarum theorema_ (splendid theorem). (Contributed by NM,
mpsyl4anc.5 $e |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ta ) -> et ) $.
$( An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) $)
mpsyl4anc $p |- ( th -> et ) $=
- ( wi exp41 mp2 mpsyl ) CDEFIJABCEFLLGHABCEFKMNO $.
+ ( a1i syl1111anc ) DABCEFADGLBDHLCDILJKM $.
$}
$( Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM,
@@ -7170,8 +7183,8 @@ _praeclarum theorema_ (splendid theorem). (Contributed by NM,
19-Jan-2020.) $)
a2and $p |- ( ph -> ( ( ( ps /\ ch ) -> ta )
-> ( ( ps /\ rh ) -> th ) ) ) $=
- ( wa wi expd imdistand imp embantd ex com23 ) ABFIZBCIZEJZDAQSDJAQIREDAQR
- ABFCABFCHKLMAQEDJGMNOP $.
+ ( wa wi expd imdistand imim1 com3l syl6c com23 ) ABFIZBCIZEJZDAQEDJZRSDJG
+ ABFCABFCHKLSTRDREDMNOP $.
$}
${
@@ -7836,7 +7849,7 @@ an illustration of the conservativity of definitions (definitions do not
$( Theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by NM,
3-Jan-2005.) $)
pm5.14 $p |- ( ( ph -> ps ) \/ ( ps -> ch ) ) $=
- ( wi wn conax1 pm2.21d orri ) ABDZBCDIEBCABFGH $.
+ ( wi pm2.521g orri ) ABDBCDABCEF $.
$( Theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by NM,
3-Jan-2005.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) $)
@@ -16379,7 +16392,6 @@ modal logic (the other standard formulation being ~ extru ). Note: This
( spvv mpg ) ABCABCDEGFH $.
$}
-
${
$d x y $.
$( Version of ~ equs4 with a disjoint variable condition, which requires
@@ -18388,8 +18400,15 @@ Converse of the inference rule of (universal) generalization ~ ax-gen .
$( Consequence of the definition of not-free. (Contributed by Mario
Carneiro, 26-Sep-2016.) ~ df-nf changed. (Revised by Wolf Lammen,
- 11-Sep-2021.) $)
+ 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.) $)
nf5r $p |- ( F/ x ph -> ( ph -> A. x ph ) ) $=
+ ( wex wnf wal 19.8a id nfrd syl5 ) AABCABDZABEABFJABJGHI $.
+
+ $( Obsolete version of ~ nfrd as of 23-Nov-2023. (Contributed by Mario
+ Carneiro, 26-Sep-2016.) ~ df-nf changed. (Revised by Wolf Lammen,
+ 11-Sep-2021.) (Proof modification is discouraged.)
+ (New usage is discouraged.) $)
+ nf5rOLD $p |- ( F/ x ph -> ( ph -> A. x ph ) ) $=
( wex wnf wal 19.8a wi df-nf biimpi syl5 ) AABCZABDZABEZABFLKMGABHIJ $.
${
@@ -32993,13 +33012,6 @@ Such interpretation is rarely needed (see also ~ df-ral ). (Contributed
elabg $p |- ( A e. V -> ( A e. { x | ph } <-> ps ) ) $=
( cv cab wcel wb nfab1 nfel2 nfv nfbi wceq eleq1 bibi12d abid vtoclg1f )
CGZACHZIZAJDUAIZBJCDEUCBCCDUAACKLBCMNTDOUBUCABTDUAPFQACRS $.
-
- $( Obsolete version of ~ elabg as of 23-Nov-2022. Membership in a class
- abstraction, using implicit substitution. Compare Theorem 6.13 of
- [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
- (Proof modification is discouraged.) (New usage is discouraged.) $)
- elabgOLD $p |- ( A e. V -> ( A e. { x | ph } <-> ps ) ) $=
- ( nfcv nfv elabgf ) ABCDECDGBCHFI $.
$}
${
@@ -33154,13 +33166,6 @@ Such interpretation is rarely needed (see also ~ df-ral ). (Contributed
cv ) DACEGZHDIHZDEHZBJZDTKUBUABDEKLCSZEHZAJUCCDTIUDDMUEUBABUDDENFOACEPQR
$.
- $( Obsolete version of ~ elrab as of 23-Nov-2022. Membership in a
- restricted class abstraction, using implicit substitution. (Contributed
- by NM, 21-May-1999.) (Proof modification is discouraged.)
- (New usage is discouraged.) $)
- elrabOLD $p |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) $=
- ( nfcv nfv elrabf ) ABCDECDGCEGBCHFI $.
-
$( Membership in a restricted class abstraction, using implicit
substitution. (Contributed by NM, 5-Oct-2006.) $)
elrab3 $p |- ( A e. B -> ( A e. { x e. B | ph } <-> ps ) ) $=
@@ -36403,7 +36408,7 @@ technically classes despite morally (and provably) being sets, like ` 1 `
eqsstr3.2 $e |- B C_ C $.
$( Substitution of equality into a subclass relationship. (Contributed by
NM, 19-Oct-1999.) $)
- eqsstr3i $p |- A C_ C $=
+ eqsstrri $p |- A C_ C $=
( eqcomi eqsstri ) ABCBADFEG $.
$}
@@ -36435,11 +36440,11 @@ technically classes despite morally (and provably) being sets, like ` 1 `
$}
${
- eqsstr3d.1 $e |- ( ph -> B = A ) $.
- eqsstr3d.2 $e |- ( ph -> B C_ C ) $.
+ eqsstrrd.1 $e |- ( ph -> B = A ) $.
+ eqsstrrd.2 $e |- ( ph -> B C_ C ) $.
$( Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.) $)
- eqsstr3d $p |- ( ph -> A C_ C ) $=
+ eqsstrrd $p |- ( ph -> A C_ C ) $=
( eqcomd eqsstrd ) ABCDACBEGFH $.
$}
@@ -36571,11 +36576,11 @@ technically classes despite morally (and provably) being sets, like ` 1 `
$}
${
- syl5sseqr.1 $e |- B C_ A $.
- syl5sseqr.2 $e |- ( ph -> C = A ) $.
+ sseqtrrid.1 $e |- B C_ A $.
+ sseqtrrid.2 $e |- ( ph -> C = A ) $.
$( Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.) $)
- syl5sseqr $p |- ( ph -> B C_ C ) $=
+ sseqtrrid $p |- ( ph -> B C_ C ) $=
( wss a1i sseqtr4d ) ACBDCBGAEHFI $.
$}
@@ -37945,7 +37950,7 @@ technically classes despite morally (and provably) being sets, like ` 1 `
Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM,
27-Apr-1994.) $)
inss2 $p |- ( A i^i B ) C_ B $=
- ( cin incom inss1 eqsstr3i ) ABCBACBBADBAEF $.
+ ( cin incom inss1 eqsstrri ) ABCBACBBADBAEF $.
${
$d x A $. $d x B $. $d x C $.
@@ -43942,7 +43947,7 @@ either the empty set or a singleton ( ~ uniintsn ). (Contributed by NM,
30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
unissint $p |- ( U. A C_ |^| A <-> ( A = (/) \/ U. A = |^| A ) ) $=
( cuni cint wss c0 wo wn wa simpl wne df-ne intssuni sylbir adantl eqssd ex
- wceq orrd cvv ssv int0 sseqtr4i inteq syl5sseqr eqimss jaoi impbii ) ABZACZ
+ wceq orrd cvv ssv int0 sseqtr4i inteq sseqtrrid eqimss jaoi impbii ) ABZACZ
DZAEQZUHUIQZFUJUKULUJUKGZULUJUMHUHUIUJUMIUMUIUHDZUJUMAEJUNAEKALMNOPRUKUJULU
KECZUHUIUHSUOUHTUAUBAEUCUDUHUIUEUFUG $.
@@ -47505,8 +47510,18 @@ holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc]
$( In the Separation Scheme ~ zfauscl , we require that ` y ` not occur in
` ph ` (which can be generalized to "not be free in"). Here we show
special cases of ` A ` and ` ph ` that result in a contradiction if that
- requirement is not met. (Contributed by NM, 8-Feb-2006.) $)
+ requirement is not met. (Contributed by NM, 8-Feb-2006.) (Proof
+ shortened by BJ, 18-Nov-2023.) $)
notzfaus $p |- -. E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) $=
+ ( cv wcel wa wb wal wn wex c0 wne csn 0ex snnz eqnetri mpbi ibar syl5rbbr
+ n0 pm5.19 bibi2d mtbiri eximii exnal nex ) BGZCGHZUJDHZAIZJZBKZCUNLZBMUOL
+ ULUPBDNOULBMDNPNENQRSBDUCTULUNUKUKLZJUKUDULUMUQUKUQAULUMFULAUAUBUEUFUGUNB
+ UHTUI $.
+
+ $( Obsolete proof of ~ notzfaus as of 18-Nov-2023. (Contributed by NM,
+ 8-Feb-2006.) (Proof modification is discouraged.)
+ (New usage is discouraged.) $)
+ notzfausOLD $p |- -. E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) $=
( cv wcel wa wb wal wn wex c0 wne csn 0ex snnz eqnetri mpbi n0 biimt iman
wi anbi2i xchbinxr syl6bb xor3 sylibr eximii exnal nex ) BGZCGHZUMDHZAIZJ
ZBKZCUQLZBMURLUOUSBDNOUOBMDNPNENQRSBDUATUOUNUPLZJUSUOUNUOUNUDZUTUOUNUBVAU
@@ -52956,7 +52971,7 @@ Contrast with domain (defined in ~ df-dm ). For alternate definitions,
$( The domain of the universe is the universe. (Contributed by NM,
8-Aug-2003.) $)
dmv $p |- dom _V = _V $=
- ( cvv cdm ssv cid dmi wss dmss ax-mp eqsstr3i eqssi ) ABZAKCADBZKEDAFLKFD
+ ( cvv cdm ssv cid dmi wss dmss ax-mp eqsstrri eqssi ) ABZAKCADBZKEDAFLKFD
CDAGHIJ $.
$}
@@ -53873,7 +53888,7 @@ the restriction (of the relation) to the singleton containing this
$( A diagonal set as a subset of a Cartesian product. (Contributed by
Thierry Arnoux, 29-Dec-2019.) (Proof shortened by BJ, 9-Sep-2022.) $)
idssxp $p |- ( _I |` A ) C_ ( A X. A ) $=
- ( cid cres cxp cin idinxpres inss2 eqsstr3i ) BACBAADZEIAFBIGH $.
+ ( cid cres cxp cin idinxpres inss2 eqsstrri ) BACBAADZEIAFBIGH $.
${
$d A x y $.
@@ -54866,7 +54881,7 @@ the restriction (of the relation) to the singleton containing this
relationship for it components. (Contributed by NM, 17-Dec-2008.) $)
ssxpb $p |- ( ( A X. B ) =/= (/) -> ( ( A X. B ) C_ ( C X. D ) <->
( A C_ C /\ B C_ D ) ) ) $=
- ( cxp c0 wne wss wa cdm wceq xpnz dmxp adantl sylbir adantr eqsstr3d syl6ss
+ ( cxp c0 wne wss wa cdm wceq xpnz dmxp adantl sylbir adantr eqsstrrd syl6ss
dmss crn dmxpss rnxp rnss rnxpss jca ex xpss12 impbid1 ) ABEZFGZUICDEZHZACH
ZBDHZIZUJULUOUJULIZUMUNUPAUKJZCUPAUIJZUQUJURAKZULUJAFGZBFGZIZUSABLZVAUSUTAB
MNOPULURUQHUJUIUKSNQCDUARUPBUKTZDUPBUITZVDUJVEBKZULUJVBVFVCUTVFVAABUBPOPULV
@@ -59996,7 +60011,7 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ).
( F |` B ) : B -onto-> D ) ->
( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) ) $=
( ccnv wfun cres wfo cdif cima wf1o wceq wb ax-mp crn df-ima wa forn syl5eq
- w3a cdm wss fofun difss fof fdmd syl5sseqr fores syl2anc cin resres reseq2i
+ w3a cdm wss fofun difss fof fdmd sseqtrrid fores syl2anc cin resres reseq2i
indif eqtri foeq1 rneqi 3eqtr4i foeq3 bitri funres11 dff1o3 biimpri syl2anr
sylib 3adant3 anim12i imadif difeq12 sylan9eq sylan2 3impb f1oeq3d mpbid )
EFGZACEAHZIZBDEBHZIZUAZABJZEWAKZEWAHZLZWACDJZWCLVOVQWDVSVQWAWBWCIZWCFGZWDVO
@@ -62051,7 +62066,7 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ).
(Contributed by FL, 25-Jan-2007.) $)
fimacnv $p |- ( F : A --> B -> ( `' F " B ) = A ) $=
( wf ccnv cima crn imassrn cdm dfdm4 fdm ssid syl6eqss syl5eqssr syl5ss wss
- fimass wfun wb ffun syl5sseqr funimass3 syl2anc mpbid eqssd ) ABCDZCEZBFZAU
+ fimass wfun wb ffun sseqtrrid funimass3 syl2anc mpbid eqssd ) ABCDZCEZBFZAU
FUHUGGZAUGBHUFUICIZACJUFUJAAABCKZALZMNOUFCAFBPZAUHPZABCAQUFCRAUJPUMUNSABCTU
FAAUJULUKUAABCUBUCUDUE $.
@@ -63129,7 +63144,7 @@ from the cartesian product of two singletons onto a singleton (case where
Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro,
22-Dec-2016.) $)
fnsnr $p |- ( F Fn { A } -> ( B e. F -> B = <. A , ( F ` A ) >. ) ) $=
- ( csn wfn wcel cfv cop wceq cres fnresdm wfun fnfun funressn eqsstr3d sseld
+ ( csn wfn wcel cfv cop wceq cres fnresdm wfun fnfun funressn eqsstrrd sseld
wss syl elsni syl6 ) CADZEZBCFBAACGHZDZFBUCIUBCUDBUBCCUAJZUDUACKUBCLUEUDQUA
CMACNROPBUCST $.
@@ -69099,7 +69114,7 @@ associative structure (such as a group). (Contributed by NM,
ofrfval $p |- ( ph -> ( F oR R G <-> A. x e. S C R D ) ) $=
( wcel vf vg cofr wbr cv cfv cdm cin wral cvv wb fnex syl2anc wceq dmeq
wfn wa ineqan12d fveq1 breqan12d raleqbidv df-ofr brabga ineq12d syl6eq
- fndmd raleqdv inss1 eqsstr3i sseli sylan2 inss2 breq12d ralbidva 3bitrd
+ fndmd raleqdv inss1 eqsstrri sseli sylan2 inss2 breq12d ralbidva 3bitrd
) AIJGUCZUDZBUEZIUFZVRJUFZGUDZBIUGZJUGZUHZUIZWABHUIEFGUDZBHUIAIUJTZJUJT
ZVQWEUKAICUPCKTWGMOCKIULUMAJDUPDLTWHNPDLJULUMVRUAUEZUFZVRUBUEZUFZGUDZBW
IUGZWKUGZUHZUIWEUAUBIJVPUJUJWIIUNZWKJUNZUQWMWABWPWDWQWRWNWBWOWCWIIUOWKJ
@@ -69115,7 +69130,7 @@ associative structure (such as a group). (Contributed by NM,
Carneiro, 20-Jul-2014.) $)
ofval $p |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) ) $=
( cfv vx wcel wa cof co cv cmpt wceq eqidd offval fveq1d adantr oveq12d
- fveq2 eqid ovex fvmpt adantl inss1 eqsstr3i sseli sylan2 inss2 3eqtrd
+ fveq2 eqid ovex fvmpt adantl inss1 eqsstrri sseli sylan2 inss2 3eqtrd
cin ) ALGUBZUCZLHIFUDUEZTZLUAGUAUFZHTZVJITZFUEZUGZTZLHTZLITZFUEZDEFUEAV
IVOUHVFALVHVNAUABCVKVLFGHIJKMNOPQAVJBUBUCVKUIAVJCUBUCVLUIUJUKULVFVOVRUH
AUALVMVRGVNVJLUHVKVPVLVQFVJLHUNVJLIUNUMVNUOVPVQFUPUQURVGVPDVQEFVFALBUBV
@@ -69125,7 +69140,7 @@ associative structure (such as a group). (Contributed by NM,
Carneiro, 28-Jul-2014.) $)
ofrval $p |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D ) $=
( wcel vx cofr wbr w3a cfv wa cv wral eqidd ofrfval biimpa wceq breq12d
- wi fveq2 rspccv syl 3impia simp1 cin inss1 eqsstr3i simp3 syl2anc inss2
+ wi fveq2 rspccv syl 3impia simp1 cin inss1 eqsstrri simp3 syl2anc inss2
sseldi 3brtr3d ) AHIFUBUCZLGTZUDZLHUEZLIUEZDEFAVHVIVKVLFUCZAVHUFUAUGZHU
EZVNIUEZFUCZUAGUHZVIVMUNAVHVRAUABCVOVPFGHIJKMNOPQAVNBTUFVOUIAVNCTUFVPUI
UJUKVQVMUALGVNLULVOVKVPVLFVNLHUOVNLIUOUMUPUQURVJALBTVKDULAVHVIUSZVJGBLG
@@ -69194,7 +69209,7 @@ associative structure (such as a group). (Contributed by NM,
$( The function operation produces a function. (Contributed by Mario
Carneiro, 20-Jul-2014.) $)
off $p |- ( ph -> ( F oF R G ) : C --> U ) $=
- ( vz cv cfv co cof ffnd wcel wa eqidd offval wral wf inss1 eqsstr3i sseli
+ ( vz cv cfv co cof ffnd wcel wa eqidd offval wral wf inss1 eqsstrri sseli
cin ffvelrn syl2an inss2 ralrimivva adantr ovrspc2v syl21anc fmpt3d ) AUA
FUAUBZKUCZVELUCZGUDZJKLGUEUDAUADEVFVGGFKLMNADHKPUFAEILQUFRSTAVEDUGZUHVFUI
AVEEUGZUHVGUIUJAVEFUGZUHVFHUGZVGIUGZBUBCUBGUDJUGZCIUKBHUKZVHJUGADHKULVIVL
@@ -69213,7 +69228,7 @@ associative structure (such as a group). (Contributed by NM,
of the operation itself. (Contributed by Mario Carneiro,
15-Sep-2014.) $)
ofres $p |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) ) $=
- ( vx co cfv wcel cvv wfn cof cv cmpt cres eqidd offval wss inss1 eqsstr3i
+ ( vx co cfv wcel cvv wfn cof cv cmpt cres eqidd offval wss inss1 eqsstrri
wa cin fnssres sylancl inss2 ssexg sylancr inidm wceq fvres adantl eqtr4d
) AFGEUAZPODOUBZFQZVCGQZEPUCFDUDZGDUDZVBPAOBCVDVEEDFGHIJKLMNAVCBRUJVDUEAV
CCRUJVEUEUFAODDVDVEEDVFVGSSAFBTDBUGZVFDTJDBCUKZBNBCUHUIZBDFULUMAGCTDCUGVG
@@ -69288,7 +69303,7 @@ associative structure (such as a group). (Contributed by NM,
ofco $p |- ( ph ->
( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) ) $=
( cfv vx vy cof co ccom cv cmpt ffvelrnda feqmptd wcel eqidd offval fveq2
- wa oveq12d fmptco wfn wf wss cin inss1 eqsstr3i fss sylancl fnfco syl2anc
+ wa oveq12d fmptco wfn wf wss cin inss1 eqsstrri fss sylancl fnfco syl2anc
wceq inss2 inidm ffnd fvco2 sylan eqtr4d ) AGHFUCZUDZIUEUAEUAUFZITZGTZVQH
TZFUDZUGGIUEZHIUEZVNUDAUAUBEDVQUBUFZGTZWCHTZFUDVTIVOAEDVPIOUHAUAEDIOUIAUB
BCWDWEFDGHJKMNPQSAWCBUJUNWDUKAWCCUJUNWEUKULWCVQVGWDVRWEVSFWCVQGUMWCVQHUMU
@@ -74888,7 +74903,7 @@ any sets (which usually are functions) and any element (even not
by AV, 28-May-2019.) $)
suppssof1 $p |- ( ph -> ( ( A oF O B ) supp Z ) C_ L ) $=
( vx cof co csupp cv cfv cmpt ffnd inidm wcel eqidd offval oveq1d feqmptd
- wa cvv eqsstr3d fvexd ffvelrnda suppssov1 eqsstrd ) ACDIUAUBZMUCUBTETUDZC
+ wa cvv eqsstrrd fvexd ffvelrnda suppssov1 eqsstrd ) ACDIUAUBZMUCUBTETUDZC
UEZVBDUEZIUBUFZMUCUBHAVAVEMUCATEEVCVDIECDKKAEJCPUGAEFDQUGRREUHAVBEUIUNZVC
UJVFVDUJUKULATBVCVDEFHIUOGLMATEVCUFZLUCUBCLUCUBHACVGLUCATEJCPUMULNUPOVFVB
CUQAEFVBDQURSUSUT $.
@@ -75527,7 +75542,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
( F o. ( x e. ( ( _V X. _V ) u. { (/) } ) |-> U. `' { x } ) ) $=
( vy vz vw cvv csn cun cv ccnv cuni wss wceq cop wcel wbr wa wex vex wb
ctpos cxp cmpt ccom cdm df-tpos cres wrel relcnv df-rel mpbi unss1 resmpt
- c0 mp2b resss eqsstr3i coss2 ax-mp eqsstri relco opelco eleq1 sneq cnveqd
+ c0 mp2b resss eqsstrri coss2 ax-mp eqsstri relco opelco eleq1 sneq cnveqd
unieqd eqeq2d anbi12d anbi2d df-mpt brab wi simplr breldm adantl eqeltrrd
eqeq1 opswap eleq1i opelcnv bitr4i eleq1d bibi12d mpbiri exlimivv biimpcd
elvv sylbi elun1 syl6 syl elun2 a1i wo simpll sylib mpjaod simpr eqbrtrrd
@@ -76689,7 +76704,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) $)
smores2 $p |- ( ( Smo F /\ Ord A ) -> Smo ( F |` A ) ) $=
( vy vx wsmo word wa cdm con0 wf cv cfv wcel wral crn wss dfsmo2 syl wceq
- adantr cres wfun simp1bi ffund funres funfnd cima df-ima imassrn eqsstr3i
+ adantr cres wfun simp1bi ffund funres funfnd cima df-ima imassrn eqsstrri
wfn frnd syl5ss df-f sylanbrc smodm cin ordin wb dmres ordeq ax-mp sylibr
ancoms sylan resss simp3bi ssralv mpsyl wel wi ordtr1 inss1 eqsstri sseli
dmss syl6 expcomd imp31 fvresd ad2antlr eleq12d ralbidva mpbird syl3anbrc
@@ -77043,7 +77058,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
( ( x g u /\ x h v ) -> u = v ) ) $=
( vz va vw cv wcel cfv wceq wral wa con0 wfn cres wrex wi tfrlem3a reeanv
wbr vex w3a cin fveq2 eqeq12d onin 3ad2ant1 wfun cdm wss simp2ll syl fndm
- fnfun inss1 syl5sseqr simp2rl inss2 simp2lr ssralv simp2rr tfrlem1 simp3l
+ fnfun inss1 sseqtrrid simp2rl inss2 simp2lr ssralv simp2rr tfrlem1 simp3l
jca mpsyl fnbr syl2anc simp3r rspcdva funbrfv sylc 3eqtr3d 3exp rexlimivv
elind sylbir syl2anb ) GNZEOWEKNZUAZLNZWEPZWEWHUBIPQZLWFRZSZKTUCZHNZMNZUA
ZWHWNPZWNWHUBIPQZLWORZSZMTUCZANZDNZWEUGZXBCNZWNUGZSZXCXEQZUDZWNEOABKLEFIW
@@ -78841,7 +78856,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
(Contributed by NM, 6-Dec-2004.) $)
oaword1 $p |- ( ( A e. On /\ B e. On ) -> A C_ ( A +o B ) ) $=
( con0 wcel wa c0 coa co wceq oa0 adantr wss 0ss 0elon oaword 3com13 mp3an3
- wb mpbii eqsstr3d ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBAJKUCFBLZUDUELZBMUAUBFCDZU
+ wb mpbii eqsstrrd ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBAJKUCFBLZUDUELZBMUAUBFCDZU
FUGRZNUHUBUAUIFBAOPQST $.
$( An ordinal is less than or equal to its sum with another. Theorem 21 of
@@ -79215,7 +79230,7 @@ A C_ ( A .o B ) ) $=
omword2 $p |- ( ( ( A e. On /\ B e. On ) /\ (/) e. B ) ->
A C_ ( B .o A ) ) $=
( con0 wcel wa c0 c1o comu co wceq om1r ad2antrr wss eloni ordgt0ge1 biimpa
- word sylan adantll wi 1on omwordri mp3an1 ancoms adantr mpd eqsstr3d ) ACDZ
+ word sylan adantll wi 1on omwordri mp3an1 ancoms adantr mpd eqsstrrd ) ACDZ
BCDZEZFBDZEZAGAHIZBAHIZUHUMAJUIUKAKLULGBMZUMUNMZUIUKUOUHUIBQZUKUOBNUQUKUOBO
PRSUJUOUPTZUKUIUHURGCDUIUHURUAGBAUBUCUDUEUFUG $.
@@ -79904,7 +79919,7 @@ A C_ ( B .o A ) ) $=
( con0 wcel co va vd ve c2o cdif c1o wa coe comu coa wceq w3a wb wss csuc
eldifi adantr ad2antrr simprl oecl syl2anc om1 syl c0 csn df1o2 wne dif1o
simprbi ad2antll onelon on0eln0 mpbird snssd syl5eqss 1on omwordi syl3anc
- wi a1i eqsstr3d omcl simplrl oaword1 simplrr sseqtrd sstrd oeeulem simp3d
+ wi a1i eqsstrrd omcl simplrl oaword1 simplrr sseqtrd sstrd oeeulem simp3d
mpd simp1d suceloni ontr2 mp2and simplll oeord simp2d word eloni ordsucss
onsssuc sylc dif20el syl21anc omword syl31anc mpbid oaord eqeltrrd oa1suc
oen0 odi oveq2d 3eqtr3d eleqtrrd sseldd oesuc ad2antrl jca eqeltrd simprr
@@ -80301,7 +80316,7 @@ general ordinal versions of these theorems (in this case ~ oa0r ) so
(Revised by Mario Carneiro, 15-Nov-2014.) $)
nnaword1 $p |- ( ( A e. _om /\ B e. _om ) -> A C_ ( A +o B ) ) $=
( com wcel wa c0 coa co wceq adantr wss 0ss wb peano1 nnaword 3com13 mp3an3
- nna0 mpbii eqsstr3d ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBARJUCFBKZUDUEKZBLUAUBFCD
+ nna0 mpbii eqsstrrd ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBARJUCFBKZUDUEKZBLUAUBFCD
ZUFUGMZNUHUBUAUIFBAOPQST $.
$( Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) $)
@@ -85208,7 +85223,7 @@ equivalent to the law of the excluded middle (LEM), and in ILE the LEM
cid cuni cpw csn cxp c1st f1f1orn 3ad2ant1 simp2 rnexg syl pwexg 1stconst
uniexg cen wbr wa difexg 3ad2ant3 disjen syl2anc simpld disjdif a1i f1oun
3syl syl22anc undif2 wf frnd ssequn1 sylib syl5eq f1oeq3d mpbid f1ocnv wb
- f1f f1oeq1 sylibr wfo f1ofo forn mpbird ssun1 syl5sseqr cores dmres f1odm
+ f1f f1oeq1 sylibr wfo f1ofo forn mpbird ssun1 sseqtrrid cores dmres f1odm
ssid ineq2d syl6eq wrel relres reldm0 uneq2d cnvun eqtri reseq1i resundir
cdm df-rn reseq2i relcnv resdm uneq1i 3eqtrri un0 3eqtr3g coeq1d f1cocnv1
eqtrd syl5eqr 3jca ) ABCUAZAEHZBFHZUBZBDIZDJZAYBKDCLZUDAMZNYAYCBABCIZUCZA
@@ -88456,7 +88471,7 @@ intersections of elements of the argument (see ~ elfi2 ). (Contributed
ssexi cmpt mpoeq12 anidms rneqd frsucmpt sylancl sylib ex ralrimiva sseq2
a1i imbi12d sseq1 rspc2v syl3c sseld simprr onsseleq rzal biantrud bitr3d
syl2an2 ssfii csn id inidm abeq2i ssriv simpl inss1 cbvmpov cbvmptv eqtri
- rdgeq1 reseq1i syl5sseqr sstr2 syl5com ralimdv ralsn jctird df-suc raleqi
+ rdgeq1 reseq1i sseqtrrid sstr2 syl5com ralimdv ralsn jctird df-suc raleqi
ralunb bitri syl6ibr fiin ss2ralv mpi jctild expimpd finds2 impcom simprd
frnd r19.21bi eqimss jaod sylbid ssun1 ssun2 peano2 ssiun2s 3syl 2rexbidv
eqeq1 elab syl6eleqr eleqtrrd sseldd syl6eleq syl2and rexlimdvva sylan2br
@@ -92074,7 +92089,7 @@ of all inductive sets (which is the smallest inductive set, since
ad2antrl pm2.27 cantnfvalf ffvelrni ad2antlr ad3antrrr suppssdm fssdm
coa sucidg sseldd oif onelon oecl oaword2 cantnfsuc ad4ant13 sseqtr4d
simpr omcl expcom adantrr syld expr fveq2d f1ocnvfv2 ad2antrr eqtr3d
- oveq2d oveq12d oaword1 eqsstr3d a1dd jaod syl5bi expimpd com23 finds2
+ oveq2d oveq12d oaword1 eqsstrrd a1dd jaod syl5bi expimpd com23 finds2
vtoclga mpcom mpd cantnfval om0 0ss syl6eqss pm2.61ne ) ACEUCUDZEHUEZ
UFUDZHCDUGUDUEZRUUSSUFUDZUVBRUUTSUUTSUHUVAUVCUVBUUTSUUSUFUIUJAUUTSUKZ
ULZUVAIUMZJUEZUVBUVEEIUNZUEZUVFTZUVAUVGRZUVEHSUOUDZUVFEUVHUVEUVFUVLIU
@@ -92351,7 +92366,7 @@ of all inductive sets (which is the smallest inductive set, since
ffvelrnd oicl ordelon ontri1 sucidg eleqtrrd isorel 3bitr3g f1ocnvfv2
nnon syl12anc mtbid onss sstrd imnan onelon elsni notbid eldifn velsn
elun2 sylbir nsyl ssun1 sscon sseli sylan2 elun mpjaod ioran sylanbrc
- ordn2lp ordtri3 mpbird oveq2d nnord sssucid syl5sseqr wfo f1ofo foima
+ ordn2lp ordtri3 mpbird oveq2d nnord sssucid sseqtrrid wfo f1ofo foima
cres ffn fnsnfv sneqd difeq12d cin disjsn disj3 difun2 syl6eqr df-suc
syl6eq difeq1d eqtr4d imaeq2d dff1o3 simprbi imadif ssneldd c1o dif1o
wfun mpbiran bicomd anbi2d sylbird mpand syl5bir necon1bd mpd fveqeq2
@@ -92606,7 +92621,7 @@ of all inductive sets (which is the smallest inductive set, since
mpbir2and cdif wn eldifn adantl iffalsed suppss2 ifeq1da syl3anc sseldd
fveq2 eleq1w ifbieq1d eqid fvex fvmpt ifeq2d syl6reqr mpteq2ia cantnfp1
ifex eqtr3d ifor omsuc word eloni simp2d ordsucss sylc suceloni omwordi
- mpd eqsstr3d cantnflt2 oaord eqeltrd ) AFOUGUHZOLUIZUJUHZONUKZUIZULMUIZ
+ mpd eqsstrrd cantnflt2 oaord eqeltrd ) AFOUGUHZOLUIZUJUHZONUKZUIZULMUIZ
BGBUMZOUNZUVLKUIZUOUPZUQZFGURUHZUIZAUVHUVHUVJMUIZVCUHZUVKAUVHUSUTZUVSUS
UTZUVHUVTUNAUVFUSUTZUVGUSUTZUWAAFUSUTZOUSUTZUWCRAGUSUTZOGUTZUWFSAUWHOKU
IZUVGUTZOEUMZUTUWKKUIUWKLUIVAVDEGVBZABCDEFGHIKLOPQRSTUAUBUCUDVEZVFZGOVG
@@ -93430,7 +93445,7 @@ a Cantor normal form (and injectivity, together with coherence
( vv vu wss cv com cfv wrex c0 wcel wceq fveq2 wtr wa wal ciun peano1 cvv
wi cuni cun cmpt crdg cres fveq1i fr0g ax-mp eqtr2i eqimssi sseq2d rspcev
mp2an ssiun sseqtr4i dftr2 eliun anbi2i r19.42v bitr4i elunii ssun2 uniex
- csuc fvex unex unieq uneq12d frsucmpt2 mpan2 syl5sseqr sseld syl5 reximia
+ csuc fvex unex unieq uneq12d frsucmpt2 mpan2 sseqtrrid sseld syl5 reximia
weq id sylbi peano2 eleq2d syl rexlimiv cbvrexv sylibr ax-gen mpgbir treq
ex mpbir wral sseq1d eqtri sseq1i biimpri adantr uniss df-tr sstr2 syl5bi
wb anc2li unss syl6ib biimprd syl9r com23 adantld finds2 ralrimiv cbviunv
@@ -93587,7 +93602,7 @@ a Cantor normal form (and injectivity, together with coherence
$( Defining property of the transitive closure function: it contains its
argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.) $)
tcid $p |- ( A e. V -> A C_ ( TC ` A ) ) $=
- ( vx wcel cv wss wtr wa cab cint ctc cfv ssmin tcvalg syl5sseqr ) ABDACEZ
+ ( vx wcel cv wss wtr wa cab cint ctc cfv ssmin tcvalg sseqtrrid ) ABDACEZ
FPGZHCIJAAKLQCAMCABNO $.
$( Defining property of the transitive closure function: it is transitive.
@@ -93874,7 +93889,7 @@ of the previous layer (and the union of previous layers when the
( vy cr1 wcel cfv wa con0 wss word ax-mp syl wi wceq eleq1 eleq2d imbi12d
fveq2 wb syl2anc vx cdm csuc simpl wlim wfun r1funlim simpri limord sseli
ordsson onelon sylan suceloni ordsucss imp cv cpw fvex pwid limsuc r1sucg
- eloni sylbir syl5eleqr a1i wtr r1tr dftr4 mpbi syl5sseqr a2i syl5bir wral
+ eloni sylbir syl5eleqr a1i wtr r1tr dftr4 mpbi sseqtrrid a2i syl5bir wral
sseld ciun wrex simprl simplr sucelon sylibr ad2antrr mpbird mpbid simprr
ordelsuc ordtr1 rspcev eliun simpll r1limg eleqtrrd expr a1d tfindsg impr
syl22anc ex ) BDUBZEZABEZADFZBDFZEZWTXAGZBHEZAUCZHEZXGBIZWTXDXEWTXFWTXAUD
@@ -93928,7 +93943,7 @@ of the previous layer (and the union of previous layers when the
value at the successor. JFM CLASSES1 Th. 39. (Contributed by FL,
20-Apr-2011.) $)
r1sssuc $p |- ( A e. On -> ( R1 ` A ) C_ ( R1 ` suc A ) ) $=
- ( con0 wcel cr1 cfv cpw csuc wtr wss r1tr dftr4 mpbi r1suc syl5sseqr ) ABCA
+ ( con0 wcel cr1 cfv cpw csuc wtr wss r1tr dftr4 mpbi r1suc sseqtrrid ) ABCA
DEZFZOAGDEOHOPIAJOKLAMN $.
${
@@ -93974,7 +93989,7 @@ of the previous layer (and the union of previous layers when the
nfv biimpac wfun r1funlim simpri limsuc sylibr r1sucg eqtrd vex syl5eleqr
sucid ssiun2 ex a1d rexlimd imp r1limg wral wtr r1tr dftr4 mpbi ralrimivw
ss2iun adantrl w3o word limord ordsson sseli onzsl sylib mpjao3dan ordtr1
- ancoms ordelord mpan adantr ordelsuc syl2anc mpbid simpl r1ord3g eqsstr3d
+ ancoms ordelord mpan adantr ordelsuc syl2anc mpbid simpl r1ord3g eqsstrrd
mpd ralrimiva iunss eqssd ) BCUAZDZBCEZABAUBZCEZUCZFZXDBGHZXEXIIZBXFUDZHZ
AJUEZBUFDZBRZKZXDXJKZXEGXIXRXEGCEGXRBGCXDXJLMUGUHGXIIXRXIUINOXDXNXKXDXMXK
AJXDAUNAXEXIAXEUJABXHUKULXDXMXKPXFJDXDXMXKXDXMKZXEXHXIXSXEXLCEZXHXSBXLCXD
@@ -94497,7 +94512,7 @@ of the previous layer (and the union of previous layers when the
22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) $)
onwf $p |- On C_ U. ( R1 " On ) $=
( vx con0 cr1 cdm cima cuni wceq r1fnon fndm ax-mp cv wcel crnk rankonidlem
- wfn cfv simpld ssriv eqsstr3i ) BCDZCBEFZCBOTBGHBCIJATUAAKZTLUBUALUBMPUBGUB
+ wfn cfv simpld ssriv eqsstrri ) BCDZCBEFZCBOTBGHBCIJATUAAKZTLUBUALUBMPUBGUB
NQRS $.
${
@@ -94838,7 +94853,7 @@ suc suc ( ( rank ` A ) u. ( rank ` B ) ) $=
rankr1id $p |- ( A e. dom R1 <-> ( rank ` ( R1 ` A ) ) = A ) $=
( cr1 cdm wcel cfv crnk wceq wss ssid con0 cima cuni csuc cpw fvex r1sucg
wb pwid syl5eleqr r1elwf syl rankr1bg mpancom mpbii biimpi onssr1 rankssb
- rankonid sylc eqsstr3d eqssd id rankdmr1 syl6eqelr impbii ) ABCZDZABEZFEZ
+ rankonid sylc eqsstrrd eqssd id rankdmr1 syl6eqelr impbii ) ABCZDZABEZFEZ
AGZUQUSAUQURURHZUSAHZURIURBJKLDZUQVAVBQUQURAMZBEZDVCUQURURNVEURABORAPSURV
DTUAZURAUBUCUDUQAAFEZUSUQVGAGAUHUEUQVCAURHVGUSHVFAUFAURUGUIUJUKUTAUSUPUTU
LURUMUNUO $.
@@ -95050,7 +95065,7 @@ C_ suc suc suc ( rank ` ( A u. B ) ) $=
wlim limuni2 cun cv 0ellim n0i unieq uni0 syl6eq con3i 3syl rankon onsuci
elexi sucid ontri1 mp2an con2bii mpbi rankxpu sstr mpan2 reeanv wi simprl
mto simpr rankuni unieqi eqtri wne df-ne xpex rankeq0 notbii bitr2i sylib
- unixp syl fveq2d syl5reqr eqimss eqsstr3d adantrr limuni sseqtr4d cvv vex
+ unixp syl fveq2d syl5reqr eqimss eqsstrrd adantrr limuni sseqtr4d cvv vex
word onordi orduni ax-mp ordelsuc sylibr limsuc mpbid eqeltrd ordsucelsuc
onsucuni2 mpan ad2antll eleqtrd onsucssi ex a1d rexlimdvv syl5bir mtoi wo
ianor un00 animorl sylbir xpeq0 unex w3o ordzsl 3ori sylan orim12d syl5bi
@@ -96702,7 +96717,7 @@ several of their earlier lemmas available (which would otherwise be
dmex rnex cres imadmres ccnv cin crab inss2 cpr ssun1 cop elinel2 1st2nd2
wral syl elinel1 eqeltrrd opeldm sseldi ssun2 opelrn prssd ordunpr sseldd
rgen ssrab mpbir2an dmres fdmi ineq2i eqtri mptpreima 3sstr4i wfun funmpt
- wb resss dmss ax-mp funimass3 mp2an mpbir eqsstr3i ssexi fnwe epse vuniex
+ wb resss dmss ax-mp funimass3 mp2an mpbir eqsstrri ssexi fnwe epse vuniex
cuni cpw pwex xpex cab df-rab adantr elssuni adantl elpw jca unssad elxp6
unssbd sylanbrc abssi eqsstri fnse mptru ) JJUAZEUBZUUIEUCZKLUUJUUKLCDIUU
IJMFEAUUIAUDZNOZUULPOZUFZUEZECUDZUUIQZDUDZUUIQZKZUUQNOZUUQPOZUFZUUSNOZUUS
@@ -99498,7 +99513,7 @@ our Axiom of Choice (in the form of ~ ac2 ). The proof does not depend
dju1dif $p |- ( ( A e. V /\ B e. ( A |_| 1o ) )
-> ( ( A |_| 1o ) \ { B } ) ~~ A ) $=
( wcel c1o cdju wa csn cdif cop cen wbr cvv simpl 1oex djuex cxp 0ex wceq
- c0 sylancl simpr cun df1o2 xpeq2i xpsn eqtri ssun2 eqsstr3i opex snss mpbir
+ c0 sylancl simpr cun df1o2 xpeq2i xpsn eqtri ssun2 eqsstrri opex snss mpbir
wss df-dju eleqtrri a1i difsnen syl3anc difeq1i xp01disjl disj3 mpbi difun2
cin difeq2i 3eqtr2i eqtr4i xpsnen2g sylancr eqbrtrid entr syl2anc ) ACDZBAE
FZDZGZVNBHIZVNETJZHZIZKLZVTAKLVQAKLVPVNMDZVOVRVNDZWAVPVMEMDWBVMVONZOAECMPUA
@@ -99852,7 +99867,7 @@ Because we use a disjoint union for cardinal addition (as explained in the
unctb $p |- ( ( A ~<_ _om /\ B ~<_ _om ) -> ( A u. B ) ~<_ _om ) $=
( com cdom wbr wa cun cdju cvv wcel ctex undjudom sylancl domtr syl2anc cxp
omex wss c2o mp2an syl2an djudom1 sylan2 simpr djudom2 cen xpex xp2dju word
- ordom 2onn ordelss xpss1 ax-mp eqsstr3i ssdomg mp2 xpomen domentr ) ACDEZBC
+ ordom 2onn ordelss xpss1 ax-mp eqsstrri ssdomg mp2 xpomen domentr ) ACDEZBC
DEZFZABGZABHZDEZVDCDEZVCCDEUTAIJBIJZVEVAAKBKZABIILUAVBVDCCHZDEZVICDEZVFVBVD
CBHZDEZVLVIDEZVJVAUTVGVMVHACBIUBUCVBVACIJVNUTVAUDQBCCIUEMVDVLVINOVICCPZDEZV
OCUFEVKVOIJVIVORVPCCQQUGVISCPZVOCUHSCRZVQVORCUISCJVRUJUKCSULTSCCUMUNUOVIVOI
@@ -100025,7 +100040,7 @@ Because we use a disjoint union for cardinal addition (as explained in the
wf com co ccrd cdm w3a cv oveq2 breq2 anbi2d abbidv breq12d wne brrelex1i
cxp simpl2 brrelex2i xpcomeng simpl3 simpr mapdom3 numdom simpl1 infxpabs
syl3anc syl22anc entr ssenen relen abid2 elmapi fssxp wfun ffun vex ensym
- fundmen 3syl fdm breqtrd jca ss2abi eqsstr3i ssdomg domentr crn cmpt ovex
+ fundmen 3syl fdm breqtrd jca ss2abi eqsstrri ssdomg domentr crn cmpt ovex
mpisyl mptex rnex wrex wex wf1o ad2antll bren f1of adantl simplrl fssd wb
sylib elmapd ad2antrr mpbird wfo f1ofo eqcomd ex eximdv mpd df-rex sylibr
forn ss2abdv eqid rnmpt syl6sseqr mpsyl wfn wral rgenw mp1i fodomnum sylc
@@ -100220,7 +100235,7 @@ Because we use a disjoint union for cardinal addition (as explained in the
cv wn w3a simpr1 ackbij1lem3 simpr3 ackbij1lem1 syl6eqss csn ackbij1lem10
inss2 con0 ffvelrni nnon onpsssuc 4syl ackbij1lem14 psseq2d mpbird simpll
inss1 ackbij1lem11 sylancl cun ssun1 simpr2 ackbij1lem2 psssstrd sspsstrd
- syl5sseqr pssned necomd neneqd ) CHIUAJZKZDVOKZLZFUBZHKZVSCKZVSDKUCZUDZLZ
+ sseqtrrid pssned necomd neneqd ) CHIUAJZKZDVOKZLZFUBZHKZVSCKZVSDKUCZUDZLZ
CVSMZJZENZDWEJZENZWDWIWGWDWIWGWDWIVSENZWGWDVSVOKZWHVSOWIWJOWDVTWKVRVTWAWB
UEZVSUFPZWDWHDVSJZVSWDWBWHWNQVRVTWAWBUGVSDUHPDVSULUIABWHVSEGRSWDWJVSUJZEN
ZWGWDWJWPTWJWJMZTZWDWKWJHKWJUMKWRWMVOHVSEABEGUKUNWJUOWJUPUQWDWPWQWJWDVTWP
@@ -100439,7 +100454,7 @@ Because we use a disjoint union for cardinal addition (as explained in the
bitr4i df-rex funfn mpbi rdgdmlim limomss ax-mp fvelimab mp2an f1ofo forn
wfn 3syl eqsstrd rneq sseq1d syl5ibcom rexlimiv sylbi sselda exlimiv csuc
wfo peano2 fnfvima mp3an12i cvv vex cardnn wbr simpri sseli onssr1 ssdomg
- cdom mpsyl con0 nnon onenon finnum carddom2 mpbird eqsstr3d 4syl eleqtrrd
+ cdom mpsyl con0 nnon onenon finnum carddom2 mpbird eqsstrrd 4syl eleqtrrd
sucssel eleq1 eleq2d anbi12d spcev impbii 3bitri dff1o5 mpbir2an f1oeq1
eqriv eqtri ) KLUBUCZLEMZUULLDUDUEZLUBZUCZMZUUQUULLUUPNZUUPUGZLOUURILIUFZ
KPZUHZLILUUTUUNPZUHZNZUVALUVCNZUVCJUFZUUNPZQZUVHUVCQZUIZJLUJZRUVEILIJLUVC
@@ -101060,7 +101075,7 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality
eliun 3adant2 mpbird 3expa ralrimiva expl rexbiia syl6bb anbi12d syl21anc
cfflb ontri1 mpbid pm2.21dd expcomd 3impib 3adant1l ordunel eqeltrd 3impa
jaod syldc tfis2 mpcom onssi fneq1i mpbiran onordi sucid eliuni syl6eleqr
- a1i mpan2 elun2 eleqtrrd 3eltr4d rgen issmo2 mp3an23 sylc ssun1 syl5sseqr
+ a1i mpan2 elun2 eleqtrrd 3eltr4d rgen issmo2 mp3an23 sylc ssun1 sseqtrrid
sseq12d vtoclga sstr reximia ralimi ad2antlr fnex smoeq 3anbi123d exlimdv
) DMNZDUAOZDFUBZUCZAUBZBUBZUYEOZPZBUYDUDZADQZRZFUEUYDDEUBZUFZUYNUGZUYGUYH
UYNOZPZBUYDUDZADQZUHZEUEZABDFUIUYCUYMVUBFUYMUYCVUBUYMUYCRUYDDHUFZHUGZUYGU
@@ -103102,7 +103117,7 @@ version of the pigeonhole principle (for aleph-null pigeons and 2 holes)
ccnv dmfex fssdm sselpwd cres ccom wfun wf1o fin1a2lem4 f1cnv f1ofo fofun
wf1 ax-mp resex cofunexg mp2an cdm wss fores sylancl foimacnv mpan2 foeq3
f1f frn syl mpbid foco fowdom cnvex imaex con2bii sylib fin1a2lem6 f1ocnv
- isfin3-2 difss syl5sseqr syl2anc funcnvcnv imadif imaeq2d fimacnv difeq1d
+ isfin3-2 difss sseqtrrid syl2anc funcnvcnv imadif imaeq2d fimacnv difeq1d
fdmd 3eqtr3rd difexg con2bid wa eleq1 difeq2 eleq1d orbi12d notbid syl6bb
3syl ioran rspcev syl12anc rexnal exlimiv sylbi con2i syl5ibr imp ) CFJZB
UAZKJZCYGLZKJZUBZBCUEZUCZCKJZYMYNYFMCUDUFZNYOYMYOCMIUAZOZIUGZYMNZUHMJMUHU
@@ -103394,7 +103409,7 @@ version of the pigeonhole principle (for aleph-null pigeons and 2 holes)
~ trcl . (Contributed by Stefan O'Rear, 11-Feb-2015.) $)
itunitc $p |- ( TC ` A ) = U. ran ( U ` A ) $=
( va vb vc cvv wcel ctc cfv crn cuni wceq cv fveq2 wss c0 com eqeq12d wtr
- rneqd unieqd ituni0 elv fvssunirn eqsstr3i dftr3 wrex wfn itunifn fnunirn
+ rneqd unieqd ituni0 elv fvssunirn eqsstrri dftr3 wrex wfn itunifn fnunirn
wb vex mp2b elssuni csuc itunisuc syl6ss rexlimivw sylbi mprgbir wa tcmin
wi mp2an unissb fvelrnb itunitc1 sseq1 syl5ibcom rexlimiv eqssi vtoclg wn
a1i rn0 unieqi uni0 eqtr2i fvprc 3eqtr4a pm2.61i ) CIJZCKLZCDLZMZNZOZFPZK
@@ -104484,7 +104499,7 @@ a sequence whose values can only be shown to exist (but cannot be
dcomex expcom mtod vex difss eqsstri ssdomg mp2 jctil sylibr entr mpancom
bren2 ensym wf1o bren cuni wf csuc f1of peano1 ffvelrn eldifn eleq2s fvex
co elsn notbii neq0 bitr2i syl w3a cxp cpw elunii sylan2 ffvelrnda difabs
- difeq1i 3eqtr4i pwuni ssdif eqsstr3i sseli ralrimivw ralrimiva fmpo sylib
+ difeq1i 3eqtr4i pwuni ssdif eqsstrri sseli ralrimivw ralrimiva fmpo sylib
adantl difexi eqeltri uniex axdc4 syl2anc suceq fveq2d fveq2 3ad2ant3 imp
exlimiv 3adant1 3adant2 3ad2ant1 3simpb eximi ex mpcom velsn eleq2i eldif
necon3bbii sylbbr sylan2br simpl wrex wfo f1ofo foelrn sylan ccnv oveq12d
@@ -105498,7 +105513,7 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario
ttukeylem7 $p |- ( ph -> E. x e. A ( B C_ x /\ A. y e. A -. x C. y ) ) $=
( cfv wcel wss wn wa c0 con0 wceq cuni cdif ccrd wpss wral wrex csuc fvex
va cv sucid ttukeylem6 mpan2 ttukeylem4 w3a cardon 0ss 3pm3.2i ttukeylem5
- 0elon eqsstr3d wi simprr cun ssun1 undif1 sseqtr4i ccnv simpl wf1o f1ocnv
+ 0elon eqsstrrd wi simprr cun ssun1 undif1 sseqtr4i ccnv simpl wf1o f1ocnv
wf f1of 3syl adantr eldifi ad2antll simprll elunii eldifn eldifd ffvelrnd
syl2anc onelon sylancr suceloni syl a1i word onordi ordsucss syl13anc csn
mpsyl ssun2 eloni ordunisuc fveq2d f1ocnvfv2 eqtr2d velsn ordelss eqsstrd
@@ -109460,7 +109475,7 @@ prove that every set is contained in a weak universe in ZF (see
weq pweq preq12d preq2 cbvmptv preq1 mpteq2dv syl5eq rneqd cbviunv mpteq1
id uneq2d iuneq12d frsucmpt2 sylancl sseqtr4d fvssunirn rexlimdvaa syl5bi
sseqtr4i ralrimiv dftr3 sylibr con0 1on unexg mpan2 fveq1i fr0g syl6eqssr
- syl unssbd 1n0 ssn0 syl5sseqr sstrd unssad vpwex vuniex prss simprd fveq2
+ syl unssbd 1n0 ssn0 sseqtrrid sstrd unssad vpwex vuniex prss simprd fveq2
ssiun2s sseq2d vtoclga findsg sseldd wi imbi12d eqeltri simpld word ordom
simplrl ordunel mp3an2i ssidd suceq fveq2d sseq12d ad2antrr sstr2 syl5com
syl2anc simplrr biantrud bicomd eleq1w anbi2d sseq1 anbi12d sseq1d chvarv
@@ -109557,7 +109572,7 @@ prove that every set is contained in a weak universe in ZF (see
$( The weak universe closure of a set contains the set. (Contributed by
Mario Carneiro, 2-Jan-2017.) $)
wuncid $p |- ( A e. V -> A C_ ( wUniCl ` A ) ) $=
- ( vu wcel cv wss cwun crab cint cwunm cfv ssintub wuncval syl5sseqr ) ABD
+ ( vu wcel cv wss cwun crab cint cwunm cfv ssintub wuncval sseqtrrid ) ABD
ACEFCGHIAAJKCAGLCABMN $.
$( The weak universe closure of a set is a weak universe. (Contributed by
@@ -110353,7 +110368,7 @@ values in the universe (see ~ gruiun for a more intuitive version).
( cgru wcel wfun cima wss w3a wa cdm cin cres wrel wceq simpl2 wf syl3anc
crn 3syl funrel resres resdm reseq1d syl5eqr rneqd df-ima syl6reqr simpl1
simpr inss2 a1i gruss wfn wfo funforn fof sylbi fssres sylancl ffn simpl3
- inss1 eqsstr3d df-f sylanbrc grurn eqeltrd ex ) BDEZCFZCAGZBHZIZABEZVLBEV
+ inss1 eqsstrrd df-f sylanbrc grurn eqeltrd ex ) BDEZCFZCAGZBHZIZABEZVLBEV
NVOJZVLCCKZALZMZSZBVPVKCNZVLVTOVJVKVMVOPZCUAWAVTCAMZSVLWAVSWCWAVSCVQMZAMW
CCVQAUBWAWDCACUCUDUEUFCAUGUHTZVPVJVRBEZVRBVSQZVTBEVJVKVMVOUIZVPVJVOVRAHZW
FWHVNVOUJWIVPVQAUKULAVRBUMRVPVSVRUNZVTBHWGVPVKVRCSZVSQZWJWBVKVQWKCQZVRVQH
@@ -124305,7 +124320,7 @@ subset of complex numbers ( ~ nnsscn ), in contrast to the more elementary
dfnn3 $p |- NN =
|^| { x | ( x C_ RR /\ 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $=
( vz cv cr wss c1 wcel caddc wral wa cab cint cn eleq2 raleqbi1dv anbi12d
- wceq dfnn2 pm3.2i co w3a wb eqeq2i sylbir nnssre eqsstr3i peano2nn intabs
+ wceq dfnn2 pm3.2i co w3a wb eqeq2i sylbir nnssre eqsstrri peano2nn intabs
1nn rgen 3anass abbii inteqi 3eqtr4ri ) ADZEFZGUPHZBDZGIUAZUPHZBUPJZKZKZA
LZMVCALMUQURVBUBZALZMNVCGCDZHZUTVHHZBVHJZKZGNHZUTNHZBNJZKZACEUPVHRURVIVBV
KUPVHGOVAVJBUPVHUPVHUTOPQUPVLCLMZRUPNRZVCVPUCNVQUPCBSZUDVRURVMVBVOUPNGOVA
@@ -147802,7 +147817,7 @@ Proper unordered pairs and triples (sets of size 2 and 3)
by AV, 12-Nov-2021.) $)
hashdmpropge2 $p |- ( ph -> 2 <_ ( # ` dom F ) ) $=
( va wcel vb cdm cvv cv wne wrex chash cfv cle wbr dmexd cpr wss cop wceq
- c2 dmpropg syl2anc dmss eqsstr3d wa wb prssg wi neeq1 neeq2 rspc2ev 3expa
+ c2 dmpropg syl2anc dmss eqsstrrd wa wb prssg wi neeq1 neeq2 rspc2ev 3expa
syl expcom sylbird mpd hashge2el2difr ) AFUBZUCTSUDZUAUDZUEZUAVNUFSVNUFZU
PVNUGUHUIUJAFKPUKABCULZVNUMZVRAVSBDUNCEUNULZUBZVNADITEJTWBVSUONOBDCEIJUQU
RAWAFUMWBVNUMRWAFUSVIUTAVTBVNTZCVNTZVAZVRABGTCHTWEVTVBLMBCVNGHVCURABCUEZW
@@ -151427,7 +151442,6 @@ computer programs (as last() or lastChar()), the terminology used for
YNIZYMUVTYNYMYSYLYMWFQUYHUWBYSUVTYNUWBYSUWCQYSYNXJUWDWGYEWSQYCUWERWHWHTXM
UVEUVFUXSXNXOFUUCUUEUWRYIRXRYFYJWS $.
-
$( The subword of a concatenation is either a subword of the first
concatenated word or a subword of the second concatenated word or a
concatenation of a suffix of the first word with a prefix of the second
@@ -154809,7 +154823,7 @@ the symbol at any position is repeated at multiples of L (modulo the
cotr2 $p |- ( ( R o. R ) C_ R
<-> A. x e. A A. y e. B A. z e. C
( ( x R y /\ y R z ) -> x R z ) ) $=
- ( crn cdm cin incom eqsstr3i cotr2g ) ABCGGGDEFHGKZGLZMRQMERQNIOJP $.
+ ( crn cdm cin incom eqsstrri cotr2g ) ABCGGGDEFHGKZGLZMRQMERQNIOJP $.
$}
${
@@ -155129,7 +155143,7 @@ the symbol at any position is repeated at multiples of L (modulo the
$( The transitive closure of a relation has a lower bound. (Contributed by
RP, 28-Apr-2020.) $)
trclfvlb $p |- ( R e. V -> R C_ ( t+ ` R ) ) $=
- ( vr wcel cv wss ccom wa cab cint ctcl cfv ssmin trclfv syl5sseqr ) ABDAC
+ ( vr wcel cv wss ccom wa cab cint ctcl cfv ssmin trclfv sseqtrrid ) ABDAC
EZFPPGPFZHCIJAAKLQCAMCABNO $.
$}
@@ -155758,7 +155772,7 @@ the symbol at any position is repeated at multiples of L (modulo the
by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) $)
rtrclreclem1 $p |- ( ph -> ( _I |` U. U. R ) C_ ( t*rec ` R ) ) $=
( vr vn cuni crtrcl cfv wss wi cvv cn0 cv crelexp co ciun cc0 wcel wceq
- cres cmpt wrex 0nn0 ssid relexp0d syl5sseqr oveq2 sseq2d rspcev sylancr
+ cres cmpt wrex 0nn0 ssid relexp0d sseqtrrid oveq2 sseq2d rspcev sylancr
cid ssiun syl nn0ex ovex iunex oveq1 iuneq2d fvmptg sylancl sseqtr4d wb
eqid df-rtrclrec fveq1 imbi2d ax-mp mpbir ) AULBGGUAZBHIZJZKZAVJBELFMEN
ZFNZOPZQZUBZIZJZKZAVJFMBVOOPZQZVSAVJWBJZFMUCZVJWCJARMSVJBROPZJZWEUDAVJV
@@ -160519,7 +160533,7 @@ Superior limit (lim sup)
( y <_ z -> ( abs ` ( B - C ) ) < R ) ) $=
( vx cv wbr clt wral cr cc wf wcel cle cmin co cabs cfv wi wrex crp breq2
wceq imbi2d rexralbidv cmpt crli cdm rlimf syl eqid fmpt sylib fdmd feq2d
- mpbid sylibr wss rlimss eqsstr3d rlimcl rlim2 rspcdva ) ABMCMUANZEFUBUCUD
+ mpbid sylibr wss rlimss eqsstrrd rlimcl rlim2 rspcdva ) ABMCMUANZEFUBUCUD
UEZLMZONZUFZCDPBQUGZVKVLGONZUFZCDPBQUGLUHGVMGUJZVOVRBCQDVSVNVQVKVMGVLOUIU
KULACDEUMZFUNNZVPLUHPKALBCDEFADRVTSZERTCDPAVTUOZRVTSZWBAWAWDKFVTUPUQAWCDR
VTADHVTAEHTCDPDHVTSICDHEVTVTURZUSUTVAZVBVCCDREVTWEUSVDADWCQWFAWAWCQVEKFVT
@@ -160532,7 +160546,7 @@ Superior limit (lim sup)
( y <_ z -> ( abs ` ( B - C ) ) < R ) ) $=
( cv wbr co wral wrex cr wss cle cmin cabs cfv wi cpnf cico rlimi wcel wb
clt cmpt cdm wfn wceq eqid fnmpt fndm 3syl crli rlimss syl rexico syl2anc
- eqsstr3d mpbird ) ABNCNUAOEFUBPUCUDHUKOZUECDQZBGUFUGPRZVHBSRZABCDEFHIJKLU
+ eqsstrrd mpbird ) ABNCNUAOEFUBPUCUDHUKOZUECDQZBGUFUGPRZVHBSRZABCDEFHIJKLU
HADSTGSUIVIVJUJADCDEULZUMZSAEIUICDQVKDUNVLDUOJCDEVKIVKUPUQDVKURUSAVKFUTOV
LSTLFVKVAVBVEMVGDGBCVCVDVF $.
$}
@@ -160591,7 +160605,7 @@ Superior limit (lim sup)
lo1bdd $p |- ( ( F e. <_O(1) /\ F : A --> RR ) -> E. x e. RR
E. m e. RR A. y e. A ( x <_ y -> ( F ` y ) <_ m ) ) $=
( clo1 wcel cr wf wa cv cle wbr cfv wi wral wrex simpl wss wb wceq adantl
- simpr cdm fdm lo1dm adantr eqsstr3d ello12 syl2anc mpbid ) EFGZCHEIZJZULA
+ simpr cdm fdm lo1dm adantr eqsstrrd ello12 syl2anc mpbid ) EFGZCHEIZJZULA
KBKZLMUOENDKLMOBCPDHQAHQZULUMRUNUMCHSULUPTULUMUCUNCEUDZHUMUQCUAULCHEUEUBU
LUQHSUMEUFUGUHABCDEUIUJUK $.
$}
@@ -160736,7 +160750,7 @@ Superior limit (lim sup)
o1bdd $p |- ( ( F e. O(1) /\ F : A --> CC ) -> E. x e. RR
E. m e. RR A. y e. A ( x <_ y -> ( abs ` ( F ` y ) ) <_ m ) ) $=
( co1 wcel cc wf wa cv cle wbr cfv cabs wi wral cr wrex wss simpl wb wceq
- simpr cdm fdm adantl o1dm adantr eqsstr3d elo12 syl2anc mpbid ) EFGZCHEIZ
+ simpr cdm fdm adantl o1dm adantr eqsstrrd elo12 syl2anc mpbid ) EFGZCHEIZ
JZUNAKBKZLMUQENONDKLMPBCQDRSARSZUNUOUAUPUOCRTUNURUBUNUOUDUPCEUEZRUOUSCUCU
NCHEUFUGUNUSRTUOEUHUIUJABCDEUKULUM $.
@@ -161062,7 +161076,7 @@ Superior limit (lim sup)
( vj vk wbr clt wa cr wrex wcel cc syl adantr cv cle cfv cmin cabs cdiv
co c2 wi wral wceq crli rlimcl ad2antrr subcld abscld ffvelrnda adantlr
ltnrd abssubd breq1d anbi1d abs3lem syl22anc sylbid imim2d impcomd mtod
- wn nrexdv r19.29r nsyl cxr csup cpnf wss wb fdmd rlimss eqsstr3d ressxr
+ wn nrexdv r19.29r nsyl cxr csup cpnf wss wb fdmd rlimss eqsstrrd ressxr
cdm syl6ss supxrunb1 mpbird r19.29 ex wne wf ffvelrn ralrimiva absrpcld
simpr subne0d rphalfcld feqmptd eqbrtrrd rexanre mpbir2and necon1bd mpd
cmpt rlimi ) AJUAZKUAZUBLZXEEUCZCUDUGUEUCZCDUDUGZUEUCZUHUFUGZMLZXGDUDUG
@@ -161517,7 +161531,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
nfel1 csbeq1a eleq1d rspc sylc rlimi ad2antrr rpred wss ad4ant14 sseldd
ad3antrrr subcld abscld nfcv nfral oveq2 fveq2d breq12d ralbidv fvoveq1
nfbr breq2d rspcv lensymd imp nsyl nrexdv cxr csup cpnf cdm eqid dmmptd
- id wb rlimss eqsstr3d ressxr syl6ss supxrunb1 mpbird r19.29 expcom mtod
+ id wb rlimss eqsstrrd ressxr syl6ss supxrunb1 mpbird r19.29 expcom mtod
r19.21bi condan ) AGHQZPUAZBUAZRUBZFGUCUDZUEUFZCGIUGZUHUBZUIZBESZPTUJAX
MULZUMZPBEFGXSHAFHQZBESYCAYEBEOUKUNYDGUOHUPZQZIUQQZCYFSZXSUQQZYDGUOHYDB
EFURZGUSUBZGUOQZAYLYCKUNZGYKUTVAZAYCVBVCZAYIYCAYHCYFMUKUNYHYJCGYFCXSUQC
@@ -161658,7 +161672,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
bounded function. (Contributed by Mario Carneiro, 12-May-2016.) $)
o1co $p |- ( ph -> ( F o. G ) e. O(1) ) $=
( vn wcel cle cfv cr wrex wa vz ccom co1 cv wbr cabs wi wral cc wf wss wb
- cdm fdmd o1dm syl eqsstr3d elo12 syl2anc mpbid reeanv ad3antrrr ffvelrnda
+ cdm fdmd o1dm syl eqsstrrd elo12 syl2anc mpbid reeanv ad3antrrr ffvelrnda
wceq breq2 2fveq3 breq1d imbi12d rspcva sylan an32s adantr fveq2d sylibrd
fvco3 imim2d ralimdva expimpd ancomsd reximdva mpand rexlimdva mpd mpbird
syl5bir fco ) AGHUBZUCOZBUDZCUDZPUEZWJWGQZUFQZNUDZPUEZUGZCEUHZNRSZBRSZAFU
@@ -161711,7 +161725,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
fveq2 co cabs clt wi wral cr wrex crp wcel fvexd ralrimiva simpr eqbrtrrd
wa cvv ad2antrr rlimi wceq fvoveq1 imbrov2fvoveq simplrr ad4ant14 rspcdva
breq1d imim2d ralimdva reximdv expr mpid rexlimdva mpd syldan fdmd rlimss
- cdm wss syl eqsstr3d ffvelrnd rlim2 mpbird eqbrtrd ) AGHUBOEOQZHRZGRZUCZF
+ cdm wss syl eqsstrrd ffvelrnd rlim2 mpbird eqbrtrd ) AGHUBOEOQZHRZGRZUCZF
GRZUDAOUAEIWNUAQZGRWOHGAEIWMHJUEZAOEIHJUFZAUAIUGGMUFWRWNGUJUHAWPWQUDSPQWM
UISZWOWQTUKULRBQZUMSZUNZOEUOZPUPUQZBURUOAXFBURAXBURUSZVDZDQZFTUKULRZCQZUM
SZXIGRWQTUKULRXBUMSUNZDIUOZCURUQXFNXHXNXFCURXHXKURUSZVDZXNXAWNFTUKULRZXKU
@@ -161759,7 +161773,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
rlimcn2 $p |- ( ph -> ( z e. A |-> ( B F C ) ) ~~>r ( R F S ) ) $=
( vc va vb co cmpt crli wbr cv cle cmin cabs cfv clt wi wral cr wrex wcel
crp wa ralrimiva adantr simprl rlimi simprr reeanv r19.26 prth wb simplrl
- cif simplrr wss cdm eqid dmmptd rlimss syl eqsstr3d ad2antrr sselda maxle
+ cif simplrr wss cdm eqid dmmptd rlimss syl eqsstrrd ad2antrr sselda maxle
syl3anc imbi1d syl5ibr ralimdva ifcl ancoms ad2antlr adantlr wceq fvoveq1
jca breq1d anbi1d oveq1 fvoveq1d imbi12d anbi2d oveq2 rspc2va sylan an32s
imim2d breq1 rspceaimv syl6an ex com23 syld syl5bir rexlimdvva mp2and imp
@@ -162356,7 +162370,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
(Contributed by Mario Carneiro, 26-May-2016.) $)
o1add2 $p |- ( ph -> ( x e. A |-> ( B + C ) ) e. O(1) ) $=
( cmpt caddc co co1 cvv cr wss wcel syl eqidd cof wral ralrimiva dmmptg
- cdm wceq o1dm eqsstr3d reex ssex offval2 o1add syl2anc eqeltrrd ) ABCDK
+ cdm wceq o1dm eqsstrrd reex ssex offval2 o1add syl2anc eqeltrrd ) ABCDK
ZBCEKZLUAMZBCDELMKNABCDELUOUPOFFACPQCORACUOUEZPADFRZBCUBURCUFAUSBCGUCBC
DFUDSAUONRZURPQIUOUGSUHCPUIUJSGHAUOTAUPTUKAUTUPNRUQNRIJUOUPULUMUN $.
@@ -162364,7 +162378,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
(Contributed by Mario Carneiro, 26-May-2016.) $)
o1mul2 $p |- ( ph -> ( x e. A |-> ( B x. C ) ) e. O(1) ) $=
( cmpt cmul co co1 cvv cr wss wcel syl eqidd wral wceq ralrimiva dmmptg
- cof cdm o1dm eqsstr3d reex ssex offval2 o1mul syl2anc eqeltrrd ) ABCDKZ
+ cof cdm o1dm eqsstrrd reex ssex offval2 o1mul syl2anc eqeltrrd ) ABCDKZ
BCEKZLUEMZBCDELMKNABCDELUOUPOFFACPQCORACUOUFZPADFRZBCUAURCUBAUSBCGUCBCD
FUDSAUONRZURPQIUOUGSUHCPUIUJSGHAUOTAUPTUKAUTUPNRUQNRIJUOUPULUMUN $.
@@ -162372,7 +162386,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
(Contributed by Mario Carneiro, 15-Sep-2014.) $)
o1sub2 $p |- ( ph -> ( x e. A |-> ( B - C ) ) e. O(1) ) $=
( cmpt cmin co co1 cvv cr wss wcel syl eqidd wral wceq ralrimiva dmmptg
- cof cdm o1dm eqsstr3d reex ssex offval2 o1sub syl2anc eqeltrrd ) ABCDKZ
+ cof cdm o1dm eqsstrrd reex ssex offval2 o1sub syl2anc eqeltrrd ) ABCDKZ
BCEKZLUEMZBCDELMKNABCDELUOUPOFFACPQCORACUOUFZPADFRZBCUAURCUBAUSBCGUCBCD
FUDSAUONRZURPQIUOUGSUHCPUIUJSGHAUOTAUPTUKAUTUPNRUQNRIJUOUPULUMUN $.
$}
@@ -162384,7 +162398,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
bounded. (Contributed by Mario Carneiro, 26-May-2016.) $)
lo1add $p |- ( ph -> ( x e. A |-> ( B + C ) ) e. <_O(1) ) $=
( vc vm vn wcel cle wi wral cr wrex wa vp cmpt clo1 caddc co wbr reeanv
- cv wss wb cdm wceq ralrimiva dmmptg syl eqsstr3d adantr rexanre readdcl
+ cv wss wb cdm wceq ralrimiva dmmptg syl eqsstrrd adantr rexanre readdcl
lo1dm adantl lo1mptrcl adantlr simplrl simplrr le2add syl22anc ralimdva
imim2d imbi2d ralbidv rspcev syl6an reximdv sylbird rexlimdvva ello1mpt
breq2 syl5bir rexcom syl6bb anbi12d readdcld 3imtr4d mp2and ) ABCDUBZUC
@@ -162404,7 +162418,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
(Contributed by Mario Carneiro, 26-May-2016.) $)
lo1mul $p |- ( ph -> ( x e. A |-> ( B x. C ) ) e. <_O(1) ) $=
( vc vm vn wcel cle wbr cr wrex wa vp cmpt clo1 cmul co wral reeanv wss
- cv wi wb cdm ralrimiva dmmptg syl lo1dm eqsstr3d adantr rexanre cc0 cif
+ cv wi wb cdm ralrimiva dmmptg syl lo1dm eqsstrrd adantr rexanre cc0 cif
wceq simprl simprr ifcl sylancl remulcld simplrr max2 sylancr lo1mptrcl
0re adantlr syl3anc mpan2d jca simplrl lemul12b syl22anc sylan2d imim2d
letr max1 breq2 imbi2d ralbidv rspcev syl6an reximdv sylbird rexlimdvva
@@ -162444,7 +162458,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
o1dif $p |- ( ph ->
( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> C ) e. O(1) ) ) $=
( cmpt co1 wcel cmin co cof syl cvv cc cr eqidd caddc wi o1sub expcom wss
- wral wceq cv wa subcld ralrimiva dmmptg o1dm eqsstr3d reex offval2 nncand
+ wral wceq cv wa subcld ralrimiva dmmptg o1dm eqsstrrd reex offval2 nncand
cdm ssex mpteq2dva eqtrd eleq1d sylibd o1add ex npcand impbid ) ABCDIZJKZ
BCEIZJKZAVHVGBCDELMZIZLNMZJKZVJAVLJKZVHVNUAHVHVOVNVGVLUBUCOAVMVIJAVMBCDVK
LMZIVIABCDVKLVGVLPQQACRUDCPKACVLUQZRAVKQKZBCUEVQCUFAVRBCABUGCKUHZDEFGUIZU
@@ -162734,7 +162748,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
18-May-2016.) $)
rlimneg $p |- ( ph -> ( k e. A |-> -u B ) ~~>r -u C ) $=
( cc0 cmin co cmpt cneg crli cc wcel cr wss wbr syl cv wa 0cnd rlimmptrcl
- cdm wral wceq ralrimiva dmmptg rlimss eqsstr3d 0cn sylancl rlimsub df-neg
+ cdm wral wceq ralrimiva dmmptg rlimss eqsstrrd 0cn sylancl rlimsub df-neg
rlimconst mpteq2i 3brtr4g ) AEBICJKZLIDJKEBCMZLDMNAEBICIDOAEUABPUBUCABCDE
FGHUDABQRIOPEBILINSABEBCLZUEZQACFPZEBUFVBBUGAVCEBGUHEBCFUITAVADNSVBQRHDVA
UJTUKULEBIUPUMHUNEBUTUSCUOUQDUOUR $.
@@ -162771,7 +162785,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
rlimsqzlem $p |- ( ph -> ( x e. A |-> C ) ~~>r E ) $=
( vz vy wbr wcel wa cr cmpt crli cv cle cmin co cabs cfv clt wi wral cpnf
cico wrex crp ad3antrrr wb ad2antrr elicopnf syl simprbda adantrr wss cdm
- eqid dmmptd rlimss eqsstr3d adantr sselda simplbda letrd anassrs adantllr
+ eqid dmmptd rlimss eqsstrrd adantr sselda simplbda letrd anassrs adantllr
cc simprr syldan subcld abscld ad4ant13 rlimcl rpre ad3antlr lelttr mpand
syl3anc expr an32s a2d ralimdva reximdva ralrimiva rlim3 3imtr4d mpd ) AB
CDUAZFUBQZBCEUAGUBQZKAOUCZBUCZUDQZDFUEUFZUGUHZPUCZUIQZUJZBCUKZOHULUMUFZUN
@@ -162831,7 +162845,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
function. (Contributed by Mario Carneiro, 26-May-2016.) $)
lo1le $p |- ( ph -> ( x e. A |-> C ) e. <_O(1) ) $=
( vm wcel cle wbr wi cr wrex wa vy vz cmpt clo1 cv cif simpr adantr ifcld
- wral wb ad2antrr simplr wss cdm wceq ralrimiva dmmptg syl eqsstr3d simprr
+ wral wb ad2antrr simplr wss cdm wceq ralrimiva dmmptg syl eqsstrrd simprr
lo1dm sseldd maxle syl3anc syl6bi imim1d adantlr adantrll simpl lo1mptrcl
syl2an simprll letr mpand expr adantrd a2d syld anassrs ralimdva reximdva
sylbid breq1 imbi1d rexralbidv rspcev syl6an rexlimdva ello1mpt 3imtr4d
@@ -162876,7 +162890,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
( vc vy wcel cle wbr cr wrex wa wfal c1 cc0 adantr cmpt co1 cabs cfv wral
cv wi fal cdiv co cmin cif clt cc reccld ralrimiva simpr 1re ifcl sylancl
crp 1rp a1i max1 sylancr rpgecld rpreccld crli wss wb cdm wceq dmmptg syl
- rlimi rlimss eqsstr3d cxr csup cpnf ressxr syl6ss supxrunb1 mpbird r19.29
+ rlimi rlimss eqsstrrd cxr csup cpnf ressxr syl6ss supxrunb1 mpbird r19.29
rexanre r19.29r adantlr wne subid1d fveq2d 1cnd 0le1 absidd oveq1d 3eqtrd
absdivd ad2antrr rprecred absrpcld max2 lediv2ad lensymd eqnbrtrd pm2.21d
rpred letrd expimpd ancomsd imim2d impcomd rexlimdva syl5 rexlimdvw mpand
@@ -164944,7 +164958,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/
fsumsplit $p |- ( ph -> sum_ k e. U C =
( sum_ k e. A C + sum_ k e. B C ) ) $=
( csu caddc co wcel cc0 wss cc wceq syldan wa cv cif wral cuz cfv cfn cun
- wo ssun1 syl5sseqr sselda ralrimiva olcd sumss2 syl21anc oveq12d 0cn ifcl
+ wo ssun1 sseqtrrid sselda ralrimiva olcd sumss2 syl21anc oveq12d 0cn ifcl
ssun2 sylancl fsumadd eleq2d elun syl6bb biimpa iftrue adantl wn noel cin
wi c0 elin syl5rbbr mtbii imnan sylibr imp iffalsed addid1d eqtrd addid2d
con2d jaodan sumeq2dv 3eqtr2rd ) ABDFKZCDFKZLMEFUAZBNZDOUBZFKZEWICNZDOUBZ
@@ -166413,7 +166427,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/
wb abscld adantrr rprege0 ad2antrl absid oveq2d eqtrd cmul adantlr adantr
remulcld fveq2i fsumabs eqbrtrid cun ssun2 nnred flle simprr letrd fznnfl
flge1nn mpbir2and syldan recnd absge0d chash cfn cn0 3syl reflcl breqtrd
- clt fzsplit syl5sseqr sselda mulid2d fsumge0 jca simprd syl31anc eqbrtrrd
+ clt fzsplit sseqtrrid sselda mulid2d fsumge0 jca simprd syl31anc eqbrtrrd
lemul1a hashcl nn0re elfzuz peano2nnd eluznn sylan peano2re fllep1 eluzle
cuz simpllr nfv nffv nfbr breq2 fveq2d breq1d imbi12d syl3c sylan2 fsumle
nfim fsumconst syl2anc biidd 0red mpan9 nnzd uzid rspcdva ssdomg hashdomi
@@ -170552,7 +170566,7 @@ seq n ( x. ,
fprodsplit $p |- ( ph -> prod_ k e. U C =
( prod_ k e. A C x. prod_ k e. B C ) ) $=
( cprod cmul co wcel c1 wa cc wceq adantl iffalsed cv cif iftrue prodeq2i
- ssun1 syl5sseqr sselda syldan eqeltrd cdif eldifn fprodss syl5eqr oveq12d
+ ssun1 sseqtrrid sselda syldan eqeltrd cdif eldifn fprodss syl5eqr oveq12d
cun ssun2 ax-1cn ifcl sylancl fprodmul wo eleq2d elun syl6bb biimpa c0 wn
cin disjel sylan mulid1d eqtrd ex con2d mulid2d jaodan prodeq2dv 3eqtr2rd
imp ) ABDFKZCDFKZLMEFUAZBNZDOUBZFKZEWBCNZDOUBZFKZLMEWDWGLMZFKEDFKAVTWEWAW
@@ -183689,7 +183703,7 @@ reduced fraction representation (no common factors, denominator
cn nnne0 simpr necon3ai 3syl gcdn0cl syl21anc simp2d dvdsmultr1d nnmulcld
wn simprd wi dvdslegcd syl31anc mp2and simprbi wb nnle1eq1 mpbid sylanbrc
breqtrd eqtrd dvdsmul2 dvdstr syl3anc opelxpd eqeltrd ralrimiva wfun wf1o
- cdm wfn crth f1ofn fnfun ssrab3 fndm syl5sseqr syl2an cmpt eqtri eqeltrrd
+ cdm wfn crth f1ofn fnfun ssrab3 fndm sseqtrrid syl2an cmpt eqtri eqeltrrd
eqtr3d sylib dfphi2 fveq2i syl6eqr funimass4 mpbird xpss12 sseqtr4i sseli
ccnv f1ocnvfv2 wf f1ocnv f1of ffvelrn cbvmptv opelxp funfvima2 imp syldan
rpmul eqelssd wf1 f1of1 elexi f1imaen sylancl eqbrtrrd xpfi hashen sylibr
@@ -188742,7 +188756,7 @@ with complex numbers (gaussian integers) instead, so that we only have
ord nn0nnaddcl fmptd nnex elmap sylibr addcld nnm1nn0 elfznn0 nn0mulcld
adantl add4d 1cnd add32d mul12d 3eqtr4d rspceeqv biimpar sylan2 vdwlem4
mpan2 vdwlem3 eqeltrd mptex eleq1 fveqeq2 rexlimdva 3imtr4d ssrdv ssun1
- r19.21bi cun fzsuc syl5sseqr sselda eqeq1 ifbieq2d nnred ltp1d peano2re
+ r19.21bi cun fzsuc sseqtrrid sselda eqeq1 ifbieq2d nnred ltp1d peano2re
clt ltnled mpbid breq1 notbid con2d elfzle2 impel iffalse add12d oveq1i
cr subcld ax-1cn sylancl npcand 3eqtr3d syl5eq addcomd 3eqtr2d cnvimass
subcl 3eqtrd fssdm vdwapid1 3sstr4d ex eqtr4d c0 wne eluzfz1 elfzuz3 cz
@@ -189873,7 +189887,7 @@ with complex numbers (gaussian integers) instead, so that we only have
cmin adantl eqbrtrrd cn ffvelrnd nnred 1red cr cn0 ssfid hashcl nn0re
3syl lesubaddd mpbid fveq2 snidg syl sseld eldifn syl6 mt2d hashunsng
wn wi mp2and breqan12rd mpbird crab sylancl hashbcval syl2anc simpl1l
- eqsstr3d simpr simpl3 rabid sylanbrc sseldd elpw sylibr fveq2d eqtr3d
+ eqsstrrd simpr simpl3 rabid sylanbrc sseldd elpw sylibr fveq2d eqtr3d
sylib cc breq2d oveq1 sseq1d anbi12d rspcev syl12anc snfi unfi nnnn0d
w3a simpl2 elpwid vex eleqtrrd nnm1nn0 fveqeq2 uncom syl6sseq ssundif
difexi diffi nn0cn ax-1cn pncan undif1 eldifd elpwunsn ssequn2 oveq1d
@@ -193504,7 +193518,7 @@ C_ dom ( S sSet <. I , E >. ) ) $=
(Revised by Mario Carneiro, 30-Apr-2015.) $)
ressbasss $p |- ( Base ` R ) C_ B $=
( cvv wcel cbs cfv wss cin ressbas inss2 syl6eqssr wn c0 cress reldmress
- co ovprc2 syl5eq fveq2d base0 0ss eqsstr3i syl6eqss pm2.61i ) AGHZCIJZBKU
+ co ovprc2 syl5eq fveq2d base0 0ss eqsstrri syl6eqss pm2.61i ) AGHZCIJZBKU
IUJABLBABCGDEFMABNOUIPZUJQIJZBUKCQIUKCDARTQEDARSUAUBUCULQBUDBUEUFUGUH $.
$}
@@ -193545,7 +193559,7 @@ C_ dom ( S sSet <. I , E >. ) ) $=
the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.) $)
ressinbas $p |- ( A e. X -> ( W |`s A ) = ( W |`s ( A i^i B ) ) ) $=
( wcel cvv cress co cin wceq elex wss w3a eqid ressid2 syl3an eqtr4d csts
- wn wa ssid incom df-ss biimpi syl5eq syl5sseqr inex1g 3expb cnx cbs inass
+ wn wa ssid incom df-ss biimpi syl5eq sseqtrrid inex1g 3expb cnx cbs inass
cfv cop inidm ineq2i eqtr2i opeq2i oveq2i ressval2 inss1 sstr mpan2 con3i
3eqtr4a pm2.61ian c0 reldmress ovprc1 adantr syl ) ADFAGFZCAHIZCABJZHIZKZ
ADLCGFZVLVPBAMZVQVLUAVPVRVQVLVPVRVQVLNVMCVOABVMCGGVMOZEPVRBVNMZVQVQVLVNGF
@@ -195238,7 +195252,7 @@ C_ dom ( S sSet <. I , E >. ) ) $=
( cfv cop va vc vd ve cds cmpt crn cc0 csn cun cxr clt csup cmpo cplusg
cv co cxp c2nd chom cixp c1st cco cpr wss cple wbr wa copab cvsca cmulr
wral cip cgsu ctopn ccom cpt cbs eqid prdsplusg prdsmulr prdsvsca eqidd
- prdsbas prdsval pleid cvv wcel fvexi xpex prss anbi1i opabssxp eqsstr3i
+ prdsbas prdsval pleid cvv wcel fvexi xpex prss anbi1i opabssxp eqsstrri
vex opabbii ssexi a1i cnx cts csca snsstp2 ssun1 sstri ssun2 prdsvallem
ctp ) AJCGHCCBIBUPZGUPZSZXHHUPZSZXHESZUESUQUFUGUHUIUJUKULUMUNZDUOSZFUAU
BCCURZCUCUDUBUPZUAUPZUSSZGHCCBIXJXLXMUTSUQVAUNZUQXRXTSBIXHUCUPSXHUDUPSX
@@ -195255,7 +195269,7 @@ C_ dom ( S sSet <. I , E >. ) ) $=
Mario Carneiro, 16-Aug-2015.) $)
prdsless $p |- ( ph -> .<_ C_ ( B X. B ) ) $=
( vf vg vx cv cfv cpr wss cple wbr wral wa copab cxp prdsle wcel anbi1i
- vex prss opabbii opabssxp eqsstr3i syl6eqss ) AGPSZQSZUABUBZRSZURTVAUST
+ vex prss opabbii opabssxp eqsstrri syl6eqss ) AGPSZQSZUABUBZRSZURTVAUST
VADTUCTUDRFUEZUFZPQUGZBBUHZARBCDEPQFGHIJKLMNOUIVDURBUJUSBUJUFZVBUFZPQUG
VEVGVCPQVFUTVBURUSBPULQULUMUKUNVBPQBBUOUPUQ $.
$}
@@ -196336,7 +196350,7 @@ topology is based on the order and not the extended metric (which would
eliun opth syl5bi eqeq2 biimprd syl6 3expa rexlimdvva sylbid ralrimivva
impd alrimiv mo2icl fofn opeq2 breq1d mobidv ralrn ralbidv mpbird opeq1
breq1 ralxp sylibr ssralv sylc dffun7 sylanbrc eqimss2 iunss sylib snss
- opeldm sylbir ralimi sylbi eleq1d eleq1 dfss3 eqsstr3d eqssd df-fn ) AC
+ opeldm sylbir ralimi sylbi eleq1d eleq1 dfss3 eqsstrrd eqssd df-fn ) AC
UBZCUCZBBUDZQCUUFUEACULZNUFZOUFZCUGZOUHZNUUERZUUDAUUGHFGFHUFZEUIZGUFZEU
IZUJZUUMUUODUKZEUIZUJZUMZUNZUNZULZUVDUVBULZHFRUVEHFUVEUVAULZGFRUVFGFUUQ
UUSUUNUUPUOZUUREVEZUPUQGFUVAURUSUQHFUVBURUSACUVCMUTVAAUUEEVBZUVIUDZVCUU
@@ -196470,7 +196484,7 @@ topology is based on the order and not the extended metric (which would
weq mobidv mpbird ralxp sylanbrc eqtr3d adantl elsni simpl2im ex sylan2br
impel eqtr4d anassrs rexlimdva sylbid alrimiv mo2icl breq1d ssralv dffun7
fofn breq1 sylc eqimss2 r19.21bi adantrl syl5eqssr simprl opelxpi sylancl
- snid sseldd eleq1d eleq1 dfss3 eqsstr3d eqssd df-fn ) ADUHZDUIZIBUJZUKDUV
+ snid sseldd eleq1d eleq1 dfss3 eqsstrrd eqssd df-fn ) ADUHZDUIZIBUJZUKDUV
QULADUMZUDUNZUEUNZDUOZUEUPZUDUVPUQZUVOAUVRLJMUDILUNZGURZVBZMUNZUWDEUSZGUR
ZUTZVAZUMZUWLUWJUMZLJUQUWMLJUWJIUWFUJZULZUWMMUDIUWFUWIUWJUWJVCZUWHGVNZVDZ
UWNUWJVEVFVGLJUWJVHVIADUWKAUDBCDEFGHIJKLMOPQRSTUAUBVJZVKVLAUVPIGVMZUJZVOU
@@ -197572,7 +197586,7 @@ the closure of the set with the element removed (Section 0.6 in
$( A set is closed iff it is equal to its closure. (Contributed by Stefan
O'Rear, 31-Jan-2015.) $)
mrcidb $p |- ( C e. ( Moore ` X ) -> ( U e. C <-> ( F ` U ) = U ) ) $=
- ( cmre cfv wcel wceq mrcid wa simpr mrcssv adantr eqsstr3d mrccl eqeltrrd
+ ( cmre cfv wcel wceq mrcid wa simpr mrcssv adantr eqsstrrd mrccl eqeltrrd
wss syldan impbida ) ADFGHZBAHBCGZBIZABCDEJUAUCKZUBBAUAUCLZUAUCBDRUBAHUDB
UBDUEUAUBDRUCABCDEMNOABCDEPSQT $.
@@ -197589,7 +197603,7 @@ the closure of the set with the element removed (Section 0.6 in
$( The closure of a set is a superset. (Contributed by Stefan O'Rear,
31-Jan-2015.) $)
mrcssid $p |- ( ( C e. ( Moore ` X ) /\ U C_ X ) -> U C_ ( F ` U ) ) $=
- ( vs cmre cfv wcel wss wa cv crab cint ssintub mrcval syl5sseqr ) ADGHIBD
+ ( vs cmre cfv wcel wss wa cv crab cint ssintub mrcval sseqtrrid ) ADGHIBD
JKBFLJFAMNBBCHFBAOABCDFEPQ $.
$( A set is closed iff it contains its closure. (Contributed by Stefan
@@ -197853,7 +197867,7 @@ the closure of the set with the element removed (Section 0.6 in
mrieqv2d $p |- ( ph -> ( S e. I <->
A. s ( s C. S -> ( N ` s ) C. ( N ` S ) ) ) ) $=
( vx wcel wpss cfv wa w3a 3ad2ant1 adantr cvv cv wi wal wn wex pssnel csn
- 3ad2ant3 cdif cmre wceq simprr difsnb sylib simpl3 pssssd ssdifd eqsstr3d
+ 3ad2ant3 cdif cmre wceq simprr difsnb sylib simpl3 pssssd ssdifd eqsstrrd
simpl2 mrissd ssdifssd mrcssd difssd mrcssidd ismri2dad sspsstrd exlimddv
simprl sseldd ssnelpssd 3expia alrimiv ex wral wne elfvexd wss difexg syl
ssexd simp1r difsnpss simp2 psseq1d mpbird simp3 mpd fveq2d mpbid spcimdv
@@ -198038,7 +198052,7 @@ u C_ ( N ` ( v u. t ) ) /\ ( u u. t ) e. I )
( ( F \ { Y } ) u. ( H u. { g } ) ) e. I ) ) $=
( cv wcel csn cdif wn cun wa wex wrex cfv wss cmre simpr ssun2 difundir
adantr wceq cin c0 incom ssdifin0 syl syl5eqr minel difsnb sylib uneq2d
- syl2anc syl5eq syl5sseqr mrissd ssdifssd mrcssidd sstrd mrcssvd mrcidmd
+ syl2anc syl5eq sseqtrrid mrissd ssdifssd mrcssidd sstrd mrcssvd mrcidmd
unssd mrcssd sseqtrd sseldd sseldi ismri2dad pm2.65da nss simprl simprr
ssun1 ssneldd unass wral cpw difss unss1 mp1i mrissmrid fveq2d neleqtrd
difss2d mreexmrid syl5eqelr jca32 ex eximdv mpd df-rex sylibr ) AEUCZGU
@@ -198096,7 +198110,7 @@ f C_ ( N ` ( g u. h ) ) /\ ( f u. h ) e. I ) ->
csn wral wss animorrl mreexexlem3d wne n0 biimpi adantl wn mreexexlem2d
wex simpr w3a 3anass cvv ad2antrr elfvexd simpr2 difsnb ssdifssd ssdifd
sylib difun1 syl6sseqr simpr1 uncom uneq2i unass difsnid uneq1d syl5eqr
- eqsstr3d syl5eq syl fveq2d sseqtr4d simpr3 csuc wo com wi simplr dif1en
+ eqsstrrd syl5eq syl fveq2d sseqtr4d simpr3 csuc wo com wi simplr dif1en
3anan12 sylbir expcom syl2anc orim12d mpd mreexexlemd ad3antrrr difss2d
wal ssexd simprl simplr1 snssd unssd sselpwd ad3antlr cin simprrl en2sn
elpwid el2v a1i incom disjdif ssdifin0 unen syl22anc eqbrtrrd rexlimddv
@@ -202241,7 +202255,7 @@ is always a subcategory (and it is full, meaning that all morphisms of
wf funcf1 eqidd subcfn subcss1 fssresd simpld rescbas feq2d mpbid wfn cxp
fnmpti eqcomd fndm sqxpeqd fneq12d mpbii wss simprl simprr funcf2 subcss2
resf2nd feq1d mpbird reschom oveq12d feq23d eleq2d biimpar subcid fveq12d
- oveqd eqsstr3d sselda funcid subcidcl 3eqtr4d simp21 simp22 simp23 simp3l
+ oveqd eqsstrrd sselda funcid subcidcl 3eqtr4d simp21 simp22 simp23 simp3l
simp3r subccocl funcco rescco opeq12d fveq1d oveq123d isfuncd df-br sylib
w3a eqeltrd ) ADEUETZDUFIZEUGZUGZJZYPUHIZUIZBEUJTZCUKTZAYPYTHYRHULZDUHIZI
ZUUEEIZJZUMZUIZUUBAHDEBCUKTZBUPIZFGUNZAUUAUUJYTAUUAUUKUHIZUUJAYPUUKUHUUNU
@@ -205326,7 +205340,7 @@ is always a subcategory (and it is full, meaning that all morphisms of
( cfunc co wcel chomf cfv ccat ccomf wceq eqid vf cv cxp cres cresc eqidd
cress cbs setccat syl adantr wss setcbas sseqtrd fullresc simpld resssetc
wa eqtr3d simprd funcrcl adantl fullsubc subccat funcpropd csubc funcres2
- eqsstr3d simpr sseldd ex ssrdv ) AUAECLMZEBLMZAUAUBZVMNZVOVNNAVPURZVMVNVO
+ eqsstrrd simpr sseldd ex ssrdv ) AUAECLMZEBLMZAUAUBZVMNZVOVNNAVPURZVMVNVO
VQVMEBBOPZFFUCUDZUEMZLMZVNVQEEVTCQVQEOPUFVQERPUFVQBFUGMZOPZVTOPZCOPZVQWCW
DSZWBRPZVTRPZSZVQBUHPZBWBFVTVRWJTZVRTZABQNZVPADGNWMJBDGHUIUJUKZAFWJULVPAF
DWJKABDGHJUMUNUKZWBTVTTZUOZUPVQWCWESZWGCRPZSZAWRWTURVPABCDFGHIJKUQUKZUPUS
@@ -213207,7 +213221,7 @@ net proof size (existence part)? $)
cv acsmred simplr elin1d elpwid mrissmrid ralrimiva dfss3 sylibr csn cdif
cun elfpw sylib simpld difss2d snssd unssd simprd snfi unfi sylanbrc wceq
sylancl ad4antr simpllr snidg 3syl eleqtrrd ismri2dad ad3antrrr neldifsnd
- elun2 ssneldd difsnb ssun1 syl5sseqr ssdifd eqsstr3d sstrd eqsstrd mrcssd
+ elun2 ssneldd difsnb ssun1 sseqtrrid ssdifd eqsstrrd sstrd eqsstrd mrcssd
wi ssdifssd sseld mtod rspcimdv syl5bi impancom ralrimiv acsficl2d notbid
ex wb wrex ralnex syl6bbr mpbird an32s ismri2dd impbida ) ACDMZCUBZNUCZDO
ZAXHPZKUGZDMZKXJUDZXKXLXNKXJXLXMXJMZPZBCXMDEFABFUEQMZXHXPABFGUHZUFHIAXHXP
@@ -214161,7 +214175,7 @@ net proof size (existence part)? $)
( vy vz cdir wcel cv wbr wa wex cuni eleq2d wral ccom wss wceq wrex dirdm
cdm syl5eq anbi12d cxp ccnv wrel cid cres isdir simprrd codir sylib breq1
eqid ibi anbi1d exbidv anbi2d rspc2v syl5com sylbid crn reldir relelrn ex
- sylan cun ssun2 dmrnssfld sstri syl5sseqr sseld syld adantrd ancrd eximdv
+ sylan cun ssun2 dmrnssfld sstri sseqtrrid sseld syld adantrd ancrd eximdv
df-rex syl6ibr 3impib ) DIJZBEJZCEJZBAKZDLZCWEDLZMZAEUAZWBWCWDMZWHANZWIWB
WJBDOOZJZCWLJZMZWKWBWCWMWDWNWBEWLBWBEDUCZWLFDUBUDZPWBEWLCWQPUEWBGKZWEDLZH
KZWEDLZMZANZHWLQGWLQZWOWKWBWLWLUFDUGDRSZXDWBDUHZUIWLUJDSMZDDRDSZXEWBXGXHX
@@ -214175,7 +214189,7 @@ net proof size (existence part)? $)
25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.) $)
tsrdir $p |- ( A e. TosetRel -> A e. DirRel ) $=
( ctsr wcel cdir wrel cid cuni cres wss ccom cxp ccnv cps syl jca wceq eqid
- wa syl5eqssr eqsstr3d tsrps psrel cin psref2 inss1 syl6eqssr cdm crn psdmrn
+ wa syl5eqssr eqsstrrd tsrps psrel cin psref2 inss1 syl6eqssr cdm crn psdmrn
pstr2 simpld sqxpeqd cun istsr simprbi relcoi2 cnvresid cnvss coss1 relcoi1
relcnv ax-mp relcnvfld reseq2d coss2 unssd sstrd isdir mpbir2and ) ABCZADCA
EZFAGGZHZAIZRAAJAIZVLVLKZALZAJZIZRVJVKVNVJAMCZVKAUAZAUBNZVJVTVNWAVTVMAVQUCA
@@ -216266,7 +216280,7 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is
resmhm2 $p |- ( ( F e. ( S MndHom U ) /\ X e. ( SubMnd ` T ) ) ->
F e. ( S MndHom T ) ) $=
( vx vy cmhm co wcel cfv wa cmnd cbs wf cplusg wceq c0g eqid csubmnd wral
- cv w3a mhmrcl1 submrcl anim12i mhmf submbas submss eqsstr3d syl2an mhmlin
+ cv w3a mhmrcl1 submrcl anim12i mhmf submbas submss eqsstrrd syl2an mhmlin
wss 3expb adantlr ressplusg ad2antlr oveqd eqtr4d ralrimivva adantr subm0
fss mhm0 adantl 3jca ismhm sylanbrc ) DACIJKZEBUALZKZMZANKZBNKZMAOLZBOLZD
PZGUCZHUCZAQLZJDLZVSDLZVTDLZBQLZJZRZHVPUBGVPUBZASLZDLZBSLZRZUDDABIJKVJVNV
@@ -222599,7 +222613,7 @@ permutation associated with the composition of these two elements (in
$( The center is a subset of the base field. (Contributed by Thierry
Arnoux, 21-Aug-2023.) $)
cntrss $p |- ( Cntr ` M ) C_ B $=
- ( ccntr cfv ccntz eqid cntrval cntzssv eqsstr3i ) BDEABFEZEAABKCKGZHAABKC
+ ( ccntr cfv ccntz eqid cntrval cntzssv eqsstrri ) BDEABFEZEAABKCKGZHAABKC
LIJ $.
$}
@@ -223183,7 +223197,7 @@ operation is a permutation group (group consisting of permutations), see
SO, 9-Jul-2018.) $)
symgbasfi $p |- ( A e. Fin -> B e. Fin ) $=
( vf cfn wcel cmap co mapfi anidms cv wf wf1o symgbas f1of ss2abi eqsstri
- cab wceq mapvalg syl5sseqr ssfid ) AGHZAAIJZBUEUFGHAAKLUEAAFMZNZFTZBUFBAA
+ cab wceq mapvalg sseqtrrid ssfid ) AGHZAAIJZBUEUFGHAAKLUEAAFMZNZFTZBUFBAA
UGOZFTUIFABCDEPUJUHFAAUGQRSUEUFUIUAAAGGFUBLUCUD $.
$( The function value of a permutation. (Contributed by AV,
@@ -224185,7 +224199,7 @@ operation is a permutation group (group consisting of permutations), see
-. dom ( ( F o. G ) \ _I ) C_ X ) $=
( wf1o cid cdif cdm wss ccom wn wa ccnv wceq coass f1of syl difeq1d dmeqd
wf wxo wo excxor f1ococnv1 coeq1d fcoi2 sylan9eq syl5eqr adantr cun mvdco
- cres f1omvdcnv ad2antrr simprl eqsstrd simprr unssd syl5ss eqsstr3d con3d
+ cres f1omvdcnv ad2antrr simprl eqsstrd simprr unssd syl5ss eqsstrrd con3d
expr expimpd f1ococnv2 coeq2d fcoi1 sylan9eqr syl5eq ad2antlr jaod syl5bi
ancomsd 3impia ) AABEZAACEZBFGZHZDIZCFGZHZDIZUAZBCJZFGHZDIZKZWBVRWAKZLZVR
KZWALZUBVNVOLZWFVRWAUCWKWHWFWJWKVRWGWFWKVRLWEWAWKVRWEWAWKVRWELZLZVTBMZWCJ
@@ -227213,7 +227227,7 @@ P pGrp ( G |`s S ) ) $=
ssel syld impr sylan2b en1b fveq2d cvv vuniex ax-mp syl6eq sumeq2dv ssexg
hashsng sylancr eqeltrd sseldi syl2anc wb mpbird oveq1d 3eqtr4rd wral cdm
ssrab3 mulid1d rabexd inss2 crn cxp wrel cpr wrex relopabi relssdmrn erdm
- pwexd errn xpexd ecelqsg eqeltrrd snelpwi adantl elind eqsstr3d snss sneq
+ pwexd errn xpexd ecelqsg eqeltrrd snelpwi adantl elind eqsstrrd snss sneq
eqeq2d syl5ibrcom adantrl unieq unisn syl6req impbid1 hashen eqtr3d diffi
en3d eldifi sylan2 fsumcl subaddd breqtrrd ) AELGUCZMUDZUEZUAUFZUGUHZUAUI
ZLUGUHZMUGUHZUJUKZULABCUADEFGHIJKLMNOPQRSTUMAUWJUWGUPUWIUWGUNUKZUWHUPAUWB
@@ -227719,7 +227733,7 @@ same as the index (number of cosets) of the normalizer of the Sylow
csg cgrp ad2antrr nmzsubg subgslw syl3anc ssnmz cvv cplusg fvexi rabex2
wb ressplusg ax-mp sylow2 cnsg conjnsg sylan eqeq2 syl5ibrcom rexlimdva
nmznsg mpd eqtrd cz eqeltrd impbida 3bitr3d rabsn fveq2d hashsng oveq2d
- eqsstr3d rabbidva breqtrd cn cprime prmnn cn0 hashcl nn0zd 1zzd moddvds
+ eqsstrrd rabbidva breqtrd cn cprime prmnn cn0 hashcl nn0zd 1zzd moddvds
mpbird c2 cuz prmuz2 clt eluz2b2 cr nnre 1mod sylbi 3syl ) AEHUFUGZUHUI
ZEUJUGZUKEUJUGZUKAUVLUVMULZEUVKUKUMUGZUNUOZAEUVKUCUSZMUSZGUGZUVRULZUCHI
UPUGZUTUIZUQZMUVJURZUHUIZUMUGUVOUNADUDMEGDUSZUDUSZVAUVJVBUEUSUWFGUGUWGU
@@ -229420,7 +229434,7 @@ an extension of the previous (inserting an element and its inverse at
efgredlemf $p |- ( ph -> ( ( A ` K ) e. W /\ ( B ` L ) e. W ) ) $=
( vi cfv wcel cc0 chash cfzo co cword wf c0 csn cdm cdif cv cmin wral
c1 crn efgsdm simp1bi syl eldifad wrdf cfz fzossfz cz cn0 lencl nn0zd
- wceq fzoval syl5sseqr cn efgredlema simpld syl5eqel ffvelrnd simpl2im
+ wceq fzoval sseqtrrid cn efgredlema simpld syl5eqel ffvelrnd simpl2im
fzo0end sseldd jca ) ARHURUAUSSIURUAUSAUTHVAURZVBVCZUARHAHUAVDZUSZWSU
AHVEAHWTVFVGZAHLVHZUSZHWTXBVIZUSZUKXDXFUTHURJUSUQVJZHURXGVMVKVCZHURMU
RVNUSUQVMWRVBVCVLBCDEFGJKLMUQNOPHQTUAUDUEUFUGUHUIVOVPVQVRZUAHVSVQAUTW
@@ -229579,7 +229593,7 @@ an extension of the previous (inserting an element and its inverse at
efgredlemd $p |- ( ph -> ( A ` 0 ) = ( B ` 0 ) ) $=
( vi cc0 chash cfv c1 cmin co cfzo cres cs1 cconcat cword wcel wceq
c0 cv crn wral efgsdm simp1bi syl eldifad wf wrdf cfz fzossfz lencl
- cz nn0zd fzoval syl5sseqr cn simpld fzo0end syl5eqel ffvelrnd s1cld
+ cz nn0zd fzoval sseqtrrid cn simpld fzo0end syl5eqel ffvelrnd s1cld
cn0 sseldd wa lbfzo0 sylibr ccatval1 syl3anc simprd eqtr4d c2 caddc
clt wbr cr sseldi nn0red 2rp ltaddrp sylancl cle lem1d wb mpbir2and
fznn syl2anc fveq2d fvresd fveq2i wi fveq1 imbi12d eqeq2d csn sylbi
@@ -232657,7 +232671,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
wf inss1 ax-mp syl6eq eqtr3d oveq2d 3eqtr4d ex cplusg ccom cseq crn wfo
f1ofo ad2antll eqsstrd cores seqeq3d fveq1d fssres sylancl feq1d biimpa
forn eqid syldan resss rnssi cntzidss simprl f1of1 suppssdm fssdm ssind
- wf1 f1ss wi fex ressuppss syl5ibr adantl impcom gsumval3 syl5sseqr expr
+ wf1 f1ss wi fex ressuppss syl5ibr adantl impcom gsumval3 sseqtrrid expr
sseq2 exlimdv expimpd cfsupp wbr cfn wfun fsuppimp simprd fz1f1o mpjaod
wo ) ADHUCUDZUEUFZEDGUGZUHUDZEDUHUDZUFZYRUIUJZUKULZUMUUDUOUDZYRUAUPZUNZ
UAUQZVIZAYSUUCAYSVIZEUBBGURZHUSZUHUDZEUBBHUSZUHUDZUUAUUBAUUNUUPUFYSAUUN
@@ -232691,7 +232705,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
( co wcel wa adantr vf vk vx csupp c0 wceq cgsu chash cfv cn c1 cv wf1o
cfz wex cmpt cvv c0g fvexi a1i ssidd gsumcllem cmnd gsumz syl2anc eqtrd
oveq2d mndidcl syl eqeltrd ex cplusg ccom cseq eqid wf crn simprl f1of1
- wss wf1 ad2antll suppssdm fssdm f1ss ssid f1ofo forn syl5sseqr gsumval3
+ wss wf1 ad2antll suppssdm fssdm f1ss ssid f1ofo forn sseqtrrid gsumval3
wfo cuz nnuz syl6eleq f1f fco ffvelrnda mndcl 3expb sylan seqcl exlimdv
expr expimpd cfn wo fz1f1o mpjaod ) ADGUDQZUEUFZEDUGQZCRZXIUHUIZUJRZUKX
MUNQZXIUAULZUMZUAUOZSZAXJXLAXJSZXKGCXTXKEUBBGUPZUGQZGXTDYAEUGABCUQUBDFX
@@ -232720,7 +232734,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
wa a1i ssidd gsumcllem oveq2d f1of ffvelrnda feqmptd eqidd fmptco 3eqtr4d
wf cplusg cseq ccnv coass cid cres f1ococnv2 coeq1d wss ad2antll suppssdm
ex fssdm f1ss f1f fcoi2 3syl eqtrd syl5reqr coeq2d syl6eqr seqeq3d fveq1d
- eqid crn simprl ssid wfo f1ofo forn syl5sseqr gsumval3 fco rncoss sylancl
+ eqid crn simprl ssid wfo f1ofo forn sseqtrrid gsumval3 fco rncoss sylancl
cntzidss f1ocnv f1co csn cdif cima suppimacnv eqcomd 3sstr4d imass2 cnvco
fex imaeq1i imaco eqtri rnco2 3sstr4g f1oexrnex coexg sseq1d mpbird expr
wb exlimdv expimpd cfsupp wbr cfn wo wfun fsuppimp simprd fz1f1o mpjaod )
@@ -232995,7 +233009,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
gsumadd $p |- ( ph ->
( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) $=
( cfv wcel eqid ccmn cmnd cmnmnd syl csubmnd submid ssid wss wceq cntzcmn
- ccntz sylancl syl5sseqr gsumzadd ) ABCDCEFGHIFULSZJKLUPUAZAFUBTZFUCTZMFUD
+ ccntz sylancl sseqtrrid gsumzadd ) ABCDCEFGHIFULSZJKLUPUAZAFUBTZFUCTZMFUD
UEZNQRAUSCFUFSTUTCFJUGUEACCCUPSZCUHZAURCCUIVACUJMVBCCFUPJUQUKUMUNOPUO $.
$}
@@ -233073,7 +233087,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
wceq wn wi cin c0 noel eleq2 mtbiri elin sylnib imnan sylibr imp iffalsed
oveq12d ffvelrnda mndrid syl2an2r eqtrd con2d mndlid wo cun eleq2d syl6bb
elun biimpa mpjaodan mpteq2dva eqtr4d mndidcl eqidd offval2 reseq1d ssun1
- oveq2d syl5sseqr mpteq2ia resmpt 3eqtr4a cntzidss eldifn suppss2 gsumzres
+ oveq2d sseqtrrid mpteq2ia resmpt 3eqtr4a cntzidss eldifn suppss2 gsumzres
cdif ssun2 3eqtr4d ) AHUCBUCUDZDUEZUUAGUFZJUGZUHZUCBUUAEUEZUUCJUGZUHZFUIU
JZUKUJHUUEUKUJZHUUHUKUJZFUJHGUKUJHGDULZUKUJZHGEULZUKUJZFUJABCFGUMZHUNUFZU
OUFZUFZUUEHUUHIJKLMNOPQABCDUCGIUPJRQJUPUEAJHUQMURUSZTUTZABCEUCGIUPJRQUUTT
@@ -233135,7 +233149,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
19-Dec-2014.) (Revised by AV, 5-Jun-2019.) $)
gsumsplit2 $p |- ( ph -> ( G gsum ( k e. A |-> X ) ) =
( ( G gsum ( k e. C |-> X ) ) .+ ( G gsum ( k e. D |-> X ) ) ) ) $=
- ( cmpt cgsu co cres fmpttd gsumsplit ssun1 syl5sseqr resmptd oveq2d ssun2
+ ( cmpt cgsu co cres fmpttd gsumsplit ssun1 sseqtrrid resmptd oveq2d ssun2
cun oveq12d eqtrd ) AHGBJUAZUBUCHUODUDZUBUCZHUOEUDZUBUCZFUCHGDJUAZUBUCZHG
EJUAZUBUCZFUCABCDEFUOHIKLMNOPAGBJCQUERSTUFAUQVAUSVCFAUPUTHUBAGBDJADEULZDB
DEUGTUHUIUJAURVBHUBAGBEJAVDEBEDUKTUHUIUJUMUN $.
@@ -235235,7 +235249,7 @@ mapping the (infinite, but finitely supported) cartesian product of
dprdspan $p |- ( G dom DProd S -> ( G DProd S ) = ( K ` U. ran S ) ) $=
( vk cdprd cdm wbr co crn cuni cfv id eqidd csubg wcel wss syl wral iunss
cbs cmre cgrp cacs dprdgrp eqid subgacs acsmre 3syl cv ciun wfn wceq ffnd
- dprdf fniunfv simpl simpr dprdub ralrimiva sylibr eqsstr3d dprdssv syl6ss
+ dprdf fniunfv simpl simpr dprdub ralrimiva sylibr eqsstrrd dprdssv syl6ss
mrccl syl2anc eqimss sylib r19.21bi mrcssidd adantr sstrd dprdlub mrcsscl
wa dprdsubg syl3anc eqssd ) BAFGHZBAFIZAJKZCLZVSAWBEBAGZVSMVSWCNVSBOLZBUA
LZUBLPZWAWEQWBWDPVSBUCPWDWEUDLPWFABUEWEBWEUFZUGWDWEUHUIZVSWAVTWEVSWAEWCEU
@@ -235631,7 +235645,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
( wcel cfv co wceq wbr wa c2nd c1st csn cima cv cmpt cdprd cop df-ov wrel
1st2nd sylan simpr eqeltrrd df-br sylibr wb adantr elrelimasn mpbird eqid
syl oveq2 ovex fvmpt3i fveq2d 3eqtr4a sneq imaeq2d oveq1 mpteq12dv breq2d
- cdm wral ralrimiva wss 1stdm sseldd rspcdva dmmpti a1i dprdub eqsstr3d )
+ cdm wral ralrimiva wss 1stdm sseldd rspcdva dmmpti a1i dprdub eqsstrrd )
AIBPZUAZICQZIUBQZEBIUCQZUDZUEZWIEUFZCRZUGZQZFWNUHRWFWIWHCRZWIWHUIZCQWOWGW
IWHCUJWFWHWKPZWOWPSWFWRWIWHBTZWFWQBPWSWFIWQBABUKZWEIWQSJIBULUMZAWEUNUOWIW
HBUPUQWFWTWRWSURAWTWEJUSWIWHBUTVCVAZEWHWMWPWKWNWLWHWICVDWNVBZWIWLCVEZVFVC
@@ -235650,7 +235664,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
wceq ffun funiunfv wral c1st resss sseli dprd2dlem2 sylan2 cvv c2nd cop
wa wrel 1st2nd syl2an simpr eqeltrrd fvex opelresi simplbi ovex imaeq2d
oveq1 mpteq12dv oveq2d elrnmpt1s sylancl elssuni sstrd ralrimiva sylibr
- sneq iunss eqsstr3d cpw sselda syldan dmmpti a1i imassrn mresspw syl5ss
+ sneq iunss eqsstrrd cpw sselda syldan dmmpti a1i imassrn mresspw syl5ss
frnd sspwuni sylib mrccl syl2anc adantr oveq2 fvmpt3i adantl df-ov ffnd
wfn ad2antrr simplr wb elrelimasn biimpa df-br sylanbrc fnfvima syl3anc
syl5eqel mrcssidd eqsstrd dprdlub elpw fmpttd mrcssvd mrcssd mrcsscl
@@ -235688,7 +235702,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
c2nd sneq oveq1 mpteq12dv breq2d adantr 1stdm sylan rspcdva 3ad2antr1 a1i
sseldd cop 1st2nd simpr df-br sylibr elrelimasn mpbird sneqd eqeq12d fvex
wb opth necon3bid mpbid dprdcntz df-ov oveq2 fvmpt3i fveq2d 3eqtr4a eqtrd
- oveq1d oveq2d sseqtr4d ad2antrr eqsstr3d eqsstrd sstrd cdif cin cun snssd
+ oveq1d oveq2d sseqtr4d ad2antrr eqsstrrd eqsstrd sstrd cdif cin cun snssd
cuni sylib syl6eq eleq1d simplbi crn imassrn frnd syl5ss sspwuni mrcssidd
mrccl difss ax-mp elrnmpt1s unissd mrcssd syl6eqr syl3anc sseqtrd subg0cl
c0 dprdspan ccntz cgrp dprdgrp resiun2 iunid reseq2i eqtr3i wrel relssres
@@ -235697,7 +235711,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
dprdub subgacs acsmre ssequn1 syl5req reseq2d resundi difeq1d difundir wn
undif2 eqtr3d neirr brresi sylbir syl6bi mtoi disjsn disj3 eqcomd imaundi
eldifsni uneq2d unieqd uniun cpw mresspw unss12 lsmunss imass2 mpan2 rgen
- sseqtri ralbidv ralsn wfun ffund resss eqsstr3i fdmd syl5sseqr funimassov
+ sseqtri ralbidv ralsn wfun ffund resss eqsstrri fdmd sseqtrrid funimassov
wf biimpa ffvelrnd fmpttd wfn fnmpti fnressn opeq2d dprdsn simprd disjdif
dprdsubg dprdcntz2 adantlr dprd2dlem1 resmpt oveq2i lsmsubg mrcsscl sslin
lsmlub mpbi2and dprdres simpld df-ima unieqi fveq2i eqimss ss2in dprddisj
@@ -235877,9 +235891,9 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
( ( Y e. I -> ( X =/= Y -> ( S ` X ) C_ ( Z ` ( S ` Y ) ) ) ) /\
( ( S ` X ) i^i ( K ` U. ( S " ( I \ { X } ) ) ) ) C_ { .0. } ) ) $=
( vy wcel wa wne cfv wss wi csn cdif cima cuni cin wo adantr eleq2d
- cun wceq elun syl6bb cdprd cdm wbr ad2antrr csubg syl5sseqr fssresd
+ cun wceq elun syl6bb cdprd cdm wbr ad2antrr csubg sseqtrrid fssresd
cres ssun1 fdmd simplr simprl simprr dprdcntz fvres ad2antlr fveq2d
- ad2antrl 3sstr3d exp32 co dprdub eqsstr3d cbs dprdssv ssun2 cntz2ss
+ ad2antrl 3sstr3d exp32 co dprdub eqsstrrd cbs dprdssv ssun2 cntz2ss
eqid sylancr jaod sylbid clsm cmre cgrp cacs dprdgrp subgacs acsmre
sstrd syl difundir difeq1d c0 simpr snssd sslin incom syl5eqr sseq0
3syl syl2anc disj3 sylib uneq2d 3eqtr4a imaeq2d syl6eq unieqd uniun
@@ -235943,7 +235957,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.) $)
dmdprdsplit2 $p |- ( ph -> G dom DProd S ) $=
( cfv cvv wcel vx csubg cmrc eqid cres cdprd cdm wbr cgrp dprdgrp syl
- vy cun ssun1 syl5sseqr fssresd dprddomcld ssun2 unexg syl2anc eqeltrd
+ vy cun ssun1 sseqtrrid fssresd dprddomcld ssun2 unexg syl2anc eqeltrd
fdmd cv wne wss wo wi eleq2d elun syl6bb wa cdif cima dmdprdsplit2lem
csn cin c0 incom syl5eqr uncom syl6eq dprdsubg cntzrecd jaodan simpld
cuni co ex sylbid 3imp2 biimpa simprd syldan dmdprdd ) AUAULDEFEUBRZU
@@ -235964,7 +235978,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
( G DProd ( S |` C ) ) C_ ( Z ` ( G DProd ( S |` D ) ) ) /\
( ( G DProd ( S |` C ) ) i^i ( G DProd ( S |` D ) ) ) = { .0. } ) ) ) $=
( cdprd wbr wa co wss wceq adantr cdm cres cfv cin csn simpr csubg fdmd
- w3a cun ssun1 syl5sseqr dprdres simpld ssun2 jca c0 dprdcntz2 dprddisj2
+ w3a cun ssun1 sseqtrrid dprdres simpld ssun2 jca c0 dprdcntz2 dprddisj2
3jca wf simpr1l simpr1r simpr2 simpr3 dmdprdsplit2 impbida ) AEDNUAZOZE
DBUBZVHOZEDCUBZVHOZPZEVJNQZEVLNQZHUCRZVOVPUDGUESZUIZAVIPZVNVQVRVTVKVMVT
VKVOEDNQZRVTBDEFAVIUFZADUAFSVIAFEUGUCZDIUHTZVTBCUJZBFBCUKAFWESZVIKTZULZ
@@ -235981,10 +235995,10 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
dprdsplit $p |- ( ph -> ( G DProd S ) =
( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) $=
( cdprd co cfv wcel wss wa wceq adantr vx cres csubg fdmd ccntz cdm ssun1
- wbr cun syl5sseqr dprdres simpld dprdsubg syl cin c0g csn w3a dmdprdsplit
+ wbr cun sseqtrrid dprdres simpld dprdsubg syl cin c0g csn w3a dmdprdsplit
ssun2 eqid mpbid simp2d lsmsubg syl3anc cv wo eleq2d syl6bb biimpa adantl
elun fvres fssresd simpr dprdub lsmub1 syl2anc sstrd lsmub2 jaodan syldan
- eqsstr3d dprdlub simprd wb lsmlub mpbi2and eqssd ) AFEMNZFEBUBZMNZFECUBZM
+ eqsstrrd dprdlub simprd wb lsmlub mpbi2and eqssd ) AFEMNZFEBUBZMNZFECUBZM
NZDNZAEWOUAFGLAGFUCOZEHUDZAWLWPPZWNWPPZWLWNFUEOZOQZWOWPPAFWKMUFZUHZWRAXCW
LWJQZABEFGLWQABCUIZBGBCUGJUJZUKZULZWKFUMUNZAFWMXBUHZWSAXJWNWJQZACEFGLWQAX
ECGCBUTJUJZUKZULZWMFUMUNZAXCXJRZXAWLWNUOFUPOZUQSZAFEXBUHZXPXAXRURLABCEFGX
@@ -236067,7 +236081,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
( S ` X ) C_ ( Z ` ( G DProd ( S |` ( I \ { X } ) ) ) ) ) $=
( cfv csn cres cdprd co wbr wss cin wceq cun dpjlem cdm wa c0g dprdf2
cdif w3a c0 disjdif a1i undif2 snssd ssequn1 syl5req eqid dmdprdsplit
- sylib mpbid simp2d eqsstr3d ) AEBKCBELZMZNOZCBDVAUFZMZNOZFKZABCDEGHIU
+ sylib mpbid simp2d eqsstrrd ) AEBKCBELZMZNOZCBDVAUFZMZNOZFKZABCDEGHIU
AACVBNUBZPCVEVHPUCZVCVGQZVCVFRCUDKZLSZACBVHPVIVJVLUGGAVAVDBCDVKFABCDG
HUEVAVDRUHSAVADUIUJAVAVDTVADTZDVADUKAVADQVMDSAEDIULVADUMUQUNJVKUOUPUR
USUT $.
@@ -236616,7 +236630,7 @@ factorization into prime power factors (even if the exponents are
odcl hashcl ssrabdv oveq12d breq2d rabbidv fvmpt3i sseqtr4d nnnn0d pcdvds
oveq1 ne0d sneq difeq2d breq12d notbid wral cmpt ssrab3 ssidd clsm fssres
difss eldif iddvdsexp sylan eqeltrrd breq1 elrab2 sylanbrc ex con3d elnn0
- impr wo ord nncnd exp0d fvex hashsng ax-mp syl6reqr snfi sylancr eqsstr3d
+ impr wo ord nncnd exp0d fvex hashsng ax-mp syl6reqr snfi sylancr eqsstrrd
sylan2b eqsstrd dprdlub lsmss2 undif2 ssequn1 syl5req jca eqtrd cfz fzfid
w3a 3ad2ant2 dvdsle syl2anr 3ad2ant1 fznn mpbir2and syl5eqss ablfac1eulem
3impia rabssdv ralrimiva rspcdva coprm rpexp1i syl31anc ccntz inss1 inass
@@ -238672,7 +238686,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be
.1. = ( 0g ` M ) ) $=
( vy wcel cbs cfv eqid wceq co cvv oveqd 3ad2antl1 eqtr3d syldan crg cmgp
wss w3a cplusg c0g simp3 mgpbas ressbas2 3ad2ant2 eleqtrd cv simp2 sselda
- eqsstr3d wa fvex syl6eqel mgpplusg ressplusg syl adantr ringlidm ringridm
+ eqsstrrd wa fvex syl6eqel mgpplusg ressplusg syl adantr ringlidm ringridm
cmulr ismgmid2 ) CUAJZABUCZDAJZUDZIEKLZEUELZDEEUFLZVKMVMMVLMVJDAVKVGVHVIU
GVHVGAVKNVIABECUBLZFBCVNVNMZGUHUIUJZUKVJIULZVKJZVQBJZDVQVLOZVQNVJVKBVQVJV
KABVPVGVHVIUMUOUNZVJVSUPZDVQCVELZOZVTVQWBWCVLDVQVJWCVLNZVSVJAPJWEVJAVKPVP
@@ -240479,7 +240493,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be
(Revised by Mario Carneiro, 5-Oct-2015.) $)
invrpropd $p |- ( ph -> ( invr ` K ) = ( invr ` L ) ) $=
( cmgp cfv cui cress co cbs wceq eqid wcel wa cvv cminusg cinvr unitpropd
- unitgrpbas a1i syl6eq cmulr cplusg unitss syl5sseqr sselda anim12dan fvex
+ unitgrpbas a1i syl6eq cmulr cplusg unitss sseqtrrid sselda anim12dan fvex
cv syldan mgpplusg ressplusg ax-mp 3eqtr3g grpinvpropd invrfval 3eqtr4g
oveqi ) AEJKZELKZMNZUAKFJKZFLKZMNZUAKEUBKZFUBKZABCVEVFVIVEVFOKPAEVEVFVEQZ
VFQZUDUEAVEVHVIOKABCDEFGHIUCFVHVIVHQZVIQZUDUFABUNZVERZCUNZVERZSZSVPVREUGK
@@ -241741,7 +241755,7 @@ nonzero elements form a group under multiplication (from which it
Stefan O'Rear, 27-Nov-2014.) $)
subrg1 $p |- ( A e. ( SubRing ` R ) -> .1. = ( 1r ` S ) ) $=
( vx cfv wcel cur cbs cmulr co wceq wa eqid crg oveqd eqeq1d syldan sylan
- csubrg cv wral subrg1cl subrgbas eleqtrd subrgss eqsstr3d sselda subrgrcl
+ csubrg cv wral subrg1cl subrgbas eleqtrd subrgss eqsstrrd sselda subrgrcl
ringidmlem ressmulr biimpa ralrimiva subrgring isringid mpbi2and syl6reqr
anbi12d wb syl ) ABUBHZIZCJHZBJHZDVDVFCKHZIZVFGUCZCLHZMZVINZVIVFVJMZVINZO
ZGVGUDZVEVFNZVDVFAVGABVFVFPZUEABCEUFZUGVDVOGVGVDVIVGIVIBKHZIZVOVDVGVTVIVD
@@ -241796,7 +241810,7 @@ nonzero elements form a group under multiplication (from which it
4-Dec-2014.) $)
subrgdvds $p |- ( A e. ( SubRing ` R ) -> E C_ .|| ) $=
( vx vy vz cfv wcel cv wbr cbs cmulr co wrex eqid csubrg reldvdsr a1i cop
- wrel wa subrgbas subrgss eqsstr3d sseld ressmulr oveqd eqeq1d rexbidv wss
+ wrel wa subrgbas subrgss eqsstrrd sseld ressmulr oveqd eqeq1d rexbidv wss
wceq wi ssrexv syl sylbird anim12d dvdsr 3imtr4g df-br 3imtr3g relssdv )
ACUALZMZIJEBEUEVHEDHUBUCVHINZJNZEOZVIVJBOZVIVJUDZEMVMBMVHVIDPLZMZKNZVIDQL
ZRZVJUPZKVNSZUFVICPLZMZVPVICQLZRZVJUPZKWASZUFVKVLVHVOWBVTWFVHVNWAVIVHVNAW
@@ -241815,7 +241829,7 @@ nonzero elements form a group under multiplication (from which it
subrguss $p |- ( A e. ( SubRing ` R ) -> V C_ U ) $=
( vx cfv wcel wa cur cdsr wbr eqid wceq adantr cmulr co coppr simpr sylib
csubrg cv isunit simpld subrg1 breqtrrd wss subrgdvds ssbrd mpd cinvr cbs
- subrgbas subrgss eqsstr3d unitcl adantl subrgring ringinvcl sylan opprbas
+ subrgbas subrgss eqsstrrd unitcl adantl subrgring ringinvcl sylan opprbas
sseldd crg dvdsrmul syl2anc opprmul unitrinv oveqd 3eqtr4d syl5eq breqtrd
ressmulr sylanbrc ex ssrdv ) ABUDJZKZIEDVTIUEZEKZWADKZVTWBLZWABMJZBNJZOZW
AWEBUAJZNJZOWCWDWAWECNJZOWGWDWACMJZWEWJWDWAWKWJOZWAWKCUAJZNJZOZWDWBWLWOLV
@@ -241838,7 +241852,7 @@ nonzero elements form a group under multiplication (from which it
subrginv $p |- ( ( A e. ( SubRing ` R ) /\ X e. U ) ->
( I ` X ) = ( J ` X ) ) $=
( cfv wcel cur cmulr co wceq adantr eqid syl2an2r csubrg crg cbs subrgrcl
- wa wss subrgbas subrgss eqsstr3d subrgring ringinvcl sseldd unitcl adantl
+ wa wss subrgbas subrgss eqsstrrd subrgring ringinvcl sseldd unitcl adantl
sylan cui subrguss sselda ringass syl13anc unitlinv ressmulr oveqd subrg1
3eqtr4d oveq1d unitrinv oveq2d 3eqtr3d ringlidm ringridm ) ABUALZMZGDMZUE
ZBNLZGELZBOLZPZGFLZVPVRPZVQVTVOVTGVRPZVQVRPZVTGVQVRPZVRPZVSWAVOBUBMZVTBUC
@@ -242012,7 +242026,7 @@ nonzero elements form a group under multiplication (from which it
( cdr wcel cfv wa cdif syl wceq wb eqid ad2antlr wss csubrg cv wral cinvr
csn cbs crg cui simpllr subrgring c0g simpr eldifsn sylib simpld subrgbas
eleqtrd simprd subrg0 neeqtrd drngunit mpbir2and syl2anc subrginv 3eltr4d
- wne ringinvcl ralrimiva subrguss isdrng simprbi ad2antrr unitss syl5sseqr
+ wne ringinvcl ralrimiva subrguss isdrng simprbi ad2antrr unitss sseqtrrid
cin sseqtrd ssind subrgss difin2 simprl sseldd simprr subrgunit mpbir3and
sseqtr4d w3a expr ralimdva imp dfss3 sylibr eqssd sneqd difeq12d sylanbrc
eqtrd impbida ) CJKZBCUALKZMZDJKZAUBZELZBKZABFUEZNZUCZWTXAMZXDAXFXHXBXFKZ
@@ -242395,7 +242409,7 @@ nonzero elements form a group under multiplication (from which it
difssd subrgss ressbas2 3syl sseqtr4d ressabs eqtr3d ciin intiin syl5reqr
cv difeq1d oveq2d cmpt crn vex difexi dfiin3 iindif1 syl5eqr difss mgpbas
cvv ax-mp fvexi ciun iinssiun csubg subrgsubg ssriv syl6ss subgint adantr
- subg0 sneqd difeq2d sselda ssdifd eqsstr3d iunssd sstrd syl5eqssr sylancr
+ subg0 sneqd difeq2d sselda ssdifd eqsstrrd iunssd sstrd syl5eqssr sylancr
wral drngmgp syl6sseq sylan eqcomd oveq12d simpr subrg0 difeq12d eqeltrrd
eqeltrd issubg syl3anbrc ralrimiva rnmptss cdm dmmptg difexg mprg eqnetrd
wa a1i dm0rn0 necon3bii sylib subggrp syl5eqel isdrng2 sylanbrc ) ADUAKZD
@@ -242448,7 +242462,7 @@ nonzero elements form a group under multiplication (from which it
( vs vx vy cdr wcel cfv wss cv co cbs eqid cmulr wceq wral syl wa syl2anc
ccrg cfield csdrg id csubrg cress issdrg simp2bi ssriv a1i sdrgid simp3bi
ne0d adantl subdrgint crg drngring cmgp ccntr ccntz ssidd cntzsdrg intss1
- cint mgpbas syl6sseq cntrss syl6ss ressbas2 eqsstr3d adantr simprl sseldd
+ cint mgpbas syl6sseq cntrss syl6ss ressbas2 eqsstrrd adantr simprl sseldd
cntrval simprr mgpplusg cntri cvv ssexd oveqdr 3eqtr3d ralrimivva iscrng2
ressmulr sylanbrc isfld ) BGHZAGHZAUAHZAUBHWGBBUCIZADCWGUDZWJBUEIZJWGDWJW
LDKZWJHZWGWMWLHZBWMUFLGHZBWMUGZUHUIUJWGWJBMIZWRBWRNZUKZUMWNWPWGWNWGWOWPWQ
@@ -244880,7 +244894,7 @@ Absolute value (abstract algebra)
lmodring adantr csubg cgrp lsssubg subggrp cv lssvscl 3impb simpll simpr1
crg w3a ad2antlr simpr2 sseldd simpr3 lmodvsdi lmodvsdir lmodvsass sselda
syl13anc lmodvs1 adantlr syldan islmodd jca simprl fvex syl6eqel a1i clss
- eqcomd eqsstr3d c0 wne lmodgrp ad2antll grpbn0 lsscl sylan islssd eqeltrd
+ eqcomd eqsstrrd c0 wne lmodgrp ad2antll grpbn0 lsscl sylan islssd eqeltrd
lss1 impbida ) DUBJZBAJZBCUCZEUBJZKZXAXBKZXCXDXBXCXAABCDGHUDZLZXFIUAUEDUF
MZNMZDUGMZXIUGMZDUHMZXIUIMZXIUJMZXIBEXBXAXCBENMZOZXGXCXQXABCEDFGUKZLULXBX
KEUGMZOZXABXKDEAFXKPZUMLXBXIEUFMZOZXABXIDEAFXIPZUNLXBXMEUHMZOZXABXMDEAFXM
@@ -245283,7 +245297,7 @@ Absolute value (abstract algebra)
(Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro,
19-Jun-2014.) $)
lspssid $p |- ( ( W e. LMod /\ U C_ V ) -> U C_ ( N ` U ) ) $=
- ( vt clmod wcel wss wa clss cfv crab cint ssintub eqid lspval syl5sseqr
+ ( vt clmod wcel wss wa clss cfv crab cint ssintub eqid lspval sseqtrrid
cv ) DHIACJKAGTJGDLMZNOAABMGAUAPGUAABCDEUAQFRS $.
$( The span of a set of vectors is idempotent. (Contributed by NM,
@@ -245556,7 +245570,7 @@ Absolute value (abstract algebra)
( wcel cfv csn co eqid lmodvneg1 sneqd fveq2d wss cgrp lspsnvsi syl3anc
clmod wa csca cur cminusg cvsca cbs simpl lmodfgrp lmod1cl syl2anc adantr
grpinvcl simpr wceq lmodvnegcl syldan lmodgrp grpinvinv sylan eqtrd eqssd
- eqsstr3d ) DUAIZECIZUBZEAJZKZBJZEKZBJZVFVIDUCJZUDJZVLUEJZJZEDUFJZLZKZBJZV
+ eqsstrrd ) DUAIZECIZUBZEAJZKZBJZEKZBJZVFVIDUCJZUDJZVLUEJZJZEDUFJZLZKZBJZV
KVFVRVHBVFVQVGVPVMVLVNACDEFGVLMZVPMZVMMZVNMZNOPVFVDVOVLUGJZIZVEVSVKQVDVEU
HZVDWEVEVDVLRIVMWDIWEVLDVTUIVMVLWDDVTWDMZWBUJWDVLVNVMWGWCUMUKULZVDVEUNVOV
PVLWDBCDEVTWGFWAHSTVCVFVKVOVGVPLZKZBJZVIVFWJVJBVFWIEVFWIVGAJZEVDVEVGCIZWI
@@ -246382,7 +246396,7 @@ Absolute value (abstract algebra)
E* g e. ( S LMHom T ) ( g |` X ) = F ) $=
( vh wss cfv wceq wa cv cres wral wcel cdm ad2antrl weq wi clmhm co eqtr3
wrmo cin inss1 dmss ax-mp wfn cbs wf eqid lmhmf ffnd adantrr syl syl5sseq
- fndm simplr clmod lmhmlmod1 adantr lmhmeql simprr lspssp syl3anc eqsstr3d
+ fndm simplr clmod lmhmlmod1 adantr lmhmeql simprr lspssp syl3anc eqsstrrd
clss eqssd expr wb ffn simpll fnreseql fneqeql syl2anc 3imtr4d ralrimivva
3syl syl5 reseq1 eqeq1d rmo4 sylibr ) GAKZGFLZAMZNZDOZGPZEMZJOZGPZEMZNZDJ
UAZUBZJBCUCUDZQDWTQWMDWTUFWJWSDJWTWTWQWLWOMZWJWKWTRZWNWTRZNZNZWRWLWOEUEXE
@@ -247409,7 +247423,7 @@ division ring is an Abelian group (vectors) together with a division ring
( wcel co csn cfv syl3anc clvec wne wa w3a clmod lveclmod 3ad2ant1 simp2l
wss simp3 lspsnvsi cinvr cmulr wceq lvecdrng simp2r eqid drnginvrl oveq1d
cur drnginvrcl lmodvsass syl13anc lmodvs1 syl2anc 3eqtr3d fveq2d lmodvscl
- cdr sneqd eqsstr3d eqssd ) GUAPZADPZAIUBZUCZHFPZUDZAHBQZRESZHRZESZVRGUEPZ
+ cdr sneqd eqsstrrd eqssd ) GUAPZADPZAIUBZUCZHFPZUDZAHBQZRESZHRZESZVRGUEPZ
VNVQVTWBUIVMVPWCVQGUFUGZVMVNVOVQUHZVMVPVQUJZABCDEFGHKMJLOUKTVRWBACULSZSZV
SBQZRZESZVTVRWJWAEVRWIHVRWHACUMSZQZHBQZCUTSZHBQZWIHVRWMWOHBVRCVIPZVNVOWMW
OUNVMVPWQVQCGKUOUGZWEVMVNVOVQUPZDCWLWOWGAIMNWLUQZWOUQZWGUQZURTUSVRWCWHDPZ
@@ -248061,7 +248075,7 @@ division ring is an Abelian group (vectors) together with a division ring
mpbid simp2l wi oveq2 eqeq2d biimpac ad2antrr lssel lmod0vrid biimpd syl7
ex exp4a 3imp eleq1 biimparc syl6an necon3bd mpd csg cabl lmodabl simp1l1
simp2r ablpncan2 syl3anc simp1rl eqeltrrd simp1l2 sselda syl22anc simp12r
- lssvsubcl lspsneleq lspsnel5a eqsstr3d mpd3an23 rexlimdvv exlimddv lsmlub
+ lssvsubcl lspsneleq lspsnel5a eqsstrrd mpd3an23 rexlimdvv exlimddv lsmlub
3exp mpbi2and eqssd ) ADEUOZEDIUDFUEZBUFZUGZUHZEYCAYAYDUIYEDEUGZYBEUGZYCE
UGZYEYAYFAYAYDUJZDEUKZRYEUAULZESZYKDSZUMZUNZYGUAYEYAYOUAVGYIUADEUPRYEYOUN
ZYKUBULZUCULZHUQUEZUFZURZUCYBUSUBDUSZYGYPYKYCSZUUBYPEYCYKAYAYDYOUTYEYLYNV
@@ -248450,7 +248464,7 @@ U C_ ( N ` { X } ) ) -> ( U = ( N ` { X } ) \/ U = { .0. } ) ) $=
jca alrimiv 3jca csn cdif wral simpr1 simpr2 simplr1 ssdifssd fvexi ssexg
wn cbs sylancl simplr3 difssd simpr neldifsn nelne1 necomd psseq1 psseq1d
fveq2 imbi12d spcgv syl3c simprbi simplr2 clss ad2antrr adantrr lspcl cun
- dfpss3 ssun1 undif1 sseqtr4i lspssid simprr snssd syl5ss syl3anc eqsstr3d
+ dfpss3 ssun1 undif1 sseqtr4i lspssid simprr snssd syl5ss syl3anc eqsstrrd
unssd lspssp expr mtod ralrimiva wb islbs2 adantr mpbir3and impbida ) EUA
KZABKZADLZACMZDUBZFUCZANZYGCMZDNZUDZFUFZOZYBYCPZYDYFYLYCYDYBABDEGHUEZUGYC
YFYBABCDEGHIUHUGYNYKFYBYCYHYJYBYCYHOZYIDLZYIDQZYJYPEUIKZYGDLZYQYBYCYSYHEU
@@ -250311,7 +250325,7 @@ zero ring (at least if its operations are internal binary operations).
29-Mar-2015.) $)
drngdomn $p |- ( R e. DivRing -> R e. Domn ) $=
( cdr wcel cnzr cbs cfv c0g csn cdif crlreg wss cdomn drngnzr cui wceq eqid
- crg isdrng simprbi drngring unitrrg syl eqsstr3d isdomn2 sylanbrc ) ABCZADC
+ crg isdrng simprbi drngring unitrrg syl eqsstrrd isdomn2 sylanbrc ) ABCZADC
AEFZAGFZHIZAJFZKALCAMUFUIANFZUJUFAQCZUKUIOUGAUKUHUGPZUKPZUHPZRSUFULUKUJKATA
UKUJUJPZUNUAUBUCUGAUJUHUMUPUOUDUE $.
@@ -250776,7 +250790,7 @@ such that every prime ideal contains a prime element (this is a
(Contributed by Mario Carneiro, 7-Jan-2015.) $)
aspssid $p |- ( ( W e. AssAlg /\ S C_ V ) -> S C_ ( A ` S ) ) $=
( vt casa wcel wss wa cv csubrg cfv clss cin crab cint ssintub eqid
- aspval syl5sseqr ) DHIBCJKBGLJGDMNDONZPZQRBBANGBUDSGABUCCDEFUCTUAUB $.
+ aspval sseqtrrid ) DHIBCJKBGLJGDMNDONZPZQRBBANGBUDSGABUCCDEFUCTUAUB $.
$}
${
@@ -251385,7 +251399,7 @@ such that every prime ideal contains a prime element (this is a
cofr 3ad2antr2 simpr2 cv cdif cfv eldifi wral simpr3 ffnd 3ad2antr1 simpl
psrbagf inidm eqidd ofrfval mpbid r19.21bi sylan2 cvv wss jca eqimss 3syl
c0ex a1i suppssr breqtrd ffvelrn syl2an nn0ge0d nn0red 0re letri3 sylancl
- cr wb mpbir2and suppss eqsstr3d ) EFIZCAIZEJDKZDCLUDMZUAZNZDOPQZDRSUBZCOP
+ cr wb mpbir2and suppss eqsstrrd ) EFIZCAIZEJDKZDCLUDMZUAZNZDOPQZDRSUBZCOP
QZWDWEWFWKWJTWGDEFUCUEWIEJHDWLRWDWEWFWGUFZWIHUGZEWLUHIZNZWNDUIZRTZWQRLMZR
WQLMZWPWQWNCUIZRLWOWIWNEIZWQXALMZWNEWLUJZWIXCHEWIWGXCHEUKWDWEWFWGULWIHEEW
QXALEDCFFWIEJDWMUMWIEJCWDWFWEEJCKZWGABCEFGUPUNZUMWDWHUOZXGEUQWIXBNZWQURXH
@@ -252519,7 +252533,7 @@ such that every prime ideal contains a prime element (this is a
by Mario Carneiro, 3-Jul-2015.) $)
resspsrbas $p |- ( ph -> B = ( Base ` P ) ) $=
( vf cbs cfv cvv c0 wss wceq wcel wa cv ccnv cn cima cfn cn0 cmap co crab
- fvex csubrg subrgbas syl eqid subrgss eqsstr3d adantr mapss sylancr simpr
+ fvex csubrg subrgbas syl eqid subrgss eqsstrrd adantr mapss sylancr simpr
psrbas 3sstr4d wn cmps reldmpsr ovprc1 syl5eq adantl fveq2d base0 3eqtr4g
0ss syl6eqss pm2.61dan ressbas2 ) ABEQRZUAZBCQRUBAISUCZWAAWBUDZHQRZPUEUFU
GUHUIUCPUJIUKULUMZUKULZDQRZWEUKULZBVTWCWGSUCWDWGUAZWFWHUADQUNAWIWBAWDFWGA
@@ -252532,7 +252546,7 @@ such that every prime ideal contains a prime element (this is a
resspsradd $p |- ( ( ph /\ ( X e. B /\ Y e. B ) ) ->
( X ( +g ` U ) Y ) = ( X ( +g ` P ) Y ) ) $=
( wcel cfv co vf wa cplusg cof eqid simprl simprr psradd cbs cv ccnv cima
- cn cfn cn0 cmap crab wss cvv fvex csubrg wceq subrgbas syl eqsstr3d mapss
+ cn cfn cn0 cmap crab wss cvv fvex csubrg wceq subrgbas syl eqsstrrd mapss
subrgss sylancr adantr cmps reldmpsr elbasov simpld psrbas 3sstr4d sseldd
ad2antrl ressplusg ofeq oveqd eqtrd fvexi mp1i 3eqtr2d ) AJBRZKBRZUBZUBZJ
KGUCSZTJKHUCSZUDZTZJKEUCSZTZJKCUCSZTWHBWJWIHGIJKNOWJUEWIUEAWEWFUFZAWEWFUG
@@ -252552,7 +252566,7 @@ such that every prime ideal contains a prime element (this is a
psrbaglefi sylan csubmnd csubrg subrgsubg syl subgsubm ad2antrr ad3antrrr
csubg cbs wf simprl psrelbas adantr elrabi ffvelrn wceq subrgbas eleqtrrd
syl2an simprr ssrab2 simplr simpr syl3anc sseldi ffvelrnd subrgmcl fmpttd
- psrbagconcl gsumsubm ressmulr oveqd mpteq2dva oveq2d wss subrgss eqsstr3d
+ psrbagconcl gsumsubm ressmulr oveqd mpteq2dva oveq2d wss subrgss eqsstrrd
eqtrd fvex mapss sylancr psrbas 3sstr4d sseldd psrmulfval 3eqtr4rd fvexi
mp1i ) AJBRZKBRZUEZUEZJKGUFSZTZJKEUFSZTZJKCUFSZTYBUAUBUGUHUIUJUKRUBULIUMT
UNZDUCUDUGUAUGZUPUOUQZUDYHUNZUCUGZJSZYIYLURUSTZKSZDUFSZTZUTZVATZUTUAYHHUC
@@ -253942,7 +253956,7 @@ series in the subring which are also polynomials (in the parent ring).
A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } ) $=
( cvv wcel wa cv cpr wss cfv wbr wceq wi wral wrex wo copab cple cnx csts
cop co eqid simprl simprr cxp adantr opsrval fveq2d cmps ovexi fvexi xpex
- cbs vex anbi1i opabbii opabssxp eqsstr3i ssexi pleid setsid mp2an 3eqtr4g
+ cbs vex anbi1i opabbii opabssxp eqsstrri ssexi pleid setsid mp2an 3eqtr4g
prss wn c0 copws reldmopsr ovprc adantl fveq1d syl5eq 0fv syl6eq reldmpsr
str0 base0 xpeq2d xp0 sseq0 sylancr eqtr4d pm2.61dan ) ANUEUFZIUEUFZUGZOB
UHZCUHZUIFUJZDUHZXIUKXLXJUKKULEUHZXLGULXMXIUKXMXJUKUMUNEHUOUGDHUPXIXJUMUQ
@@ -254929,7 +254943,7 @@ This function maps an element of the formal polynomial algebra (with
syl6 cmrc cmps casp cascl casa psrassa mvrf frnd aspval2 mplbas2 mplsubrg
mplval2 subsubrg2 fveq2d ressascl syl6reqr rneqd cmre subrgmre 3syl unssd
cpw syl3anc ad2antrr wfn wb rhmf ffn adantr uneq1d assaring submrc eqtr2d
- fveq12d eqsstr3d 3eqtr3d simpr ad2antlr mrcsscl eqsstrd fnreseql fneqeql2
+ fveq12d eqsstrrd 3eqtr3d simpr ad2antlr mrcsscl eqsstrd fnreseql fneqeql2
rhmeql simprl simprr 3imtr4d syl5 ralrimivva reseq1 rmo4 sylibr sylc reu5
rmoim sylanbrc ) AGUEZBUFZHUGZUVGKUFZIUGZUHZGDFUIUJZUKZUVLGUVMULZUVLGUVMU
MAUADUNUOZFUBUCUEUPVMUQURUSUCUTJVAUJVBZUBUEZUAUEUOHUOFVCUOZUVRIUVSVDUOZVE
@@ -263639,7 +263653,7 @@ S C_ ( ._|_ ` ( ._|_ ` S ) ) ) $=
(Contributed by Mario Carneiro, 16-Oct-2015.) $)
pjcss $p |- ( W e. PreHil -> dom K C_ C ) $=
( vx cphl wcel cdm cv wa clsm cfv cocv cbs eqid simpl clss wss co wceq ex
- pjdm2 simprbda lssss syl ocvss simplbda syl5sseqr lsmcss ssrdv ) CGHZFBIZ
+ pjdm2 simprbda lssss syl ocvss simplbda sseqtrrid lsmcss ssrdv ) CGHZFBIZ
AULFJZUMHZUNAHULUOKZACLMZUNCNMZCOMZCEUSPZURPZUQPZULUOQUPUNCRMZHZUNUSSULUO
VDUNUNURMZUQTZUSUAZUQUNBVCURUSCUTVCPZVAVBDUCZUDVCUNUSCUTVHUEUFUPUSVEURMVF
VEURUSCUTVAUGULUOVDVGVIUHUIUJUBUK $.
@@ -265886,7 +265900,7 @@ each element of the range ( ~ islindf5 ). (Contributed by Stefan
c0g wral 3adant3 f1f 3ad2ant3 fco ffdmd wne simpl2 eleq2d biimpa ffvelrnd
fdmd adantrr eldifi ad2antll eldifsni lindfind syl22anc wi wfn f1fn fvco2
oveq2d eleq1d simpl1 imassrn frnd syl5ss imaco difeq1d imaeq2d ccnv df-f1
- crn simprbi imadif syl eqtrd fnsnfv sylan difeq2d ssdifd eqsstr3d eqsstrd
+ crn simprbi imadif syl eqtrd fnsnfv sylan difeq2d ssdifd eqsstrrd eqsstrd
wfun imass2 syl5eqss lspss syl3anc syldan sseld sylbid mtod ralrimivva wb
simp1 rellindf brrelex1i 3ad2ant2 simp3 dmexd fex coexg islindf mpbir2and
f1dmex ) DGHZADIUAZCAUBZBUCZUDZABUEZDIUAZYHUBZDUFJZYHKZEUGZFUGZYHJZDUHJZU
@@ -279936,7 +279950,7 @@ _Introduction to General Topology_ (1983), p. 114) and it is convenient
istopon $p |- ( J e. ( TopOn ` B ) <-> ( J e. Top /\ B = U. J ) ) $=
( vj vb ctopon cfv wcel cvv ctop cuni wceq elfvex uniexg eleq1 syl5ibrcom
wa cv crab cpw wss imp eqeq1 rabbidv df-topon vpwex pwex rabss pwuni pweq
- wi syl5sseqr selpw sylibr mprgbir ssexi fvmpt3i eleq2d unieq eqeq2d elrab
+ wi sseqtrrid selpw sylibr mprgbir ssexi fvmpt3i eleq2d unieq eqeq2d elrab
a1i syl6bb pm5.21nii ) BAEFZGZAHGZBIGZABJZKZPZBAELVGVIVFVGVFVIVHHGBIMAVHH
NOUAVFVEBACQZJZKZCIRZGVJVFVDVNBDADQZVLKZCIRZVNHEVOAKVPVMCIVOAVLUBUCCDUDVQ
VOSZSZVRDUEUFVQVSTVPVKVSGZUJZCIVPCIVSUGWAVKIGVPVKVRTVTVPVLSVKVRVKUHVOVLUI
@@ -280965,7 +280979,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
{ x e. ~P A | ( P e. x -> x = A ) } e. ( TopOn ` A ) ) $=
( vy vz wcel wa cv wceq wi wss wral eleq2 eqeq1 imbi12d adantr syl ex cvv
cpw crab ctop ctopon cfv wal cin ssrab simprl sspwuni sylib vuniex sylibr
- cuni elpw wrex eluni2 r19.29 simpr impr elssuni eqsstr3d rexlimiva syl5bi
+ cuni elpw wrex eluni2 r19.29 simpr impr elssuni eqsstrrd rexlimiva syl5bi
ad2antll jctild eqss syl6ibr elrabd alrimiv inss1 elpwid syl5ss vex inex1
simprll elin simprlr simprrr anim12d ineq12 inidm syl6eq syl6 jca anbi12i
elrab 3imtr4g ralrimivv wb pwexg rabexg istopg mpbir2and pwidg a1d ssrab2
@@ -281442,7 +281456,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
$( A subset of a topology's underlying set is included in its closure.
(Contributed by NM, 22-Feb-2007.) $)
sscls $p |- ( ( J e. Top /\ S C_ X ) -> S C_ ( ( cls ` J ) ` S ) ) $=
- ( vx ctop wcel wss wa cv ccld cfv crab cint ccl ssintub clsval syl5sseqr
+ ( vx ctop wcel wss wa cv ccld cfv crab cint ccl ssintub clsval sseqtrrid
) BFGACHIAEJHEBKLZMNAABOLLEASPEABCDQR $.
$( A subset includes its interior. (Contributed by NM, 3-Oct-2007.)
@@ -281790,7 +281804,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
( vx c0 cpr ccld cfv cv wcel cid wss cdif wa ctop wb indistop wceq difeq2
ax-mp syl6eq syl6eqel indisuni iscld simpl dfss4 sylib wo simpr syl6eleqr
indislem elpri dif0 fvex prid2 eleqtri difid 0ex prid1 jaoi 3syl eqeltrrd
- sylbi ssriv 0cld topcld prssi mp2an eqsstr3i eqssi ) CADZEFZVIBVJVIBGZVJH
+ sylbi ssriv 0cld topcld prssi mp2an eqsstrri eqssi ) CADZEFZVIBVJVIBGZVJH
ZVKAIFZJZVMVKKZVIHZLZVKVIHVIMHZVLVQNAOZVKVIVMAUAZUBRVQVMVOKZVKVIVQVNWAVKP
VNVPUCVKVMUDUEVQVOCVMDZHVOCPZVOVMPZUFWAVIHZVQVOVIWBVNVPUGAUIZUHVOCVMUJWCW
EWDWCWAVMVIWCWAVMCKVMVOCVMQVMUKSVMWBVICVMAIULUMWFUNTWDWACVIWDWAVMVMKCVOVM
@@ -282983,7 +282997,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
( vc vd va vb ve ctop wcel wss cv wa wrex cin wceq syl2anc cvv crest cnei
w3a cfv cuni wex nfv nfre1 nfan simpl anim2i cdif cun simp-5r simp1 simp2
co restuni ad5antr sseqtr4d sstrd eltopss ssdifssd unssd simpr1l 3anassrs
- simplr simpr sseqtrd inss1 inundif simpr1r eqsstr3d unss1 syl5eqssr sseq2
+ simplr simpr sseqtrd inss1 inundif simpr1r eqsstrrd unss1 syl5eqssr sseq2
syl6ss sseq1 anbi12d rspcev syl12anc c0 indir incom disjdif eqtr3i uneq2i
syl un0 3eqtri df-ss biimpi syl5req vex difexi unex anbi2d rexbidv eqeq2d
ineq1 spcev syl21anc ad3antrrr uniexg ssexd elrest biimpa r19.29a sylanl1
@@ -283265,10 +283279,10 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
( vm vn ctsr wcel cun cvv wss vz csn cfi cfv cv cin wceq wrex w3o wb snex
ssun2 cuni ordtuni cdm dmexg syl5eqel eqeltrrd uniexb sylibr ssexg elfiun
sylancr wi ssun1 eqsstri sseli a1i ordtbas2 syl6eqss sseld wa cpw fipwuni
- fisn elpwid ad2antll unissi syl5sseqr adantr simprl syl6eleq syl sseqtr4d
+ fisn elpwid ad2antll unissi sseqtrrid adantr simprl syl6eleq syl sseqtr4d
sstrd elsni sseqin2 sylib sselda eleq1 syl5ibrcom rexlimdvva 3jaod sylbid
adantrl eqeltrd ssrdv ssfii unssad fiss sylancl unssd eqssd unass syl6eqr
- eqsstr3d ) FPQZGUBZCDRZRZUCUDZXHXIERZRZXJERXGXKXMXGUAXKXMXGUAUEZXKQZXNXHU
+ eqsstrrd ) FPQZGUBZCDRZRZUCUDZXHXIERZRZXJERXGXKXMXGUAXKXMXGUAUEZXKQZXNXHU
CUDZQZXNXIUCUDZQZXNNUEZOUEZUFZUGZOXRUHNXPUHZUIZXNXMQZXGXHSQXISQZXOYEUJGUK
XGXIXJTZXJSQZYGXIXHULZXGXJUMZSQYIXGGYKSABCDFPGJKLUNZXGGFUOSJFPUPUQURXJUSU
TZXIXJSVAVCNOXNXHXISSVBVCXGXQYFXSYDXQYFVDXGXPXMXNXPXHXMGVOZXHXLVEVFVGVHXG
@@ -283460,7 +283474,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
fibas ctop inex1g ordttop cuni ordtuni uniexb sylibr restval wral sseqin2
eqeltrrd wf sylib ctopon ordttopon psssdm fveq2d eleqtrd toponmax eqeltrd
elsni ineq1d eleq1d syl5ibrcom ralrimiv ordtrest2lem df-rn cnvtsr sseqtrd
- ccnv psrn wa adantr rabeqdv vex brcnv anbi12ci rabbii eqsstr3d ancom2s wb
+ ccnv psrn wa adantr rabeqdv vex brcnv anbi12ci rabbii eqsstrrd ancom2s wb
bicomi a1i notbid rabeqbidv mpteq12dv rneqd cnvin cnvxp ineq2i eqtri psss
ordtcnv syl5reqr eleq2d raleqbidv mpbird ralunb sylanbrc fmpt frnd tgfiss
eqsstrd eqssd ) AFEEUAZUBZUCNZFUCNZEUDUEZAFUFOZEPOZUUAUUCUGAFQOZUUDIFUIRZ
@@ -283546,7 +283560,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
fmpti cico icossxr unssi ssexi unex ssun2 fiss fvex cuni ovex unipr mnfxr
frn cpr cicc 0xr pnfxr w3a wbr mnflt0 0lepnf df-icc df-ioc xrltnle xrletr
wa xrlttr wi xrltle 3adant2 syld ixxun mpanr12 mp3an 0lt1 df-ico xrlelttr
- 1xr ixxss2 unss1 eqsstr3i iccmax uncom 3sstr3i eqssi eqtri ssun1 rspceeqv
+ 1xr ixxss2 unss1 eqsstrri iccmax uncom 3sstr3i eqssi eqtri ssun1 rspceeqv
oveq1 elrnmpti mpbir eleqtrri sselii oveq2 prssi ssfii sstri eltg3i snssi
wrex eqeltrri bastg ctb ctop fibas tgcl fitop mp2b sseqtri 2basgen eqtr4i
) GUCHZIUDZBCUEZUEZUFHZUGHZUUCUFHZUGHZGUHJUUAUUFKUIAFBCGUHIUJAFBDUKAFBCDE
@@ -283688,7 +283702,7 @@ information ensuring that it is not too fine (and of course ~ iooordt
tg2 wi wceq cvv elrnmpt elv cif mnfxr a1i simprl 0xr sylancl pnfxr xrmax1
sylancr ge0gtmnf syl2anc simpll simprr eleqtrd elioc1 mpbid simp2d 0ltpnf
ifcl w3a breq1 ifboth xrre2 syl32anc xrmax2 df-ioc xrlelttr ixxss1 simplr
- eqsstr3d sstrd oveq1 sseq1d rspcev rexlimdvaa com12 wn pnfnlt elico1 mpan
+ eqsstrrd sstrd oveq1 sseq1d rspcev rexlimdvaa com12 wn pnfnlt elico1 mpan
sylbi simp3 syl6bi mtod notbid syl5ibrcom rexlimiv pm2.21d adantrd pnfnre
eleq2 jaoi neli cuni elssuni unirnioo syl6sseqr sseld mtoi syl sylanb ) B
EUDUEZFBCGCUFZHIJZUGZUHZCGKYEUIJZUGZUHZUJZUKUHZUJZULUEZFZHBFZAUFZHIJZBLZA
@@ -283718,7 +283732,7 @@ information ensuring that it is not too fine (and of course ~ iooordt
elv w3a mtod eleq2 notbid syl5ibrcom rexlimiv pm2.21d adantrd sylbi mnfxr
cif a1i 0xr simprl mnflt0 simpll simprr eleqtrd elico1 mpbid simp3d breq2
ifcl ifboth xrmin1 0re ltpnf mp1i xrlelttrd syl32anc xrmin2 df-ico ixxss2
- xrre2 xrltletr simplr eqsstr3d sstrd oveq2 sseq1d rspcev rexlimdvaa com12
+ xrre2 xrltletr simplr eqsstrrd sstrd oveq2 sseq1d rspcev rexlimdvaa com12
syl2anc jaoi mnfnre neli cuni elssuni unirnioo syl6sseqr sseld syl sylanb
mtoi ) BEUDUEZFBCGCUFZHUGIZUHZUIZCGJYGKIZUHZUIZUJZUKUIZUJZULUEZFZJBFZJAUF
ZKIZBLZAMNZYFYQBCYJYMYOYJTYMTYOTUMUNYRYSOJDUFZFZUUDBLZOZDYPNUUCDBYPJUPUUG
@@ -283745,7 +283759,7 @@ information ensuring that it is not too fine (and of course ~ iooordt
ordtrestixx $p |- ( ( ordTop ` <_ ) |`t A ) =
( ordTop ` ( <_ i^i ( A X. A ) ) ) $=
( vz cle cordt cfv co wceq wtru cxr wcel a1i wss cv wa wbr sseli cxp ledm
- crest ctsr letsr crab cicc iccval syl2an eqsstr3d adantl ordtrest2 eqcomd
+ crest ctsr letsr crab cicc iccval syl2an eqsstrrd adantl ordtrest2 eqcomd
cin mptru ) GHICUCJZGCCUAUNHIZKLUQUPLABFCGMUBGUDNLUEOCMPLDOAQZCNZBQZCNZRZ
URFQZGSVCUTGSRFMUFZCPLVBVDURUTUGJZCUSURMNUTMNVEVDKVACMURDTCMUTDTFURUTUHUI
EUJUKULUMUO $.
@@ -284472,7 +284486,7 @@ converges to zero (in the standard topology on the reals) with this
( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) ) ) $=
( ctopon cfv wcel wa cima ccl wss cpw wral cuni wi wceq toponuni ad2antlr
ccn co wf ccnv cv cnf2 3expia elpwi adantl sseqtrd eqid cncls2i ralrimdva
- expcom syl jcad ccld cldss2 pweqd syl5sseqr sseld imim1d ad2antll imaeq2d
+ expcom syl jcad ccld cldss2 pweqd sseqtrrid sseld imim1d ad2antll imaeq2d
cldcls sseq2d ctop topontop ad2antrr cdm cnvimass ad2antrl eqtrd syl5sseq
fdm iscld4 syl2anc bitr4d expr pm5.74d sylibd ralimdv2 imdistanda sylibrd
wb iscncl impbid ) CEGHIZDFGHIZJZBCDUAUBIZEFBUCZBUDZAUEZKZCLHHZWMWNDLHHZK
@@ -285908,7 +285922,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) ->
F : X --> { P } ) $=
( ct1 wcel ccn co wa ccnv wss cfv wf ccld syl2anc csn cima ccl w3a simplr
- wceq wfn cnf ffn 3syl simpr3 simpll simpr1 t1sncld simpr2 clsss2 eqsstr3d
+ wceq wfn cnf ffn 3syl simpr3 simpll simpr1 t1sncld simpr2 clsss2 eqsstrrd
cnclima fconst3 sylanbrc ) EJKZCDELMKZNZBGKZACOBUAZUBZPZADUCQQZFUFZUDZNZC
FUGZFVFPFVECRVKVBFGCRVLVAVBVJUEZCDEFGHIUHFGCUIUJVKFVHVFVCVDVGVIUKVKVFDSQK
ZVGVHVFPVKVBVEESQKZVNVMVKVAVDVOVAVBVJULVCVDVGVIUMBEGIUNTVECDEURTVCVDVGVIU
@@ -286345,7 +286359,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
eqid ralbidv bastg adantr sspwb sylib ssralv sylbid syl5 elpwi simprr wel
simprl sselda adantrr tg2 syl2anc reximdva eluni2 elunirab r19.42v rexbii
crab expr rexcom 3bitr2i 3imtr4g ssrdv eqsstrd ssrab2 unissi simplr eqssd
- syl5sseqr wb elpw2g ad2antrr mpbiri unieq eqeq2d pweq ineq1d rexeqdv mpid
+ sseqtrrid wb elpw2g ad2antrr mpbiri unieq eqeq2d pweq ineq1d rexeqdv mpid
rspcv wex elfpw ad2antrl simplbi ssrab sseq2 ac6sfi crn frn wfo wfn dffn4
wf ffn fofi syl2an sylanbrc simplrr ciun uniiun syl5eqss ad2antll fniunfv
ss2iun sseqtrd sseqtr4d rspceeqv exlimddv rexlimdvaa syld com23 ralrimdva
@@ -286385,7 +286399,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
( J |`t S ) e. Comp ) $=
( vs vt vu vw ccmp wcel wa cv cuni wss cpw cfn wi wceq cun 3ad2ant1 sylib
cvv vv ccld cfv crest co cin wrex wral selpw w3a cdif simp1l simp2 cldopn
- csn adantl snssd unssd simp3 uniss 3ad2ant2 sstrd undif 3ad2ant3 eqsstr3d
+ csn adantl snssd unssd simp3 uniss 3ad2ant2 sstrd undif 3ad2ant3 eqsstrrd
eqid unss1 difss unss sylanblc uniexg ad2antrr 3adant3 difexg unisng 3syl
eqssd uneq2d eqtr4d uniun syl6eqr cmpcov syl3anc elfpw simp2l uncom diffi
syl6sseq ssundif ad2antll sylanbrc wex sseqtrd sselda eluni a1i wn elndif
@@ -286461,7 +286475,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
vex csbeq1 iunxsn eqtri simpl3 nfv nfel1 cbvral sylib ssun2 simprl sylibr
nfov snss rspcdva syl5eqel unexg syl2anc resttop eqid restin unieqd inss2
sseqtr4i restuni sylancl eqtr4d uneq2i ineq1i indir inss1 restabs syl3anc
- sstri a1i eqeltrd eqsstr3i uncmp syl22anc exp32 a2d syl5 findcard2 mpcom
+ sstri a1i eqeltrd eqsstrri uncmp syl22anc exp32 a2d syl5 findcard2 mpcom
a2i mpi ) DHIZBUBIZDCJKZLIZABUDZUEZBBMZDABCNZJKZLIZBUFYHYLYMYPOZYGYHYKUGY
LGUHZBMZDAYRCNZJKZLIZOZOYLPBMZDPJKZLIZOZOYLUAUHZBMZDAUUHCNZJKZLIZOZOZYLUU
HUCUHZUIZUJZBMZDAUUQCNZJKZLIZOZOZYLYQOGUAUCBYRPQZUUCUUGYLUVDYSUUDUUBUUFYR
@@ -286612,7 +286626,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
0ss 0fin elfpw mpbir2an simprr simprl unieqd eqtrd unieq rspceeqv sylancr
wn expr iineq1 0iin syl6eq eqeq1d necon3bbid mpbiri pm2.21d 2thd ad2antlr
vn0 uniss eqss eqcom ssdif0 3bitr3g iindif2 uniiun difeq2i syl6eqr bitr4d
- baib ciun imassrn cres df-ima resmpt rneqd syl5eq ad2antrr syl5sseqr wfun
+ baib ciun imassrn cres df-ima resmpt rneqd syl5eq ad2antrr sseqtrrid wfun
crn funmpt elinel2 imafi sylanbrc topopn difexg ralrimivw fnmpt ad3antrrr
wfn eqid simpr sylib simpld sseqtrd simprd fipreima syl3anc rexbii inteqd
eqeq2d rexxfrd 0ex wfo wf1o ccnv opncldf1 f1ofo forn sylibr eldifsn bitri
@@ -287261,7 +287275,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
syl mpd simpl reximi eleq2w cbvrexv sylib rabn0 sylibr rabex 0sdom ssrab2
vex cvv ssdomg mp2 cen simprrl nnenom ensymi domentr sylancl domtr fodomr
sylancr syl2anc cfz cima cint cmpt cfn ad3antrrr crn imassrn forn simprll
- ad2antll elpwid syl5ss eqsstrd cdm cin fz1ssnn fof fdmd syl5sseqr sseqin2
+ ad2antll elpwid syl5ss eqsstrd cdm cin fz1ssnn fof fdmd sseqtrrid sseqin2
adantr elfz1end ne0i adantl sylan2b eqnetrd syl2an2r imp wb oveq2 imaeq2d
weq inteqd ralrimiva eleq1d rspccva sylan fvmptd3 sseq1d fveq1 ralbidv id
imadisj necon3bii cres fzfid ffund fores fiinopn syl13anc fmpttd syl5sseq
@@ -289132,7 +289146,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
vj vs crest co ccmp cpw cin wrex wi wral simprr ssun2 wb ctopon clm wbr
lmcl snssg mpbiri adantr sseldd eluni2 sylib nnuz 1zzd ad2antrr simplrl
cuz c1 elpwid simprl lmcvg cima imassrn ssun1 sstri id syl5ss ctop frnd
- cfz resttopon topontop cres wfo fzfid wfun ffund fz1ssnn fdmd syl5sseqr
+ cfz resttopon topontop cres wfo fzfid wfun ffund fz1ssnn fdmd sseqtrrid
cdm fores fofi pwfi restsspw ssfi sylancl elind fincmp toponuni sseqtrd
wceq eqid cmpsub mpbid r19.21bi syl5 impr simprll snssd unssd vex elpw2
elin1d sylibr elin2d snfi unfi wfn ffnd ad3antrrr simprrr fveq2 rspccva
@@ -289639,7 +289653,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
( wcel wa wceq wss cvv vs vh vm ctop wf csn cv cfv cmpt ccnv cima crn cun
cmpo cfi wfn wral cuni cdif cfn wrex w3a cixp wex elpt df-3an cin ciin c0
simprr disjdif2 raleqdv biimpac ixpeq2 syl weq fveq2 unieqd cbvixpv eqtri
- syl6eqr sylan ssv iineq1 0iin syl6eq syl5sseqr adantl df-ss sylib wne cif
+ syl6eqr sylan ssv iineq1 0iin syl6eq sseqtrrid adantl df-ss sylib wne cif
eqtr4d simplll sseldi eleq12d ad2antrr rspcdva ptpjpre1 syl12anc iineq2dv
inss1 simpr adantlr cdm cnvimass eqid dmmptss sstri sseqtri rgenw sylancl
r19.2z iinss wi wn elssuni syl5ibrcom sylibr eqtrd eqssd eleq1 pm2.61dane
@@ -290623,7 +290637,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
ffn simp-4l simpllr simplr ffvelrnd toponss elin1d sseqtr4d inss1 mpsyl
syl6bb fnfvelrn intss1 syl5ss r19.26 sylan biimpd ralimia sylbir syl6an
wi sylbid ralimdaa impr eldifi syl2an ralbidv cun inundif raleqi ralunb
- bitr3i ralcom mptexg syl2anr ralbidva syl5bb ralrimivw syl5sseqr rspcev
+ bitr3i ralcom mptexg syl2anr ralbidva syl5bb ralrimivw sseqtrrid rspcev
bitrd syl12anc exlimddv ) ABUKZJNULZKUHUMZUNZFGUMZUWQUOZUPZCOEUQZUWTURZ
UWSIUOZUSZUKZGUWPUTZUKZFDUMZUPZCOGJEUQZUQZUXIURZGJUXDVAZUSZUKZDKVBZUHUW
OUWPVCUPZFUIUMZUPZUXBUXSURZUXDUSZUKZUIKVBZGUWPUTUXHUHVEUWONVCUPUWPNUSUX
@@ -290913,7 +290927,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
( vy cfv ctopn wss wceq cvv wcel vg vz vx cts ccom cpt cbs cpw cdm fnex
wfn syl2anc eqid eqidd prdstset cuni cv wral cdif cfn wrex w3a cixp wex
wa cab ctg wf topnfn dffn2 sylib fnfco sylancr ptval unieqd fvco2 sylan
- crest co topnval restsspw eqsstr3i syl6eqss fvex syl6ib ralimdva simpl2
+ crest co topnval restsspw eqsstrri syl6eqss fvex syl6ib ralimdva simpl2
sseld elpw impel ss2ixp simprr prdsbas2 adantr 3sstr4d ex exlimdv selpw
syl6ibr abssdv pwex ssex unitg 3syl eqtrd sspwuni eqsstrd sylibr topnid
syl syl6eqr eqtr3d ) AHUDOZEPBUEZUFOZAXMHPOZEAXMHUGOZUHZQXMXPRAXMXOXRAX
@@ -291020,7 +291034,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
( vx vy vz vw c0 cpr cid cfv cxp cv wcel wceq wo wn wss wa ctop wne co wi
ctx wex neq0 wrex wral wb indistop eltx mp2an sylbi cuni elssuni indisuni
rsp txunii syl6sseqr ad2antrr ne0i ad2antrl sylibr simpld neneqd indislem
- xpnz simpll syl6eleqr elpri syl ord simprd simplr xpeq12d simprr eqsstr3d
+ xpnz simpll syl6eleqr elpri syl ord simprd simplr xpeq12d simprr eqsstrrd
mpd adantll eqssd ex rexlimdvva syld exlimdv syl5bi orrd vex ssriv ctopon
elpr toptopon txtopon topgele ax-mp simpli eqssi txindislem preq2i 3eqtri
cpw mpbi ) GAHZGBHZUCUAZGAIJZBIJZKZHZGABKZIJZHGXHHXCXGCXCXGCLZXCMZXJGNZXJ
@@ -291265,7 +291279,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
frnd elin2d wfn ffnd dffn4 fofi w3a fiinopn imp elssuni syl6sseqr sseqin2
syl13anc pm2.61dane ad2antrr ciin simprrr simpl ralimi eliin mpbird elind
fniinfv eleqtrd ciun simprl uniiun syl6eq xpeq1d xpiundir wi inss2 iinss2
- eqsstr3d syl5ss xpss2 sstr2 3syl ralimdva sylc sylibr eqsstrd rspcev expr
+ eqsstrrd syl5ss xpss2 sstr2 3syl ralimdva sylc sylibr eqsstrd rspcev expr
iunss syl12anc exlimdv expimpd rexlimdva mpd ) AGUAUEZUFZUGZUUMEUBUEZUHZC
BUEZUUPUIZQZUURUUSUJZFRZSZBUUMUOZSZUBUKZSZUADULZUMUNZTZCUURQZGUURUJZFRZSZ
BETZADUPQZUCUEZUURQZCPUEZQZUURUVSUJZFRZSZPETSZBDTZUCGUOUVJKAUWEUCGAUVQGQZ
@@ -291789,7 +291803,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
ffnd rexnal df-ne simpllr ffvelrnd hausnei syl13anc expr syl5bir csn cima
adantr crab simp-4l ad4antlr simplr snssd crest cpw ccmp ctopon toptopon2
sylib restsn2 cfn snfi discmp mpbi syl6eqel simprll xkoopn simprlr imaeq1
- sseq1d ad2antrr fnsnfv simprr1 eqsstr3d elrabd simprr2 inrab cdm eleqtrrd
+ sseq1d ad2antrr fnsnfv simprr1 eqsstrrd elrabd simprr2 inrab cdm eleqtrrd
fdmd adantl simprr3 sseq0 expcom imadisj disjsn bitri syl6ib mt2d sylnibr
ralrimiva rabeq0 sylibr syl5eq eleq2 ineq1 eqeq1d 3anbi13d ineq2 3anbi23d
ssin rspc2ev syl113anc rexlimdvva syld rexlimdva sylbid sylbird ralrimivv
@@ -291902,7 +291916,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
ex ad2antrr ixpconstg eqsstrd eqtrd sseqtrd sseqin2 eqtr3d elin2d simplrr
sylib topopn ad3antrrr fvconst2g ad4ant14 eleqtrrd eldifn iffalsed eldifi
ifcld cdif unieqd eqtr4d ptopn eqeltrd eleq1 syl5ibrcom rexlimdvva abssdv
- tgfiss ptuniconst oveq2d restid xkoptsub eqsstr3d eqssd ) BHIZACIZJZBAUBZ
+ tgfiss ptuniconst oveq2d restid xkoptsub eqsstrrd eqssd ) BHIZACIZJZBAUBZ
UCKZABUDUEZUFLZUVNUVPDEUVOFMZNKZUGIZFUVOUHZBGMZDMZUIEMZOZGUVOBUJKZUHZUKZU
LZUMLUNLZUVRUVMUVOHIZUVLUVLUVPUWKPACURZUVLUVMUOZFEUVOBUWIGDUWBAUVOUPAAUQU
SZUWBQUWIQUTVAUVNUVRHIZUWJUVROUWKUVROUVNUVMAHUVQUAZUWPUVLUVMVBUVLUWQUVMAB
@@ -292059,7 +292073,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $=
sseq2d oveq2 eleq1d cmpcovf wi eqeq1d biimpar cntop2 cntop1 sspwuni wfn
frnd ffn fniunfv oveq2d simplr elin2d simpr fiuncmp ntropn iunopn txopn
eqeltrrd imaiun imaeq2d syl5eqr simpl1 3ad2ant2 simp3 r19.21bi ralbidva
- 3bitr4g mpbird eqsstr3d uniiun syl6eq simpl opelxpd anbi12i coeq1 coeq2
+ 3bitr4g mpbird eqsstrrd uniiun syl6eq simpl opelxpd anbi12i coeq1 coeq2
simprll ovmpo ntrss2 sseqtrd imass2 simprlr syl5eqss eqeltrd ralrimivva
cnco sylan2b mpofun xpss12 dmmpo sseqtr4i funimassov sseq1 expr exlimdv
sylibr syldan expimpd rexlimdva ) AFDLUKZULUMZUNZUBUOZUNZUPZUYKCUQMUKZU
@@ -292729,7 +292743,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $=
weq bitri 3bitrd rabbidva sneq xpeq2d elrab ctop ad2antrr topontop adantl
cin txtop syl2anc ad3antrrr toponmax xpexg simprr elrestr txrest syl22anc
syl3anc oveq2d restid eqtrd eleqtrd resttopon xpeq1d simprl xpss2 mpbird
- ciun snssd ssind eqsstr3d txtube toponss ssrab baib biantrud iunid xpeq2i
+ ciun snssd ssind eqsstrrd txtube toponss ssrab baib biantrud iunid xpeq2i
xpiundi eqtr3i sseq1i iunss ssin 3bitr3g anbi2d rexbidva ralrimiva eltop2
sylan2b 3syl eqeltrd imaeq2 mptpreima syl6eq syl5ibrcom rexlimdvva syl5bi
expimpd ralrimiv simpl ovex xkotf frn ax-mp ssexi cfi ctg xkoval xkotopon
@@ -292943,7 +292957,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $=
( J qTop F ) = { s e. ~P Y | ( `' F " s ) e. J } ) $=
( wcel wfo wss w3a cqtop cima cpw crab cvv wceq 3ad2ant2 syl2anc ccnv cin
co cv simp1 fof cuni uniexg 3ad2ant1 syl5eqel simp3 ssexd fex qtopval crn
- wf imassrn forn syl5sseq foima imass2 eqsstr3d eqssd pweqd cnvimass fssdm
+ wf imassrn forn syl5sseq foima imass2 eqsstrrd eqssd pweqd cnvimass fssdm
syl sstrd df-ss sylib eleq1d rabeqbidv eqtrd ) BCIZFEAJZFDKZLZBAMUCZAUAGU
DZNZDUBZBIZGADNZOZPZVTBIZGEOZPVQVNAQIZVRWERVNVOVPUEVQFEAUPZFQIWHVOVNWIVPF
EAUFSZVQFDQVQDBUGZQHVNVOWKQIVPBCUHUIUJVNVOVPUKZULFEQAUMTABCQDGHUNTVQWBWFG
@@ -294550,7 +294564,7 @@ Kolmogorov quotient is regular Hausdorff (T_3). (Contributed by Mario
`' G = ( z e. U. J |-> <. ( z |` A ) , ( z |` B ) >. ) ) $=
( vw vk cxp cuni wf1o ccnv cv cres cop cmpt wceq c1st cfv c2nd cun op1std
cmpo vex op2ndd uneq12d mpompt eqtr4i wcel wa cixp cdif wss adantl ixpeq2
- xp1st fvres unieqd mprg cvv ctop wf ssun1 syl5sseqr ssexd fssresd syl2anc
+ xp1st fvres unieqd mprg cvv ctop wf ssun1 sseqtrrid ssexd fssresd syl2anc
ptuni syl5eqr syl6eqr adantr eleqtrrd xp2nd eqcomd cin c0 uneqdifeq mpbid
wb ixpeq1d ssun2 eqtrd syl3anc eleqtrd eleq2d biimpar resixp opelxpd eqop
undifixp ad2antrl wfn adantrl ixpfn fnresdm reseq2d eqtr3d resundi syl6eq
@@ -294591,7 +294605,7 @@ Kolmogorov quotient is regular Hausdorff (T_3). (Contributed by Mario
ptunhmeo $p |- ( ph -> G e. ( ( K tX L ) Homeo J ) ) $=
( vz vk vn vf ctx ccn wcel ccnv chmeo cxp c1st cfv c2nd cun cmpt cmpo cop
co cv wceq vex op1std op2ndd uneq12d mpompt eqtr4i wfn cuni cixp cdif wss
- wa xp1st adantl cres ixpeq2 fvres unieqd mprg cvv ctop wf ssun1 syl5sseqr
+ wa xp1st adantl cres ixpeq2 fvres unieqd mprg cvv ctop wf ssun1 sseqtrrid
ssexd fssresd ptuni syl2anc syl5eqr syl6eqr adantr eleqtrrd eqcomd cin c0
xp2nd wb uneqdifeq mpbid ixpeq1d ssun2 eqtrd undifixp syl3anc ixpfn dffn5
syl sylib mpteq2dva syl5eq ctopon cpt syl5eqel toptopon txtopon wo eleq2d
@@ -295911,7 +295925,7 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro,
fdm expimpd eqcom imadisj notbii syl6ibr syld ralrimiv eleq2i 0ex elrnmpt
bitri eqid df-nel ralnex 3bitr4i sylibr imaeq2 cbvmptv elv anbi12i reeanv
ax-mp bitr4i fbasssin 3expb 3ad2antl1 adantrr rspceeqv mpan2 ad2antrl vex
- funimaex syl5bb ad2antrr mpbird imass2 inss1 inss2 ssini ineq12 syl5sseqr
+ funimaex syl5bb ad2antrr mpbird imass2 inss1 inss2 ssini ineq12 sseqtrrid
ad2antll ad2antlr sstrd sseq1 rspcev syl2anc adantlrl rexlimddv rexlimdvv
exp32 syl5bi ralrimivv 3jca isfbas2 3ad2ant3 mpbir2and ) BFUIUEKZFGDVBZGE
KZUFZCGUIUEKZCGUGZLZCMUHZMCUJZUANZUBNZUCNZUKZLZUACOZUCCULUBCULZUFZUUPCABD
@@ -296451,7 +296465,7 @@ contains the choice as a hypothesis (in the assumption that ` ~P ~P X `
( vf vg vu cufl wcel wss wa cv cufil cfv wrex co cfbas cpw syl2anc adantr
cfil syl wral cfg simpll filfbas adantl simplr sspwb sylib sstrd fbasweak
filsspw syl3anc fgcl ufli crest ssfg simprr filtop ad2antlr sseldd simprl
- wb trufil mpbird restid2 ssrest eqsstr3d sseq2 rspcev rexlimddv ralrimiva
+ wb trufil mpbird restid2 ssrest eqsstrrd sseq2 rspcev rexlimddv ralrimiva
wceq cvv ssexg ancoms isufl ) AFGZBAHZIZBFGZCJZDJZHZDBKLZMZCBSLZUAZVSWECW
FVSWAWFGZIZAWAUBNZEJZHZWEEAKLZWIVQWJASLGZWLEWMMVQVRWHUCZWIWAAOLGZWNWIWABO
LGZWAAPZHVQWPWHWQVSWABUDUEWIWABPZWRWHWAWSHZVSWABUKUEZWIVRWSWRHVQVRWHUFZBA
@@ -297310,7 +297324,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
co cdom wbr cen wex domeng wf1o bren biimpi ssufl wi cufil wrex cfil wral
simplr cfbas wf adantl f1of ad2antrr fmfil syl3anc ufli syl2anc cdm f1odm
ccnv vex dmex syl6eqelr simprl f1ocnv ad3antrrr fmufil ccom cid f1ococnv1
- cres oveq2d fveq1d fmco syl32anc fmid 3eqtr3d ufilfil 3syl eqsstr3d sseq2
+ cres oveq2d fveq1d fmco syl32anc fmid 3eqtr3d ufilfil 3syl eqsstrrd sseq2
simprr fmss rspcev rexlimddv ralrimiva wb isufl mpbird exlimiv imp syl2an
ex an12s exlimdv sylbid ) AHIZBAUAUBZBHIZXDXEBCJZUCUBZXGAKZLZCUDXFCBAHUEX
DXJXFCXDXJXFXHXDXIXFXHBXGDJZUFZDUDZXGHIZXFXDXILXHXMBXGDUGUHAXGUIXMXNXFXLX
@@ -298025,7 +298039,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
-> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. f e. ( Fil ` X )
( F " ( J fLim f ) ) C_ ( ( K fLimf f ) ` F ) ) ) ) $=
( vx ctopon cfv wcel wa ccn co wf cv cflf cflim wral wss wceq cfil cdm wb
- cima cnflf wfun ffun cuni flimelbas ssriv fdm toponuni ad2antrr syl5sseqr
+ cima cnflf wfun ffun cuni flimelbas ssriv fdm toponuni ad2antrr sseqtrrid
eqid adantl eqtrd funimass4 syl2an2 ralbidv pm5.32da bitr4d ) CEHIJZDFHIJ
ZKZBCDLMJEFBNZGOZBIBDAOZPMIZJGCVHQMZRZAEUAIZRZKVFBVJUDVISZAVLRZKGABCDEFUE
VEVFVOVMVEVFKZVNVKAVLVFBUFVEVJBUBZSVNVKUCEFBUGVPCUHZVJVQGVJVRVGVHCVRVRUOU
@@ -298571,7 +298585,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
( vx vy cfv wcel cv cfcls co c0 wne cfil wral wi wa wss wceq cvv syl ccmp
ctopon cuni eqid fclscmpi ralrimiva toponuni fveq2d raleqdv cfi cint ccld
syl5ibr wn cpw elpwi vn0 simpr inteqd int0 syl6eq neeq1d mpbiri a1d ssfii
- cfg ccl elv cfbas simplrl ad2antrr pweqd syl5sseqr sstrd simplrr toponmax
+ cfg ccl elv cfbas simplrl ad2antrr pweqd sseqtrrid sstrd simplrr toponmax
cldss2 w3a wb fsubbas mpbir3and ssfg syl5ss sselda fclssscls cldcls ssint
sylibr fgcl oveq2 rspcv 3syl ssn0 syl6an pm2.61dane expr sylan2 ralrimdva
sseqtrd com23 ctop topontop cmpfi sylibrd impbid ) BCUBFGZBUAGZBAHZIJZKLZ
@@ -298855,7 +298869,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
eleq2 eleq1 rexlimdva sylbid imp32 adantrrr elssuni fibas tgtopon ax-mp
ctopon ctb syl6eqel fiuni eqtrd fveq2d eleqtrrd toponuni sseqtr4d filss
simprrr rexlimddv ralrimiva imdistanda syl5bi flimopn sylibrd ralrimivw
- difexg unieq syl6eqr eqnetrd necon3bii sylib ciun sseq0 difss syl5sseqr
+ difexg unieq syl6eqr eqnetrd necon3bii sylib ciun sseq0 difss sseqtrrid
ssdif0 unissi eqssd jctil sseq1 eqeq2d anbi12d anbi2d pweq ineq1d mpcom
rexeqdv vtoclg mpdan uni0 syl6eq neeq2d ciin cmpt crn difssd riinn0 cab
sylan dfiin2g eqid rnmpt inteqi eldifbd difss2d ufilb mpbid fmpttd frnd
@@ -299612,7 +299626,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
syld biimpa impel syl12anc snid syl5eleqr flfnei simprd r19.21bi ad4ant13
cfil ad3antrrr 3an1rs adantllr creg simprl regsep expcom ad2antll syl5bir
reximi 3expib anim1d eltopss 3expa cfm cflim flfval cfbas uniexg syl5eqel
- elrestr cfg filfbas fgfil eleqtrrd eqid imaelfm flimclsi eqsstrd eqsstr3d
+ elrestr cfg filfbas fgfil eleqtrrd eqid imaelfm flimclsi eqsstrd eqsstrrd
sylibr adantlr expl wfun cdm cnextf ffund fdmd sseqtr4d funimass4 biimprd
reximdva cnnei mpbird ) AFGHUIUJRZGHUKUJSZUXQUAULZUMUBULZUNZUABULZUQZGUOR
ZRZUPZUBUYBUXQRZUQZHUORRZURZBEURZAUYJBEAUYBESZTZUYFUBUYIUYMUXTUYISZTZAUCU
@@ -300103,7 +300117,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
anbi12d sylib wi crn cint cin toponuni ad2antrr simplrl simplrr fvconst2g
ineq1d eleq2d ralbidva mpbid ffnfv sylanbrc frnd wfo dffn4 exlimdv simprl
fofi rintopn mptelixpg ralrn elrint inex1 ixpconstg inss2 fnfvelrn intss1
- sylancl syl5ss ralrimiva ss2ixp eqsstr3d ssrab ad2antll ssralv sylc oveq1
+ sylancl syl5ss ralrimiva ss2ixp eqsstrrd ssrab ad2antll ssralv sylc oveq1
simprbi raleqdv rspcev syl12anc ex 3adantr3 imbi1d syl5ibrcom expimpd mpd
impd ) AUAURZCUGZUBURZUVMUHZUVOCIUIUJZUHZTZUBCUKZUVPUVRULUMUBCUCURZUNUKUC
UOUPZUQZUDURZUBCUVPUSZUMZUTZUAVAZCJUIUJZUWDTZUWDHGURZVBVCZFTZGDCVDVCZVEZV
@@ -300390,7 +300404,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
cxp cima opelxpi ctx txcls syl22anc txtopon cnvimass cgrp wf tgpgrp fssdm
grpsubf subgsubcl 3expb ralrimivva fveq2 syl6eqr eleq1d ralxp sylibr wfun
cdm ffund xpss12 fdmd funimass5 mpbird clsss syl3anc ccn tgpsubcn cncls2i
- wb sstrd eqsstr3d sselda sylan2 wfn ad2antrr elpreima 4syl mpbid syl5eqel
+ wb sstrd eqsstrrd sselda sylan2 wfn ad2antrr elpreima 4syl mpbid syl5eqel
ffn simprd w3a issubg4 mpbir3and ) BUAHZABUBIZHZJZACUCIIZXPHZXSBUDIZKZXSU
EUFZELZFLZBUGIZMZXSHZFXSNEXSNZXRXSCUHZYAXRCUIHZAYJKZXSYJKXRCYAUJIHZYKXOYM
XQBCYADYAOZUKPZYACULQZXRAYAYJXQAYAKZXOYAABYNUNRZXRYMYAYJSYOYACUMQZUOZACYJ
@@ -300417,7 +300431,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
cbs ctopon tgptopon topontop ad2antlr subgss wceq toponuni sseqtrd clsss3
syl2anc sseqtr4d resmptd rneqd syl5eq ccn tgptmd simpr cnmptc cnmpt1plusg
ctmd cnmptid tgpsubcn cnmpt12f cnclsi nsgconj ad4ant234 fmpttd frnd clsss
- ctx eqsstrd syl3anc sstrd eqsstr3d wfn ovex fnmpti df-f mpbiran ralrimiva
+ ctx eqsstrd syl3anc sstrd eqsstrrd wfn ovex fnmpti df-f mpbiran ralrimiva
sylibr fmpt isnsg3 sylanbrc ) BUAGZABUBHZGZIZACUCHZHZBUDHZGZEJZFJZBUEHZKZ
XGBUFHZKZXDGFXDLZEBUNHZLXDWTGXAWSAXEGZXFABUGZABCDUHUIXBXMEXNXBXGXNGZIZXDX
DFXDXLMZUJZXMXRXSNZXDOZXTXRYAFXNXLMZXDPZXDXRYDYCXDQZNYAYCXDUKXRYEXSXRFXNX
@@ -300599,7 +300613,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
ghmgrp2 simprrr eqeltrd elrabd cnpimaex ssrab simprbi wfn wb ffnd toponss
sylan ralima syl5ib simpl1 ad2antrr fveq2 oveq2d rspccv grpcl c0g cminusg
ghmlin grpinvsub grprinv eqtr3d grpass syl13anc 3eqtr3d adantlr ralrimiva
- grplid fveq2d sylibd ralrab2 sylibr wfun ffund ssrab2 syl5sseqr funimass4
+ grplid fveq2d sylibd ralrab2 sylibr wfun ffund ssrab2 sseqtrrid funimass4
cdm fdmd mpbird eleq2 expr mpbir2and imaeq2 sseq1d anbi12d rspcev sylan2d
syl12anc rexlimdva mpd anassrs iscnp ex jcad cncnpi ancoms impbid1 eleq2d
cncnp anbi1d bitr4d ) CUELZDUELZBCDUHMLZUFZBAEFUIMZNLZAEUJZLZBEFUKMLZOZAG
@@ -301296,7 +301310,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
( vz wcel wa ad2antrr vx vu vy ctsu co cgsu csn ccl cfv cv wss cres cpw
wi cfn cin wral wrex cuni wne ctopon wceq ctps istps sylib toponuni syl
eleq2d csupp cun elfpw simplbi adantl suppssdm fssdm elinel2 cfsupp wbr
- c0 unssd fsuppimpd syl2anc sylanbrc wb ssun1 id syl5sseqr reseq2 oveq2d
+ c0 unssd fsuppimpd syl2anc sylanbrc wb ssun1 id sseqtrrid reseq2 oveq2d
unfi pm5.5 eleq1d bitrd rspcv ccmn wf ssun2 a1i sylibd rexlimdva simprr
gsumres simplrr eqeltrd expr ralrimiva rspceaimv syl2an2r impbid disjsn
sseq1 necon2abii syl6bb imbi2d ralbidva anbi12d eqid eltsms ctop gsumcl
@@ -301419,7 +301433,7 @@ discrete topology (which is Hausdorff), this theorem can be used to turn
wa resabs1 syl6eq oveq2d eleq1d rspcv syl ad2antlr syl6ss biantrud resres
imbi12d oveq2i ccmn ad2antrr wf fssresd csupp cvv syl2anc fvexi ressuppss
fex c0g sstrd a1i fdmfifsupp gsumres syl5reqr sylibrd ralrimdva rspceaimv
- sseq1 syl6an rexlimdva cun unssd syl2an wb ssun1 id syl5sseqr pm5.5 bitrd
+ sseq1 syl6an rexlimdva cun unssd syl2an wb ssun1 id sseqtrrid pm5.5 bitrd
unfi adantrr jctir indir df-ss sylib uneq2d simprr ssequn1 syl5eq reseq2d
eqtrd resabs1d eqtr4d eqtr3d biimpd expr eqid eltsms com23 impbid ralbidv
syld imbi2d anbi2d inex1g fssres wfn ffn fnresdm 3syl reseq1d feq1d mpbid
@@ -301697,7 +301711,7 @@ discrete topology (which is Hausdorff), this theorem can be used to turn
Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.) $)
tsmssplit $p |- ( ph -> ( X .+ Y ) e. ( G tsums F ) ) $=
( vk co cv wcel cfv c0g cif cmpt cof ctsu ffvelrnda cmnd ccmn cmnmnd eqid
- wa syl mndidcl adantr ifcld fmpttd cres feqmptd reseq1d wss cun syl5sseqr
+ wa syl mndidcl adantr ifcld fmpttd cres feqmptd reseq1d wss cun sseqtrrid
wceq ssun1 iftrue mpteq2ia resmpt 3eqtr4a eqtr4d oveq2d ctmd ctps cdif wn
tmdtps eldifn adantl iffalsed suppss2 tsmsres eqtrd eleqtrd ssun2 tsmsadd
wi cin c0 noel eleq2 mtbiri elin sylnib imnan sylibr imp oveq12d syl2an2r
@@ -301763,7 +301777,7 @@ discrete topology (which is Hausdorff), this theorem can be used to turn
reseq2d grpinvhmeo tgplacthmeo hmeoco hmeoima grpsubid1 sylancl tsmsi
elrnmpt1s ralrimiva sseq1 imbi1d ac6sfi cuni cun inss1 sspwuni rnxpss
frn rnss unssd wfo wfn ffn dffn4 fofi inss2 elinel2 rnfi unfi adantrr
- unifi ssun2 wb fvssunirn ssun1 sstri syl5sseqr pm5.5 bitrd rspcv cmnd
+ unifi ssun2 wb fvssunirn ssun1 sstri sseqtrrid pm5.5 bitrd rspcv cmnd
id reseq2 cmnmnd simplr jca ovexd fvexi fsuppmptdm gsumcl velsn ovres
c0g sylanbr oveq1 eqtrd gsumsn syl3anc snfi snssd xpss12 fssresd xpfi
sylancr fdmfifsupp gsumxp 3eqtr4rd elrnmpti cabl isabl ablnncan simpr
@@ -302536,7 +302550,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed
composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.) $)
ustssco $p |- ( ( U e. ( UnifOn ` X ) /\ V e. U ) -> V C_ ( V o. V ) ) $=
( cust cfv wcel wa cid cres cun ccom ssun1 coires1 wrel cdm wss wceq ustrel
- cxp ustssxp dmss syl dmxpid syl6sseq syl2anc syl5eq uneq1d syl5sseqr coundi
+ cxp ustssxp dmss syl dmxpid syl6sseq syl2anc syl5eq uneq1d sseqtrrid coundi
relssres syl6sseqr ustdiag ssequn1 sylib coeq2d sseqtrd ) ACDEFBAFGZBBHCIZB
JZKZBBKZUQBBURKZVAJZUTUQBVAJBVCBVALUQVBBVAUQVBBCIZBBCMUQBNBOZCPVDBQABCRUQVE
CCSZOZCUQBVFPVEVGPABCTBVFUAUBCUCUDBCUJUEUFUGUHBURBUIUKUQUSBBUQURBPUSBQABCUL
@@ -302650,7 +302664,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed
17-Nov-2017.) $)
ustund $p |- ( ph -> ( ( A u. B ) X. ( A u. B ) ) C_ ( V o. V ) ) $=
( cun cxp cin ccom wne xpindir inss1 syl5ss syl5eqss inss2 unssd xpindi
- c0 wceq xpco syl xpundi xpundir coss12d eqsstr3d ) ABCHZUHIZBCJZUHIZUHUJI
+ c0 wceq xpco syl xpundi xpundir coss12d eqsstrrd ) ABCHZUHIZBCJZUHIZUHUJI
ZKZDDKAUJTLUMUIUAGUHUJUHUBUCAUKDULDAUKUJBIZUJCIZHDUJBCUDAUNUODAUNBBIZCBIZ
JZDBCBMAURUPDUPUQNEOPAUOBCIZCCIZJZDBCCMAVAUTDUSUTQFOPRPAULBUJIZCUJIZHDBCU
JUEAVBVCDAVBUPUSJZDBBCSAVDUPDUPUSNEOPAVCUQUTJZDCBCSAVEUTDUQUTQFOPRPUFUG
@@ -302672,8 +302686,8 @@ unit group (that is, the nonzero numbers) to the field. (Contributed
18-Nov-2017.) $)
ustneism $p |- ( ( V C_ ( X X. X ) /\ A e. X )
-> ( ( V " { A } ) X. ( V " { A } ) ) C_ ( V o. `' V ) ) $=
- ( cxp wss wcel wa csn cima ccom ccnv c0 wne wceq snnzg adantl xpco eqsstr3i
- syl mp1i cnvxp ressn cnveqi resss cnvss ax-mp coss2 coss1 sstrd eqsstr3d
+ ( cxp wss wcel wa csn cima ccom ccnv c0 wne wceq snnzg adantl xpco eqsstrri
+ syl mp1i cnvxp ressn cnveqi resss cnvss ax-mp coss2 coss1 sstrd eqsstrrd
cres ) BCCDEZACFZGZBAHZIZUPDZUOUPDZUPUODZJZBBKZJZUNUOLMZUTUQNUMVCULACOPUPUO
UPQSUNUTURVAJZVBUSVAEUTVDEUNUSURKZVAUOUPUAVEBUOUKZKZVAVFURBAUBZUCVFBEVGVAEB
UOUDZVFBUEUFRRUSVAURUGTURBEVDVBEUNURVFBVHVIRURBVAUHTUIUJ $.
@@ -302743,7 +302757,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed
cvv vy cust cfv cxp crest co cpw wral cres ccnv w3a restsspw inxp sseqin2
a1i biimpi sqxpeqd syl5eq adantl simpl elfvex adantr ssexd xpexd ustbasel
wi elrestr syl3anc eqeltrrd cun simplr simp-4r elpwid sstrd ustssxp unssd
- xpss12 ssun2 ustssel mpi df-ss sylib uneq1d simpllr eqsstr3d eqtr2d indir
+ xpss12 ssun2 ustssel mpi df-ss sylib uneq1d simpllr eqsstrrd eqtr2d indir
ssequn2 syl6eqr rspceeqv elrest biimpar syl21anc syldanl ad2antrr r19.29a
biimpa ex ralrimiva ustincl simprl simprr ineq12d inindir reeanv sylanbrc
ineq1 r19.29vva simp-4l ustdiag cdm inss1 resss sstri iss wbr ssel2 equid
@@ -303893,7 +303907,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed
wf cvv cfilufbas syl2anc cucn co wa wbr wi isucn simprbda syl21anc fbasrn
elfvexd syl3anc cop cmpo ccnv cima cmpt wceq simplr eqid rspceeqv sylancl
crn imaeq2 wb imaexg elrnmpt 3syl ad3antrrr cbvmptv rneqi eqtri syl6eleqr
- mpbird wfn ffnd fbelss sylan ad4ant13 fmucndlem mpofun funimass2 eqsstr3d
+ mpbird wfn ffnd fbelss sylan ad4ant13 fmucndlem mpofun funimass2 eqsstrrd
wfun mpan adantl sqxpeqd sseq1d rspcev adantr simpr nfcv weq simpl fveq2d
opeq12d cbvmpo ucnprima cfiluexsm r19.29a ralrimiva iscfilu syl mpbir2and
id ) ACFUFRSZCHUGRSZUATZYCUHZOTZUIZUACUJZOFUKZABGUGRSZGHEUMZHUNSYBADGULRS
@@ -305137,7 +305151,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric
anbi1d eliun eleq2i elv elrab2 simprbi simp2d wf1 f1of1 simplr elfz1end
sylib nnuz syl6eleq eluzfz1 xrleidd fcompt reseq1d 1st2nd2 simp1d xp1st
opeq1d eqtr2d syl5eq 3eqtr4d a1d simprl simprr peano2fzr expr cun fzsuc
- imim1d elfzuz3 ad2antll fzss2 eqsstr3d unssad fssresd xleadd1a syl13anc
+ imim1d elfzuz3 ad2antll fzss2 eqsstrrd unssad fssresd xleadd1a syl13anc
ex xmettri fvoveq1 simp3d nnz ad2antrl eluzp1m1 elfzuzb rspcdva xaddcld
eqtr4d xrletr mpand syld cplusg xrex difexi ressplusg fzelp1 sylan2 cin
wn fzp1disj disjsn gsumunsn reseq12d oveq1d sylibrd animpimp2impd nnind
@@ -306277,7 +306291,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric
the base set. (Contributed by NM, 3-Sep-2006.) (Revised by Mario
Carneiro, 12-Nov-2013.) $)
mopnfss $p |- ( D e. ( *Met ` X ) -> J C_ ~P X ) $=
- ( cxmet cfv wcel cuni cpw pwuni mopnuni pweqd syl5sseqr ) ACEFGZBHZIBCIBJ
+ ( cxmet cfv wcel cuni cpw pwuni mopnuni pweqd sseqtrrid ) ACEFGZBHZIBCIBJ
NCOABCDKLM $.
$( The base set of a metric space is open. Part of Theorem T1 of
@@ -306586,7 +306600,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric
( vx cmopn cfv cpw wss wceq cdm wcel eqid syl ctopn setsmstset cxmet cuni
cts cbs crn cv cbl df-mopn dmmptss sseli wa simpr xmetunirn sylib mopnuni
ctg cxp cds cin cres dmeqd dmres syl6eq inss1 syl6eqss dmss dmxpid adantr
- syl6sseq eqsstr3d sspwuni sylibr ex wn ndmfv 0ss pm2.61d1 setsmsbas pweqd
+ syl6sseq eqsstrrd sspwuni sylibr ex wn ndmfv 0ss pm2.61d1 setsmsbas pweqd
syl5 c0 3sstr3d topnid eqtrd ) ABLMZCUEMZCUAMZABCDEFGHIJUBZAWHCUFMZNZOWHW
IPAWGFNZWHWLABLQZRZWGWMOZWOBUCUGUDZRZAWPWNWQBKWQKUHUIMUGURMLKUJUKULAWRWPA
WRUMZWGUDZFOWPWSWTBQZQZFWSBXBUCMRZXBWTPWSWRXCAWRUNBUOUPBWGXBWGSUQTAXBFOWR
@@ -307088,7 +307102,7 @@ S C_ ( P ( ball ` D ) T ) ) $=
( A. x e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\
( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> J = K ) ) $=
( cfv wcel wa cv cbl co crp wrex wral wss cle wbr wceq wi simprrr simplll
- cxmet cxr simplr simprlr rpxrd simprll simprrl syl221anc eqsstr3d simpllr
+ cxmet cxr simplr simprlr rpxrd simprll simprrl syl221anc eqsstrrd simpllr
ssbl eqsstrd jca expr anassrs reximdva r19.40 syl6 ralimdva r19.26 syl6ib
metequiv sylibrd ) BFUGKZLZCVJLZMZGNZHNZUAUBZANZVNBOKZPZVQVNCOKZPZUCZMZGQ
RZHQSZAFSWAVQVOVRPZTZGQRZHQSVSVQVOVTPZTZGQRZHQSMZAFSDEUCVMWEWLAFVMVQFLZMZ
@@ -308208,7 +308222,7 @@ the half element (corresponding to half the distance) is also in this
filsspw filtop ad3antrrr simpllr elpwid filss syl13anc ralrimiva ad2antrr
3syl ex filin syl3anc metustid ad5ant24 sstrd biimpa simprd sylan r19.29a
elfg adantr ssfg sseldd simpld cnvss cnvxp syl6sseq wceq metustsym adantl
- eqsstr3d metustexhalf ad4ant13 r19.41v sstr reximi sylbir sylancom ssrexv
+ eqsstrrd metustexhalf ad4ant13 r19.41v sstr reximi sylbir sylancom ssrexv
sylc 3jca cvv wb elfvex isust mpbir3and ) CUCUAZACUDIJZKZCCUBZBUEUFZCUGIJ
ZXIXHUHZLZXHXIJZFMZGMZLZXOXIJZUIZGXKNZXNXOUJXIJZGXINZUKCULZXNLZXNOZXIJZXO
XOUMZXNLZGXIPZQZQZFXINZXGBXHUNIJZXIXHUOIJZXLABCDEUPZBXHUQZXIXHURVGXGYLYMX
@@ -308482,7 +308496,7 @@ the half element (corresponding to half the distance) is also in this
nfv nfne ineq1 cxr wf wfun psmetf ffun respreima 4syl ad6antr eqtr4d rspe
simp2 inex1 eqid elrnmpt ax-mp simpllr ssinss1 pweq eleq2d syl6bb syl6bbr
a1i ssin ad5antlr mpbir2and inelcm elv sylib r19.29af2 ssn0 ancoms metuel
- 3adant2 simplbda adantr r19.29a r19.29an jca cun elpwid rspceeqv eqsstr3d
+ 3adant2 simplbda adantr r19.29a r19.29an jca cun elpwid rspceeqv eqsstrrd
cdif unssd 3bitri wex simprl simpl3 xpss12 difssd ad4antr cnvexg cnvimass
sstrd eqidd imaexg mpbird fssdm ssdif0 0ss syl6eqss simpr ssundif difdif2
difcom sseq1i rspcev anbi1i ancom exbii n0 df-rex 3bitr4i biimpi ad2antll
@@ -309610,7 +309624,7 @@ definition of norm (which itself uses the metric). (Contributed by
ccom adantl dmeqd dmcoss cminusg cplusg grpsubfval ovex sseqtri syl6eqssr
co dmmpo adantr dmss dmxpid syl6sseq simpr xmetunirn sylib mopnuni tngbas
ad2antlr 3sstr3d sspwuni sylibr ex syl5 wn c0 ndmfv 0ss syl6eqss pm2.61d1
- syl5eqss eqsstr3d topnid eqtrd ) CFMZEGMZUAZDBUBNZBUCNZABCDEFGHIJUDZWTXAB
+ syl5eqss eqsstrrd topnid eqtrd ) CFMZEGMZUAZDBUBNZBUCNZABCDEFGHIJUDZWTXAB
UENZUFZOXAXBPWTXADXEXCWTDAQNZXEJWTAQRZMZXFXEOZXHAUGUHUIZMZWTXIXGXJAKXJKUJ
ZUKNUHULNQKUMUNUOWTXKXIWTXKUAZXFUIZXDOXIXMARZRZCUENZXNXDXMXPXQXQUPZRZXQXM
XOXROZXPXSOWTXTXKWTXOECUQNZVAZRZXRWTYBAWSYBAPWRWSYBBURNABCYAEGHYASZUSIUTV
@@ -314045,7 +314059,7 @@ Normed space homomorphisms (bounded linear operators)
if ( x <_ B , D , E ) ) e. ( ( O tX J ) Cn K ) ) $=
( cicc co cxp cv cle wbr cif cmpo ctx cuni eqid ccld cfv crest cr iccssre
wcel wss syl2anc cioo crn ctg sseldd icccld fveq2i syl6eleqr cun iccsplit
- ssun1 wceq syl3anc syl5sseqr uniretop unieqi eqtr4i restcldi toponuni syl
+ ssun1 wceq syl3anc sseqtrrid uniretop unieqi eqtr4i restcldi toponuni syl
ctopon ctop topcld 3syl eqeltrd txcld ssun2 xpeq1d xpundir syl6eq retopon
topontop eqeltri resttopon sylancr syl5eqel txtopon eqtr3d wral ccn sstrd
cntop2 toptopon2 sylib w3a elicc2 biimpa simp3d 3adant3 iftrued mpoeq3dva
@@ -315170,7 +315184,7 @@ topological space to the reals is bounded (above). (Boundedness below
( vy vz cv co c1 crp wcel vr cle wbr cif cmpo cbl cfv wss wrex wral cmopn
eqid cxmet cmet 1rp stdbdmet sylancl ccmp cxr cc0 clt wceq rpxr mp1i 0lt1
a1i stdbdmopn syl3anc eqeltrrd sseqtrd lebnum wa simpr wi ad2antrr adantr
- ifcl syl cr rpre ad2antlr 1re min2 stdbdbl syl33anc metxmet min1 eqsstr3d
+ ifcl syl cr rpre ad2antlr 1re min2 stdbdbl syl33anc metxmet min1 eqsstrrd
ssbl syl221anc sstr2 reximdv ralimdva oveq2 sseq1d rexbidv ralbidv rspcev
syl6an rexlimdva mpd ) ABPZUAPZNOGGNPOPDQZRUBUCXDRUDUEZUFUGZQZCPZUHZCEUIZ
BGUJZUASUIXBHPZDUFUGZQZXHUHZCEUIZBGUJZHSUIZABCXEEXEUKUGZGUAXSULADGUMUGZTZ
@@ -317032,7 +317046,7 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` )
1m1e0 simp1d cbvmptv ctopon wf iitopon cnf2 mp3an2i feqmptd iirev cc cr
ax-1cn unitssre sseli recnd nncan sylancr eqtrd mpteq2ia syl6eqr oveq1d
eceq1d opeq2d mpteq2dv rneqd syl5eq cnveqd a1i unieqd mpteq12dv 3sstr4d
- oveq2d cnvss eqsstr3d eqssd pi1xfr eqeltrd jca ) AIUDZJUEYNEDUFUGZUHAYN
+ oveq2d cnvss eqsstrrd eqssd pi1xfr eqeltrd jca ) AIUDZJUEYNEDUFUGZUHAYN
JABCDEFGHIJKLMNOPQRSTUAUIAJJUDZUDZYNAJUJZYQJUEAJUKUKULUMYRAGGUPZLUNUOZU
QZHYSKLURUOZUGZUUBUGZYTUQZUKUKJEVIUOZUSZUAYTUKUHZUUAUKUHAYSUUGUHUTZLUNV
AZYSUKYTVBVCUUHUUEUKUHUUIUUJUUDUKYTVBVCVDJVEVFJVGVHAYPIUMYQYNUMAGUUGUUA
@@ -320228,7 +320242,7 @@ is an accumulation point (limit point) of subset ` y ` ". @)
copab cms
cne wceq fveq2 syl6eqr sseq2d eleq2d oveqd sseq1d anbi2d rexbidv
3anbi123d opabbidv df-neiOLD cpw cxp cvv fvex eqeltri pwex xpex df-3an
- vex elpw 3anbi1i bitr3i opabbii opabssxp eqsstr3i ssexi
+ vex elpw 3anbi1i bitr3i opabbii opabssxp eqsstrri ssexi
fvopab4 ) GHDAIZ
GIZJKZLZBIZVQMZNCIZOPZVSWAVPQKZRZVOLZSZCTUAZUBZABUCVOELZVSEMZWBVSWADQKZRZ
VOLZSZCTUAZUBZABUCZUDUEVPDUFZWHWPABWRVRWIVTWJWGWOWRVQEVOWRVQDJKZEVPDJUGFU
@@ -320245,7 +320259,7 @@ is an accumulation point (limit point) of subset ` y ` ". @)
( vz vw cv c1st cfv wcel wss cne wbr wa wceq wn wrex wi wal w3a copab
cms cacOLD fveq2 syl6eqr eleq2d sseq2d wb breq syl anbi12d imbi1d albidv
3anbi123d opabbidv df-accOLD cpw cxp cvv fvex eqeltri pwex xpex df-3an
- vex elpw 3anbi2i bitr3i opabbii opabssxp eqsstr3i ssexi
+ vex elpw 3anbi2i bitr3i opabbii opabssxp eqsstrri ssexi
fvopab4 ) HIEAJZ
HJZKLZMZBJZVSNZCJZVSNZWCVQVROLZPZQZDJZWAMWHVQRSQDWCTZUAZCUBZUCZABUDVQFMZW
AFNZWCFNZWCVQEOLZPZQZWIUAZCUBZUCZABUDZUEUFVRERZWLXAABXCVTWMWBWNWKWTXCVSFV
@@ -320263,7 +320277,7 @@ is an accumulation point (limit point) of subset ` y ` ". @)
( vz vw cv c1st cfv wcel wss cc0 clt wbr wceq wn wa cblOLD co wrex wi cr
wral w3a copab cms cac fveq2 syl6eqr eleq2d sseq2d wb oveqd rexeq syl
imbi2d ralbidv 3anbi123d opabbidv df-acc cpw cxp cvv fvex eqeltri pwex
- xpex df-3an vex elpw 3anbi2i bitr3i opabbii opabssxp eqsstr3i ssexi
+ xpex df-3an vex elpw 3anbi2i bitr3i opabbii opabssxp eqsstrri ssexi
fvopab4 ) HIEAJZHJZKLZMZBJZWCNZOCJZPQZDJZWEMWIWARSTZDWAWGWBUALZUBZUCZUDZC
UEUFZUGZABUHWAFMZWEFNZWHWJDWAWGEUALZUBZUCZUDZCUEUFZUGZABUHZUIUJWBERZWPXDA
BXFWDWQWFWRWOXCXFWCFWAXFWCEKLZFWBEKUKGULZUMXFWCFWEXHUNXFWNXBCUEXFWMXAWHXF
@@ -327601,7 +327615,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically
cxr cop 1st2nd2 fveq2d df-ov syl6eqr eqtrd df-pr preq12d syl5eqr 3eqtr4rd
uneq12d iuneq2dv cpw ffn inss2 rexpssxrxp sstri fss sylancl fnfco sylancr
iccf fniunfv iunun ioof wfun cdm wfo fo1st fofn ssv fndm eqimss2 dfimafn2
- fnfun 3syl syl2anc fnima eqtr3d rnco2 syl6eq fo2nd 3eqtr3d syl5sseqr cdom
+ fnfun 3syl syl2anc fnima eqtr3d rnco2 syl6eq fo2nd 3eqtr3d sseqtrrid cdom
syl5eq ovolficcss ssdifssd cen ccrd cres con0 omelon nnenom ensymi isnumi
mp2an fofun fof fdmi sseqtr4i fores ffnd dffn4 sylib fodomnum mpsyl unctb
foco domentr ctex ssid syl5sseq ssundif ssdomg sylc domtr ovolctb2 jca )
@@ -327842,7 +327856,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically
cr ccom crn cuni cc0 wceq simpld cn cle cxp wf sstrd ovolsscl syl3anc
syl difssd c1 cfz cv wral wa c1st elfznn syl2an sseldi fveq2d syl6eqr
c2nd ralrimiva sylibr readdcld cuz cima cun wfn cxr sylancl ffnd cmin
- nnuz peano2nnd syl6eleq syl5eq cc nncnd ax-1cn eqtrd syl6eq syl5sseqr
+ nnuz peano2nnd syl6eleq syl5eq cc nncnd ax-1cn eqtrd syl6eq sseqtrrid
ovollb unss1 ax-mp sseqtrd unss2 eqsstri syl6ss ovolss syl2anc ovolun
csup wbr syl22anc letrd eqid resubcld nnaddcl cz adantr adantl wb csu
nnzd mpbird simp2d simp1d recnd fsumser ad2antrr oveq12d lelttrd cicc
@@ -327960,7 +327974,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically
wa c0 wdisj mpbid cmul eqsstri csup uniioombllem1 ssid ovollecl ssun2
ovollb peano2nnd syl6eleq uzsplit ax-1cn pncan oveq2d iuneq1d cpw wfn
iunxun rexpssxrxp sstri ffn fniunfv sylan indi 3eqtr4ri eqtr3i uneq2d
- 3eqtr4d syl5sseqr syl6ss rpred ovolun syl22anc eqbrtrd iunss r19.21bi
+ 3eqtr4d sseqtrrid syl6ss rpred ovolun syl22anc eqbrtrd iunss r19.21bi
fsumrecl cfn jca ovolfiniun cdiv nndivred cdm c1st cop ffvelrn elin2d
c2nd 1st2nd2 df-ov ioombl syl6eqel adantlr finiunmbl syl5eqss ioossre
inmbl syl6eqss ovolfcl ovolioo simp2d resubcld mblsplit 3sstr4i sslin
@@ -330764,7 +330778,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
opelxpd eqcomd rspceeqv ralrimiva ssabral rnmpt eldifi simprr inss1 wbr
eldifn eliniseg brelrn sylbi nsyl pm2.21d ssrdv covol 0mbl mblvol ovol0
elv ss0 eqtri mul01d expr cif iftrue c0ex ovmpoa mp2an 0cn mul01i f1ofo
- pm2.61d2 an32s dfin4 eqsstr3i sseli syl6eqel mul02d sylan2 3eqtr2d
+ pm2.61d2 an32s dfin4 eqsstrri sseli syl6eqel mul02d sylan2 3eqtr2d
elsni ) AGHNUDOZUEUFZHUGZDUGZPUHZUIZUAUKZVUECUKZUJOZVUFIOZULOZUAUMZCUMZ
VUAGUGZBUKZVUFNOZVUMVUFIOZULOZBUMZCUMVULVUAVUPCUMBUMAUYTVUDVUEUYSUNVUEU
HUOZUPUFZULOZUAUMZVUDVUAVUICUMZUAUMVUKAUYSUEUQZQVUDURQZUYSUGZVUCUIVUDUS
@@ -330837,7 +330851,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
inmbl ssdifssd eldifsni ad2antlr necon3ai itg1addlem3 syl21anc eqeltrrd
wne simpl itg1addlem1 iunin2 mblss iunid imaeq2i imaiun cnvimarndm fdmd
3eqtr3i syl5eq sseqtr4d df-ss sylib syl5req fveq2d 3eqtr4d oveq2d dfin4
- fsummulc2 difssd eqsstr3i sseli elsni oveq1d 0re syl6eqel mul02d cuz wo
+ fsummulc2 difssd eqsstrri sseli elsni oveq1d 0re syl6eqel mul02d cuz wo
olcd sumz sylan2 fsumss 3eqtrd inss1 simpr incom iuneq2i eqtri cnvimass
syl5sseq cc anasss fsumcom oveq12d ) AEUAZFUAZLUBZUUAMUBZGNZUCNZMOZYTUU
BUUCUCNZMOZUDNZLOZYSUUELOZYSUUGLOZUDNEFUDUENUIUFZEUIUFZFUIUFZUDNAYSUUEU
@@ -332217,7 +332231,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
eqidd i1fadd eqeltrrd cdif i1ff eldifi ffvelrn syl2an leidd iftrue adantl
wf wn eldifn elin sylnib imnan sylibr iffalse oveq12d recnd addid1d eqtrd
imp breqtrrd ad2antrr syl6bb ffnd adantlr 0e0iccpnf ifclda fmptd r19.21bi
- cc eldif nfcv cid fveq2d breqtrd syldan pm2.61dan eleq1w syl5sseqr sselda
+ cc eldif nfcv cid fveq2d breqtrd syldan pm2.61dan eleq1w sseqtrrid sselda
ifbieq1d nfv nfan ofrfval2 iftrued 3brtr4d 0le0 a1d ralrimi mpbird itg2ub
wb syl3anc wo cun eleq2d elun notbid ioran biimpar simprr cpnf cicc inidm
wfn ofrfval mpbid sylan2 nfmpt1 nfcxfr nffv nfeq1 fveqeq2 fvmpt2i 0cn fvi
@@ -332285,7 +332299,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
cof simprll simprrl itg1add cin i1fadd inss1 cvol mblss syl5ss covol wceq
wss cdif nfv nfcv cmpt nfmpt1 nfcxfr nfbr nfan eldifi cvv i1ff ffnd inidm
reex eqidd ofval sylan2 ffvelrn syl2an iccssxr sseldi 0red simprrr wfn wb
- fvexd syl5sseqr sselda syldan simpr dffn5 sylib ofrfval2 r19.21bi breqtrd
+ fvexd sseqtrrid sselda syldan simpr dffn5 sylib ofrfval2 r19.21bi breqtrd
mpbid adantl recnd syl2anc breqtrrd xrletrd fveq2 expr ralrimiva resubcld
itg2leub mpbird cun ssun2 eldifn elin sylnib sylibr imp iffalsed leadd2dd
imnan addid1d simprlr ssun1 iftrued fvmpt2 iftrue 3eqtr4d iffalse addid2d
@@ -334700,7 +334714,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
by Mario Carneiro, 11-Aug-2014.) $)
itgsplit $p |- ( ph -> S. U C _d x = ( S. A C _d x + S. B C _d x ) ) $=
( vk cc0 co cr wcel wa cif adantr c3 cfz ci cv cexp cdiv cre cfv cle cmpt
- wbr citg2 cmul csu caddc citg cvol cdm cibl cmbf iblmbf syl cun syl5sseqr
+ wbr citg2 cmul csu caddc citg cvol cdm cibl cmbf iblmbf syl cun sseqtrrid
ssun1 sselda syldan mbfdm2 ssun2 cin covol wceq cxr cpnf cicc eleq2d elun
cc wo syl6bb biimpa mbfmptcl jaodan adantlr ax-icn elfznn0 adantl sylancr
cn0 expcl wne ine0 elfzelz expne0i mp3an12i divcld recld 0re ifcl sylancl
@@ -335326,7 +335340,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
limcdif $p |- ( ph ->
( F limCC B ) = ( ( F |` ( A \ { B } ) ) limCC B ) ) $=
( vx vz co cc wss wcel wa wceq adantr wf ex cfv cmpt eqid sylbi climc csn
- cdif cres cv cdm w3a limcrcl adantl simp2d eqsstr3d simp3d jca cun undif1
+ cdif cres cv cdm w3a limcrcl adantl simp2d eqsstrrd simp3d jca cun undif1
fdmd difss fssres sylancl snssd unssd syl5eqssr unssad wb cif ctopn crest
ccnfld ccnp simprl simprr ellimc eqcomi oveq2i mpteq1i wo velsn orbi2i wn
elun pm5.61 fvres ifeq2da mpteq2ia eqtr4i ssdifssd bitr4d pm5.21ndd eqrdv
@@ -335360,7 +335374,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
sseq1d sylancl ralrimivw fnmpt fmpt df-f simplrr elinel2 sylbi syl5ibrcom
baib 3syl ralrimiv undif1 ineq2i indi eqtr3i raleqi inss2 3bitrd eldifsni
ralunb bitr3d wne ifnefalse ralbiia syl6bb cres df-ima resmpt syl5eq wfun
- rneqd ffund difss sstri fdmd syl5sseqr funimass4 syl2anc 3bitr4d rexbidva
+ rneqd ffund difss sstri fdmd sseqtrrid funimass4 syl2anc 3bitr4d rexbidva
cdm pm5.74da bitrd ralbidva pm5.32da syl5bb ) FGEUBUCZNZFONZUVJPAUVKFCUJZ
NZEBUJZNZGUVNDEUDZUEZUFZUGUVLQZPZBHUHZUKZCHRZPUVJUVKUVIOFEGUIULUMAUVKUVJU
WCAUVKPZUVJMDUVPUNZMUJZEUOZFUWFGUPZUQZURZEHUWEVDUCZHUSUCUPNZEUWJUPZUVLNZE
@@ -335429,7 +335443,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
imass2 reximdv syl5com impr rexlimiva ralrimdva r19.29r ad3antrrr imaeq2d
syl6 rpxrd ineq1 sseq1d anbi12d rspcev expr syl2anc rexlimdva anim2d syl9
impd syl5 expd expdimp com23 impbid wb wfun wf ad2antrr ffund inss2 difss
- cdm fdmd syl5sseqr syl5ss funimass4 a1i simplrr sselda syl22anc cnmetdval
+ cdm fdmd sseqtrrid syl5ss funimass4 a1i simplrr sselda syl22anc cnmetdval
elbl3 breq1d bitrd simplrl simpllr eldifi ffvelrn ralbidva imbi1i ralbii2
impexp syl2an elin biancomi bitr2i eldifsn imbi2i 3bitr4i 3bitr3g anassrs
rexbidva pm5.32da ) AGHFUBMNGONZGUAUCZNZFLUCZNZHUUOEFUDZUEZUIZUFZUUMPZQZL
@@ -335717,7 +335731,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
wf wss cnfldtopon resttopon sylancr syl5eqel cnpf2 syl3anc wa cuni cnprcl
a1i eqid syl toponuni eleqtrrd ad2antrr wn elun elsni orim2i sylbi adantl
wo orcomd orcanai ffvelrnd ifclda fvco3 syl2anc ifeq2da syl6eqr mpteq2dva
- cofmpt fvif eqtr4d fssd cdm fdmd w3a limcrcl simp2d eqsstr3d simp3d mpbid
+ cofmpt fvif eqtr4d fssd cdm fdmd w3a limcrcl simp2d eqsstrrd simp3d mpbid
ellimc cnfldtop fmpttd snssd unssd feq2d toponunii cnprest2 oveq2i fveq1i
ctop wb syl6eleqr iftrue ssun2 snssg mpbiri fvmptd3 fveq2d cnpco eqeltrrd
fco mpbird ) ADGQZGFUAZCUBRSPBCUCZUDZPUEZCUFZXTYDYAQZUGZUHZCIYCUIRZIUJRQZ
@@ -335759,7 +335773,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
mp2an resttopon sylancr syl5eqel toponuni eleqtrrd opelxp simpld ad2antrr
sylib wn simpll simpr elun ord elsni syl6 con1d imp ifclda simprd opelxpd
wo eqidd wf cnpf2 syl3anc feqmptd fveq2 df-ov ovif12 eqtr3i syl6eq fmptco
- a1i cdm dmmptd w3a limcrcl simp2d eqsstr3d simp3d snssd unssd ssun2 snssg
+ a1i cdm dmmptd w3a limcrcl simp2d eqsstrrd simp3d snssd unssd ssun2 snssg
mpbiri adantr sseldd limcmpt mpbid txcnp topontopi fmpttd feq2d toponunii
wb ctop cnprest2 oveq2i fveq1i syl6eleqr iftrue opeq12d opex fvmpt fveq2d
cnpco eqeltrrd fovrnd mpbird ) AEFIUDZBCGHIUDZUEDUFUDUGBCDUHZUIZBUSZDUJZU
@@ -335801,7 +335815,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
limcco $p |- ( ph -> D e. ( ( x e. A |-> T ) limCC X ) ) $=
( co wcel cc csn cun cv wceq cif cmpt ccom climc ccnfld ctopn crest wa wo
cfv wn wne expr necon1bd wb limccl sseldi adantr elsn2g sylibrd orrd elun
- syl sylibr fmpttd cdm eqid dmmptd wf wss w3a limcrcl eqsstr3d snssd unssd
+ syl sylibr fmpttd cdm eqid dmmptd wf wss w3a limcrcl eqsstrrd snssd unssd
simp2d ccnp limcmpt mpbid limccnp iftrue ssun2 snssg mpbiri fvmptd3 eqidd
eqeq1 ifbieq2d fmptco ifid anassrs ifeq1da mpteq2dva eqtrd oveq1d 3eltr3d
syl5reqr ) AFCEFUAZUBZCUCZFUDZGIUEZUFZUNXGBDHUFZUGZKUHRGBDJUFZKUHRADKFXCX
@@ -336516,7 +336530,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( cc wcel co cc0 cdvn cfv cdm cn0 3ad2ant3 cuz syl22anc nn0cnd wss cvv wf
syl3an3 cr cpr cpm cfz w3a cmin caddc wceq simp1 elfznn0 elfzuz3 uznn0sub
simp2 dvnadd elfzuz2 nn0uz syl6eleqr pncan3d fveq2d eqtrd dmeqd cnex dvnf
- syl a1i dvnbss wa elpmi 3ad2ant2 simprd sstrd elpm2r syl3anc eqsstr3d ) A
+ syl a1i dvnbss wa elpmi 3ad2ant2 simprd sstrd elpm2r syl3anc eqsstrrd ) A
UAEUBZFZBEAUCGZFZCHDUDGFZUEZDABIGZJZKDCUFGZACWAJZIGJZKZWDKZVTWEWBVTWECWCU
GGZWAJZWBVTVPVRCLFZWCLFZWEWIUHVPVRVSUIZVPVRVSUMVSVPWJVRCDUJZMZVTDCNJFZWKV
SVPWOVRCHDUKMCDULVDZABCWCUNOVTWHDWAVTCDVTCWNPVTDVTDHNJZLVSVPDWQFVRCHDUOMU
@@ -336594,7 +336608,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( cc wcel cn0 cfv wss wa cv wi co wceq fveq2 sseq1d imbi2d cdm wf syl3anc
syl vn vm vf cr cpr cuz ccpn c1 caddc ssid 2a1i cpm cdvn ccncf simprl cdv
recnprss ad2antrr adantr simplll eluznn0 adantll dvnf dvnbss dvnp1 simprr
- cz eqeltrrd cncff fdmd wb cnex elpm2g sylancr mpbid simprd sstrd eqsstr3d
+ cz eqeltrrd cncff fdmd wb cnex elpm2g sylancr mpbid simprd sstrd eqsstrrd
cvv dvbss eqssd feq2d dvcn syl31anc peano2nn0 elcpn syl2anc 3imtr4d ssrdv
jca ex sstr2 expcom a2d uzind4 com12 3impia ) AUDDUEZEZBFEZCBUFGZEZCAUGGZ
GZBXCGZHZXBWSWTIZXFXGUAJZXCGZXEHZKXGXEXEHZKXGUBJZXCGZXEHZKXGXLUHUILZXCGZX
@@ -336916,7 +336930,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( ( S X. { A } ) oF x. ( S _D F ) ) ) $=
( vx cmul co cdv cc wf syl cc0 cfv wceq cmpt csn cxp cof caddc wcel snssd
fconstg fssd c0ex fconst ccnfld ctopn crest cnt wss cr cpr recnprss ssidd
- cres cdm dvbsss a1i eqsstr3d eqid dvres resmptd fconstmpt reseq1i 3eqtr4g
+ cres cdm dvbsss a1i eqsstrrd eqid dvres resmptd fconstmpt reseq1i 3eqtr4g
syl22anc oveq2d dvconst dmeqd fdmi syl6eq sseqtr4d dvres3 xpssres reseq1d
eqtrd 3eqtr3d ctop cnfldtopon resttopon sylancr topontop toponuni sseqtrd
ctopon ntrss2 syl2anc dvbssntr eqssd reseq12d 3eqtr4d feq1d mpbiri dvmulf
@@ -337378,7 +337392,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
wf cr dmeqd wral ralrimiva dmmptg eqtrd dvbsss syl6eqssr sstrd syl22anc
dvres resmptd oveq2d reseq1d reseq2d ctop cuni crest cnfldtopon sylancr
ctopon resttopon syl5eqel topontop toponuni sseqtrd eqid ntrss2 syl2anc
- eqsstr3d 3eqtrd 3eqtr3d ) AEBICUAZKUBZUCUDZEWHUCUDZKFUEUFUFZUBZEBKCUAZU
+ eqsstrrd 3eqtrd 3eqtr3d ) AEBICUAZKUBZUCUDZEWHUCUDZKFUEUFUFZUBZEBKCUAZU
CUDBJDUAZAETUGZITWHUOIEUGKEUGWJWMUHAEUPTUIUJWPLEUKULZABICTMUMAIWKUNZEAW
RBIDUAZUNZIAWKWSOUQADHUJZBIURWTIUHAXABINUSBIDHUTULVAEWHVBVCZAKIEPXBVDZI
KEFWHGRQVFVEAWIWNEUCABIKCPVGVHAWMWSWLUBWSJUBWOAWKWSWLOVIAWLJWSSVJABIJDA
@@ -337413,7 +337427,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( cmul co cmpt cc0 cc wcel cfv cdv caddc cv adantr wa 0cnd ccnfld ctopn
crest dvmptc cdm dmeqd wral ralrimiva dmmptg syl eqtrd dvbsss syl6eqssr
wceq eqid cnt ctop cuni wss ctopon cnfldtopon cr cpr recnprss resttopon
- sylancr topontop toponuni sseqtrd ntrss2 fmpttd dvbssntr eqsstr3d eqssd
+ sylancr topontop toponuni sseqtrd ntrss2 fmpttd dvbssntr eqsstrrd eqssd
syl2anc dvmptres2 dvmptmul mul02d oveq1d dvmptcl mulcld addid2d mulcomd
3eqtrd mpteq2dva ) AFBHECNOPUAOBHQCNOZDENOZUBOZPBHEDNOZPABEQCDFRGHIAERS
ZBUCZHSZMUDZAWRUEZUFABEQFUGUHTZFUIOZXARFHHIAWPWQFSZMUDAXCUEUFABEFIMUJAH
@@ -337522,7 +337536,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( S _D ( x e. X |-> A ) ) = ( S _D ( x e. Y |-> A ) ) ) $=
( cdv co cfv wss wceq cc cmpt cres ctop wcel cuni ctopon crest cnfldtopon
cnt resttopon sylancr syl5eqel topontop syl toponuni sseqtrd eqid syl2anc
- ntridm fveq2d eqtr3d reseq2d fmpttd dvres syl22anc eqsstr3d sstrd 3eqtr4d
+ ntridm fveq2d eqtr3d reseq2d fmpttd dvres syl22anc eqsstrrd sstrd 3eqtr4d
wf ntrss2 ssid resmpt mp1i oveq2d resmptd ) ADBGCUAZHUBZOPZDVPOPZDBHCUAZO
PADVPGUBZOPZVRVSAVSGEUIQZQZUBZVSHWCQZUBZWBVRAWDWFVSAWDWCQZWDWFAEUCUDZGEUE
ZRZWHWDSAEDUFQZUDZWIAEFDUGPZWLLAFTUFQUDDTRZWNWLUDFMUHIDFTUJUKULZDEUMUNZAG
@@ -339702,7 +339716,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( ( F ` z ) / ( G ` z ) ) ) limCC B ) ) $=
( vr cv cmin cr cres cfv co wss crp cdiv cmpt climc wcel cioo crn syl3anc
eqid a1i wa wceq syl6sseqr syl sseldd cdif cin adantr rexrd wf cun uneq1i
- cxr clt wbr syl5eq c0 wb syl2anc wn eqtri sylancl mpbid syl5sseqr cdv cdm
+ cxr clt wbr syl5eq c0 wb syl2anc wn eqtri sylancl mpbid sseqtrrid cdv cdm
cc ax-resscn fss sstrd fssresd ioossre dvres syl22anc ctop retop iooretop
cnt isopn3i mp2an reseq2i syl6eq dmeqd ssdmres sylib limcresi sseldi cima
eqtrd df-ima imass2 syl5eqssr ssneldd rneqd syl6eqr eqsstrd resmptd fvres
@@ -339863,7 +339877,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
dvcnvrelem2 $p |- ( ph -> ( ( F ` C ) e. ( ( int ` T ) ` Y ) /\
`' F e. ( ( N CnP M ) ` ( F ` C ) ) ) ) $=
( cfv cnt wcel ccnv ccnp co cmin caddc cicc cima ctop cr wss cioo retop
- crn ctg eqeltri wf1o wfo wceq f1ofo forn 3syl ccncf wf eqsstr3d imassrn
+ crn ctg eqeltri wf1o wfo wceq f1ofo forn 3syl ccncf wf eqsstrrd imassrn
cncff frn syl5sseq cuni uniretop unieqi eqtr4i ntrss dvcnvrelem1 fveq2i
mp3an2i fveq1i syl6eleqr sseldd cres crest wfun f1of ffun funcnvres ccn
f1ocnv cdv cdm dvbsss syl6eqssr ax-resscn syl6ss cncfss syl2anc wf1 syl
@@ -339918,7 +339932,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( x e. Y |-> ( 1 / ( ( RR _D F ) ` ( `' F ` x ) ) ) ) ) $=
( cr cfv eqid cc wcel wceq wss co sylancr wa wb vr vy ccnfld cioo crn ctg
ctopn tgioo2 cpr reelprrecn a1i cnt ctop retop wf1o f1ofo forn 3syl ccncf
- wfo wf cncff frn eqsstr3d uniretop ntrss2 ccnv f1ocnvfv2 sylan crest ccnp
+ wfo wf cncff frn eqsstrrd uniretop ntrss2 ccnv f1ocnvfv2 sylan crest ccnp
cv cabs cmin ccom cxp cres cbl crp cxmet wrex rexmet cdv dvbsss ax-resscn
cdm syl sylancl dvbssntr eqssd isopn3 mpbird f1ocnv ffvelrnda cmopn tgioo
fss f1of mopni2 mp3an2ani c2 cdiv ad2antrr cc0 wn rphalfcl ad2antrl caddc
@@ -341628,7 +341642,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
simpll wf ffvelrnda nn0cnd cfsupp wbr psrbagfsupp ancoms eqbrtrrd disjdif
cin c0 incom eqtri a1i difsnid eqcomd ad2antrl gsumsplit2 3eqtrd ad2antrr
cun cn difexg csubmnd nn0subm eldifi ffvelrn syl2an fmpttd wfun csupp wss
- mptexd funmpt cres difss resmpt ax-mp resss eqsstr3i funsssuppss mp3an12i
+ mptexd funmpt cres difss resmpt ax-mp resss eqsstrri funsssuppss mp3an12i
mptexg fsuppsssupp syl22anc gsumsubmcl cmnd ringmnd simprl ffvelrnd fveq2
gsumsn syl3anc wo simprr sylib elnn0 orel2 sylc eqeltrd nn0nnaddcl nnne0d
syl2anc eqnetrd expr syl5bir rexlimdva necon4bd wfn wb ffnd 0nn0 fnconstg
@@ -344686,7 +344700,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
adddird eqtrd sumeq2dv wss cuz cz nn0zd ifcld cr nn0red syl2anc syl3anbrc
max1 eluz2 fzss2 syl cdif csn c1 cima wn eldifn adantl cun wo eldifi cmin
nn0uz peano2nn0 syl6eleq uzsplit syl5eq nn0cnd ax-1cn pncan oveq2d uneq1d
- sylancl ad2antrr eleqtrd elun sylib ord mpd wi wfun ffund ssun2 syl5sseqr
+ sylancl ad2antrr eleqtrd elun sylib ord mpd wi wfun ffund ssun2 sseqtrrid
cdm fdmd sseqtr4d funfvima2 elsni mul02d fsumss max2 oveq12d 3eqtr4d
mpteq2dva eqtr4d ) AGHUAUBZUCBUDUEIUFUCZFUGZCUHZBUGZUUEUIUCZUJUCZFUKZUEJU
FUCZUUEDUHZUUHUJUCZFUKZUAUCZULBUDUEIJUMUNZJIUOZUFUCZUUECDUUCUCUHZUUHUJUCZ
@@ -344731,7 +344745,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
cuz wss nn0cnd ad2antrr adantl addsubd cz fznn0sub nn0uz syl6eleq eluzadd
nn0zd syl2anc eqeltrd addid2d fveq2d eleqtrd fzss2 adantlr cdif csn c1 wn
eldifn cun wo eldifi peano2nn0 uzsplit syl5eq ax-1cn pncan sylancl uneq1d
- cima eqtrd ad3antrrr elun sylib ord mpd wi wfun cdm ffund ssun2 syl5sseqr
+ cima eqtrd ad3antrrr elun sylib ord mpd wi wfun cdm ffund ssun2 sseqtrrid
fdmd sseqtr4d funfvima2 elsni oveq1d simplr syl2an mul02d fsumss sumeq2dv
fzfid eqtr4d mul01d fsum2mul addcomd fsumcl olcd 3eqtr3d simpll fsummulc1
sumz mul4d expaddd ad2antlr pncan3d eqtr3d 3eqtr4rd cbvsumv oveq2i syl6eq
@@ -347564,7 +347578,7 @@ of all kernels (preimages of ` { 0 } ` ) of all polynomials in
cuni wss ccrd cdm wfo omelon nn0ennn nnenom entri ensymi isnumi mp2an wfn
con0 cc0 wceq c0p wne cdgr cle ccoe cabs wral w3a cz cply crab wrex rabex
cnex fnmpti dffn4 mpbi fodomnum domentr wb fvelrnb ax-mp aannenlem1 eleq1
- mp2 syl5ibcom rexlimiv sylbi ssriv wi aasscn eqsstr3i iunfictbso mp3an12i
+ mp2 syl5ibcom rexlimiv sylbi ssriv wi aasscn eqsstrri iunfictbso mp3an12i
soss eqbrtrid cnso exlimiiv cvv ssexi cq nnssq qssaa sstri ssdomg sbth )
JKLMZKJLMZJKUAMJNLMZNKUAMXGOHPZUBZXIHXKJBUCZUEZNLABCDEFGUDZXLNLMZXLQUFXKX
MXJUBZXMNLMXLRLMZRNUAMXORUGUHSZRXLBUIZXQNURSNRUAMXRUJRNRKNUKULUMZUNNRUOUP
@@ -348705,7 +348719,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
mpteq2dv imbi2d weq fveq2 oveq1 cbvmptv syl6eq cuz ccnfld ctopn crest cnt
cdv wbr wa eqid fvmpt syl cn0 nnnn0d nn0uz syl6eleq eluzfz2b cdm eleqtrrd
ovex dvntaylp0 oveq2d cr cpr cpm wf cvv wss cnex a1i elpm2r syl22anc dvnf
- syl3anc ffvelrnd subidd 3eqtrd wfun wb funmpt sylancr mpbid cfzo eqsstr3d
+ syl3anc ffvelrnd subidd 3eqtrd wfun wb funmpt sylancr mpbid cfzo eqsstrrd
dmmpti eqssd feq2d dvnp1 eqtr3d feqmptd sselda ctayl syl2anc eqtrd taylpf
ffvelrnda dmeqd dvntaylp feq1d mpbird syldan dvn0 syl6eqr ctopon subcld
syl6eleqr funbrfvb nnm1nn0 dvnbss fzo0end elfzofz 3syl dvn2bss nncnd 1cnd
@@ -348805,7 +348819,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
( ( x - B ) ^ ( M + 1 ) ) ) ) limCC B ) ) $=
( co cr cc wcel vy vk cc0 cv c1 caddc cmin cdvn cmpt cexp cdiv climc wa
cfv cdm wss cn0 cfz syl sseldi fznn0sub2 elfznn0 syl3anc cpm reelprrecn
- wf dvnfre a1i cvv cnex ax-resscn sylancl dvnbss fssdmd dvn2bss eqsstr3d
+ wf dvnfre a1i cvv cnex ax-resscn sylancl dvnbss fssdmd dvn2bss eqsstrrd
fss eqssd feq2d mpbid ffvelrnda sselda cres adantl ccnfld nnnn0d sseldd
wceq simpr adantr ffvelrnd dvnply2 eqeltrrd syldan resubcld fmpttd 1nn0
cn nn0addcld reexpcld crn retop uniretop sylancr cdv nncnd fveq2d dvnp1
@@ -348817,7 +348831,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
cdgr cfa faccld nndivred taylply2 simpld plyreres cuz elfzouz syl6eleqr
nnuz cioo ctg cnt ctop ntrss2 nppcan2d fzonnsub elfzofz tgioo2 dvbssntr
ctopn isopn3 mpbird difss dvnf fvexd recn cnfldtopon toponmax cin df-ss
- mp1i sylib ssid cmap mapsspm elmap sylibr dmmpti syl5sseqr addcld prid2
+ mp1i sylib ssid cmap mapsspm elmap sylibr dmmpti sseqtrrid addcld prid2
wb mulcld elfznn ovexd dvmptid 0cnd dvmptc 1m0e1 mpteq2i dvexp mpteq2dv
pncand dvmptco mulid1d dvntaylp0 subidd ctx ccn subcn dvcn plycn syl6ss
syl31anc cncfmptid cncfmpt1f cncfmpt2f 0expd wrex ssdifssd wne eldifsni
@@ -349645,7 +349659,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
( wcel wa cfv vy vv vu vr vx vn vm vj vs vw cv cdv wbr ccnfld ctopn crest
co cnt csn cdif cmin cdiv cmpt climc wss wceq biidd ulmdvlem2 cc recnprss
cdm cr cpr adantr cmap wf ffvelrnda elmapi dvbsss syl6eqssr eqid dvbssntr
- syl eqsstr3d ralrimiva cuz cz uzid syl6eleqr rspcdva sselda cabs clt wral
+ syl eqsstrrd ralrimiva cuz cz uzid syl6eleqr rspcdva sselda cabs clt wral
wi crp wrex sylancr cvv ovex syl2anc mpbid uztrn2 weq fveq2 oveq2d fveq1d
c2 oveq12d breq1d ralbidv ad2antrr rphalfcl adantl simplr oveq1d fvoveq1d
fvmpt imbi2d simpllr wb ad3antrrr eldv simprd dvlem fmpttd sseldd ellimc3
@@ -349766,7 +349780,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
wb sylancr eqeltrd ralrimiva ffnfv sylanbrc ad2antrr uztrn2 simprl subcld
ffvelrnd abscld fzfid ssun2 simprr elfzuzb adantlr syldan cr recnd fvmpt2
cun mpan2 sylan9eq oveq12d fsumser fsumcl fsumsplit mvrladdd fveq2d eqidd
- letrd ralimdva fzsplit syl5sseqr sselda fsumrecl resubcld eqeq12d rspcdva
+ letrd ralimdva fzsplit sseqtrrid sselda fsumrecl resubcld eqeq12d rspcdva
serfre sylan2 cin c0 eluzelre ltp1d fzdisj 3eqtr2d fsumabs eqbrtrd simpll
ad4ant14 anass1rs fsumle eqtr3d eqeltrrd cc0 absge0d fsumge0 absidd eqtrd
0red breqtrrd simpllr rpred lelttr syl3anc ralrimdva anassrs reximdva mpd
@@ -350419,7 +350433,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
cr clt abscld absge0d crp simp2d cxr w3a wb 0re simp1d elico2 sylancr
rpxrd mpbir3and wf wfn ffn elpreima mp2b sylanbrc wceq eqid cnbl0 syl
eleqtrd icossicc imass2 mp1i cpnf iccssxr adantr sseldi simp3d df-ico
- eqsstr3d radcnvcl df-icc xrlelttr ixxss2 syl2anc syl6sseqr 3jca ) AMU
+ eqsstrrd radcnvcl df-icc xrlelttr ixxss2 syl2anc syl6sseqr 3jca ) AMU
EZFTZUFZXMUGKUHUIUJZUKULUMZTXQUHUNZUGKUOUMZUPZUQXTFUQXOXMXRUGKURUMZUP
ZXQXOXMUSTZXMUHULZYATZXMYBTZAFUSXMFUSUQAFXRUGEURUMZUPZUSRYHUHUTUSUHYG
VAUSVJUHVBVCVDVEVFVGZXOYEYDVJTZUGYDVHVIZYDKVKVIZXOXMYIVLXOXMYIVMXOKVN
@@ -352961,7 +352975,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is
wa wb fniniseg mp2b sylanbrc syl6eleqr fmpti wss frn cpi cres df-ima cmpt
c2 reseq1i clt wbr cle cxr w3a 0xr 2re pire remulcli elioc2 mp2an simp1bi
ssriv resmpt rneqi wf1o 0re eqid caddc recni addid2i oveq2i eqcomi efif1o
- f1ofo forn imassrn eqsstr3i eqssi df-fo mpbir2an ) FBCGCFHZCIZBJFBCUAZXDA
+ f1ofo forn imassrn eqsstrri eqssi df-fo mpbir2an ) FBCGCFHZCIZBJFBCUAZXDA
FBKAUBZUDLZUCUEZCDXGFMZXINUFOUGUHZBXJXIPMZXINUEOJZXIXKMZXJXHPMZXLXJKPMXGP
MXOUIXGUJKXGUKULXHUMUNXGUOPFNUANPHXNXLXMURUSUPPFNUQPOXINUTVAVBEVCVDZFBCUQ
QXEBXFXEBVEXPFBCVFQBCRVKVGUDLZSLZUHZXEXSCXRVHZIZBCXRVIYAAXRXIVJZIZBXTYBXT
@@ -361565,7 +361579,7 @@ already know is total (except at ` 0 ` ). There are branch points at
( cpnf wcel wa cc wss ad2antrr vy vz vr vk vw cmpt crli wbr ccnp co wf cv
cfv cima wrex wi wral fmpttd adantr wceq eqid csn ssun2 pnfex snid sselii
cun syl5eleqr eleq1d ralrimiva rspcdva fvmptd3 cabs cmin cbl cxmet cnxmet
- ccom crp cnfldtopn mopni2 mp3an1 cle clt cr ssun1 syl5sseqr ssralv simprl
+ ccom crp cnfldtopn mopni2 mp3an1 cle clt cr ssun1 sseqtrrid ssralv simprl
sylc simplr rlimi cioc cin cordt crest ctop cvv letop cxr a1i syl mp3an2i
pnfxr rexrd syl3anc elind crab wb sylancl syl2anc syld ad3antrrr r19.21bi
sselda breq1d bitrd simplrl eleq1 sylibr mpbird eqtri inss2 sylancr eleq2
@@ -361812,7 +361826,7 @@ exponential function (see also ~ dfef2 ). (Contributed by Mario
wf wceq eqid fmpt sylibr simpr rprecred rpcxpcld rlimi rlimmptrcl adantlr
abscld absge0d rpred rpge0d cxplt2d subid1d fveq2d breq1d abscxp2 syl2anc
ad2antrr cmul rpcnd rpne0d recid2d oveq2d simplr cxpmuld 3eqtr3rd breq12d
- cxp1d 3bitr4d biimpd imim2d ralimdva reximdv cxpcld rlimss eqsstr3d rlim0
+ cxp1d 3bitr4d biimpd imim2d ralimdva reximdv cxpcld rlimss eqsstrrd rlim0
mpd wss mpbird ) AEBCDKLZUBMUCNJUGEUGZUDNZXFUEUFZUAUGZONZUHZEBPZJQUIZUAUJ
PAXNUAUJAXJUJRZUKZXHCMULLZUEUFZXJUMDUNLZKLZONZUHZEBPZJQUIXNXPJEBCMXTSACSR
ZEBPZXOABSEBCUBZVBZYEAYFUOZSYFVBZYGAYFMUCNZYIHMYFUPUQAYHBSYFACFRZEBPYHBVC
@@ -361839,7 +361853,7 @@ exponential function (see also ~ dfef2 ). (Contributed by Mario
o1bdd cmul simpr eqid fvmpt2 oveq1d cvv ovex sylancl eqtr4d nffvmpt1 nfcv
ce nfv nfov fveq2 eqeq12d cbvral sylib r19.21bi ad2ant2r fveq2d ffvelrnda
nfeq ad2antrr simprr 0re ifcl adantr abscld max2 letrd abscxpbnd eqbrtrrd
- expr imim2d ralimdva wss o1mptrcl cxpcld fmpttd o1dm eqsstr3d simprl max1
+ expr imim2d ralimdva wss o1mptrcl cxpcld fmpttd o1dm eqsstrrd simprl max1
sylancr recld recxpcld pire remulcl reefcld remulcld elo12r syl22anc syld
3expia rexlimdvva mpd ) AUAUCZKUCZLMZYCBCDUDZNZUENZUBUCZLMZUFZKCUGZUBOUJU
AOUJZBCDEPUHZUDZUIQZAYEUIQZCRYEUKZYLJAYEULZRYEUKZYQAYPYSJYEUMUNAYRCRYEADF
@@ -362913,7 +362927,7 @@ group sum in the additive group (i.e. the sum of the elements). This is
simprd fsumabs absdivd wceq nnrpd rprege0d absid syl oveq2d eqtrd breqtrd
sumeq2dv cin c0 cn0 rpdivcld flge0nn0 nn0red ltp1d fzdisj cuz cun nn0p1nn
syl6eleq rpred wb jca rpregt0d lediv2 syl211anc mpbid recnd div1d flword2
- nnuz syl3anc syl2anc fsumsplit ssun1 syl5sseqr sselda syldan ssun2 fznnfl
+ nnuz syl3anc syl2anc fsumsplit ssun1 sseqtrrid sselda syldan ssun2 fznnfl
fzsplit2 simplbda adantr lemuldiv2 lemuldivd bitr3d mpd ledivmul2d mpbird
wi ex cc fsumle fsumless letrd nnrecred crp wn cmin rpge0d sylancl eleq2d
divrecd eqeltrrd noel elin syl5bbr mtbiri imnan sylibr con2d baibd syl2an
@@ -366164,7 +366178,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100
cr eluzel2 2z ifcl a1i zred min2 eluz2 syl3anbrc ppisval2 syl2anc eluzelz
2re flid oveq2d sumeq1d clt ltp1d fzdisj inindir 0in 3eqtr3g min1 elfzuzb
cun id sylanbrc fzsplit indir syl6eq wss fzfid inss1 ssfi wa simpr elin2d
- cn prmnn nnrpd relogcld recnd fsumsplit oveq12d mp2an cc syl5sseqr sselda
+ cn prmnn nnrpd relogcld recnd fsumsplit oveq12d mp2an cc sseqtrrid sselda
fzfi ssun1 syldan fsumcl ssun2 pncan2d ) BAUADEZBUBDZAUBDZUCFAGUDHZAGUEZA
IFZJKZCUFZUGDZCLZAUHUIFZBIFZJKZXQCLZUIFZXRUCFYBXIXJYCXKXRUCXIXJUJBUKFJKZX
QCLZYCXIBUOEZXJYEMABULZBCUMNXIYEXMBIFZJKZXQCLYCXIYDYIXQCXIYDXMBUNDZIFZJKZ
@@ -368832,7 +368846,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100
crg ccrg cn0 dchrrcl nnnn0d zncrng crngring eqid unitsubm syl2anc cc0 csn
cn resmhm cdif crn wss wb cnring cnfldbas cnfld0 cndrng drngui ax-mp cima
df-ima cv wral wa wne cbs dchrf unitss sseli ffvelrn syl2an adantr adantl
- wf simpr dchrn0 mpbird eldifsn sylanbrc ralrimiva wfun cdm fdmd syl5sseqr
+ wf simpr dchrn0 mpbird eldifsn sylanbrc ralrimiva wfun cdm fdmd sseqtrrid
ffund funimass4 syl5eqssr resmhm2b sylancr mpbid cgrp wceq unitgrp ablgrp
cabl cnmgpabl ghmmhmb sylancl eleqtrrd ) AHCUBZEFUCUDZEFUEUDZAXOEUFUGQZUC
UDRZXOXPRZAHIUGQZXRUCUDZRCYAUHQRZXSABYBHBDGIJKLUIPUJAIUKRZYCAIULRZYDAGUMR
@@ -372762,7 +372776,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and
csu chash oveq2i cin wceq gausslemma2dlem0b nnred rehalfcld flcld syl cc0
c0 wo cun cn0 cle cr nnrpd rpge0d flge0nn0 syl2anc rphalflt cz nnzd mpbid
crp cuz syl6eleq syl6eq oveq1 0p1e1 oveq1d fzfid gausslemma2dlem0a adantr
- wb wa cn elfznn adantl sylancr remulcld oveq2d c1st cdvds wn cc syl5sseqr
+ wb wa cn elfznn adantl sylancr remulcld oveq2d c1st cdvds wn cc sseqtrrid
a1i sselda syldan fsumnn0cl expaddd cfn wss cxp sylancl ssrab2 hashcl csn
crab ssfi wrel bitri wne ad2antrr elfzle2 elfzelz flge 2re 2pos syl112anc
mpbird peano2rem eqbrtrid cprime eldifad prmz mtod syl3anc simpr nnmulcld
@@ -375782,7 +375796,7 @@ to the second component (see, for example, ~ 2sqreunnltb and
( vi crp wcel csb cr wi cpnf cico co cc0 cle wbr wral ralrimiva nfel1
nfcsb1v cv wceq csbeq1a eleq1d rspc syl5com wa cmpt cfl cfv caddc cuz
c1 csn cxp eqid wb nnred elicopnf syl simprbda flcld peano2zd cn nnuz
- cli 1zzd cdm nnrp ssriv dmmptd syl5sseqr rlimclim1 adantr cc 0red clt
+ cli 1zzd cdm nnrp ssriv dmmptd sseqtrrid rlimclim1 adantr cc 0red clt
nngt0d simplbda ltletrd elrpd sylc recnd ssid fvex climconst2 syl2anc
cz cn0 rpge0d flge0nn0 nn0p1nn eluznn sylan nnrpd ad2antrr weq fvmpts
eqeltrd fvconst2g 3expia ralrimivva nfcv nfv nfbr nfral breq2 anbi12d
@@ -375922,7 +375936,7 @@ to the second component (see, for example, ~ 2sqreunnltb and
breq2 imbi12d cseq cin fzodisj eluzp1p1 elfzuzb sylanbrc fzosplit
cun nnz cpnf cico simpld nnrp impel fsumsplit eluzelz fzval3 3syl
nnzd 3eqtr4d elfznn simpr nfov 2fveq3 oveq12d fsumser fzfid ssun2
- c0 fvmptf syl5sseqr sselda pncan2d fveq2d 2re nfel1 subcld impcom
+ c0 fvmptf sseqtrrid sselda pncan2d fveq2d 2re nfel1 subcld impcom
peano2nn syl2an mulcomd nnnn0 nn0uz elfzelz adantlr eqtr2d oveq2d
resubcld fzosump1 sumeq2dv elfzuz eluznn rspccva fsumparts eluzle
syl2an2r telfsumo fsumrecl eqeltrrd eluzelre wb nnred elicopnf ex
@@ -377391,7 +377405,7 @@ a multiplicative function (but not completely multiplicative).
( cv crp wcel wa c1 cfl cfv cfz co caddc c2 cexp cdiv csqrt cabs cseq
csu cmin cmul fzfid cc cun ssun2 cuz wceq cn cn0 cr cc0 cle wbr simpr
rprege0d flge0nn0 syl nn0p1nn adantr syl6eleq dchrisum0lem1a fzsplit2
- nnuz simprd syl2anc syl5sseqr sselda csn cdif ssrab3 sseldi ad3antrrr
+ nnuz simprd syl2anc sseqtrrid sselda csn cdif ssrab3 sseldi ad3antrrr
cz adantl dchrzrhcl elfznn nnrpd rpsqrtcld rpcnd rpne0d divcld syldan
eldifad abscld cmpt weq 2fveq3 fveq2 oveq12d ad2antrr rpregt0d simpld
wb rpred letrd ffvelrnd subcld cpnf rerpdivcld fsumser eqtr3d eqbrtrd
@@ -377465,7 +377479,7 @@ a multiplicative function (but not completely multiplicative).
nnred cn0 reflcl fllep1 eluzle lemul1d ledivmuld mpbird nnre nndivred
sylbid nnge1 pm3.2i lediv2 mp3an2i div1d breqtrd simpl nndivre syl2an
0lt1 sylbird 3bitr3d 3bitr4d ex pm5.21ndd cun syl6eleq dchrisum0lem1a
- ssun2 nnuz fzsplit2 syl5sseqr sselda csn cdif ssrab3 sseldi ad3antrrr
+ ssun2 nnuz fzsplit2 sseqtrrid sselda csn cdif ssrab3 sseldi ad3antrrr
eldifad elfzelz dchrzrhcl wne rpcnne0d anasss fsumcom2 cmin 2cn mulcl
subcld cpnf elrege0 sylib rerpdivcld adddird pncan3d mulassd divcan2d
cico 2cnd 3eqtr3d mulcld wss o1const cdm eqid divsqrsum rlimdmo1 mp1i
@@ -380703,7 +380717,7 @@ a multiplicative function (but not completely multiplicative).
c2 csu clo1 wa crp cr elioore adantl 1rp a1i 1red clt wbr eliooord simpld
ltled rpgecld pntrf ffvelrni abscld relogcld remulcld rplogcld rerpdivcld
syl recnd 2re fzfid adantr cn elfznn nnrpd rpdivcld fsumrecl resubcld cun
- ssun2 pntrlog2bndlem6a syl5sseqr sselda syldan rpne0d divdird subsubd cin
+ ssun2 pntrlog2bndlem6a sseqtrrid sselda syldan rpne0d divdird subsubd cin
subdid ssun1 c0 wceq reflcl ltp1d fzdisj fsumsplit mvrraddd oveq2d eqtr3d
oveq1d mpteq2dva pntrlog2bndlem5 wss ioossre readdcld cle rpge0d nnrecred
rpred 2rp nndivred cc absge0d elfzle2 cz mpbird mpbid lemul2ad ledivmul2d
@@ -381998,7 +382012,7 @@ a multiplicative function (but not completely multiplicative).
flge mpbird jca pntlemn fsumge0 eqbrtrd cfzo eqid pntlemi elfzoelz zred
nncnd cun ssun1 peano2zd 1re ltle mpd peano2uz leexp2ad lediv2d flword2
uzid eluzp1p1 flcld simprd elfzofz pntlemh sylan2 lttrd ltled syl3anbrc
- flwordi eluz2 fzsplit2 syl5sseqr sstrd adantlr ssun2 le2add 1cnd subcld
+ flwordi eluz2 fzsplit2 sseqtrrid sstrd adantlr ssun2 le2add 1cnd subcld
mpand adddid addcomd addsubd mulid1d 3eqtr3d cin c0 reflcl ltp1d fzdisj
zcnd fsumsplit sylibrd expcom a2d fzind2 mpcom fsumless letrd ) AJLVBVC
ZOLVDVEVCZVFVCZVGVDUUCZFVFVCZVHVCZUAVIVJZVDVEVCZVFVCZVFVCZVVOOLVFVCZVKV
@@ -403008,7 +403022,7 @@ other vertices (of the graph), or equivalently, if all other vertices
-> ( ( G NeighbVtx K ) = ( V \ { K } )
-> ( S NeighbVtx K ) = ( V \ { N , K } ) ) ) $=
( vn wcel wa csn cdif cnbgr co wceq cupgr cpr wss nbgrssovtx upgrres1lem2
- cvtx cfv eqid difpr eqcomi a1i difeq1d syl5eq syl5sseqr adantr w3a anim1i
+ cvtx cfv eqid difpr eqcomi a1i difeq1d syl5eq sseqtrrid adantr w3a anim1i
cv simpl df-3an sylibr wne dif32 eqtri eleq2i eldifsn bitri simplbi eleq2
wi syl5ibr adantl imp nbupgrres sylc eqelssd ex ) EUANGHNOZFHGPZQZNZOZEFR
SZHFPZQZTZAFRSZHGFUBQZTWBWFOZMWGWHWBWGWHUCWFWBAUFUGZWDQZWGWHAWJFWJUHUDWBW
@@ -411643,7 +411657,7 @@ is finite (in the graph). (Contributed by Alexander van der Vekens,
umgr2adedgwlk $p |- ( ph -> ( F ( Walks ` G ) P /\ ( # ` F ) = 2
/\ ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) ) ) $=
( cfv cwlks wbr chash c2 wceq cc0 c1 w3a cvtx wne wcel cumgr cpr sylanbrc
- wa 3anass umgr2adedgwlklem syl simprd simpld wss ssid syl5sseqr jca 2wlkd
+ wa 3anass umgr2adedgwlklem syl simprd simpld wss ssid sseqtrrid jca 2wlkd
eqid cs2 fveq2i s2len eqtri a1i s3fv0 s3fv1 s3fv2 3anim123i fveq1i eqeq2i
cs3 eqcom bitri 3anbi123i sylibr 3jca ) AGEHUATUBGUCTZUDUEZBUFETZUEZCUGET
ZUEZDUDETZUEZUHZABCDEGHIJKHUITZONABCUJCDUJUOZBWMUKZCWMUKZDWMUKZUHZAHULUKZ
@@ -411658,7 +411672,7 @@ is finite (in the graph). (Contributed by Alexander van der Vekens,
30-Jan-2021.) $)
umgr2adedgwlkon $p |- ( ph -> F ( A ( WalksOn ` G ) C ) P ) $=
( wcel cvtx cfv wne wa w3a cumgr cpr 3anass sylanbrc umgr2adedgwlklem syl
- simprd simpld wss ssid syl5sseqr jca eqid 2wlkond ) ABCDEGHIJKHUAUBZONABC
+ simprd simpld wss ssid sseqtrrid jca eqid 2wlkond ) ABCDEGHIJKHUAUBZONABC
UCCDUCUDZBUTTCUTTDUTTUEZAHUFTZBCUGZFTZCDUGZFTZUEZVAVBUDAVCVEVGUDVHPQVCVEV
GUHUIBCDFHLUJUKZULAVAVBVIUMAVDJIUBZUNVFKIUBZUNAVDVDVJVDUORUPAVFVFVKVFUOSU
PUQUTURMUS $.
@@ -411688,7 +411702,7 @@ between two (different) vertices. (Contributed by Alexander van der
1-Feb-2018.) (Revised by AV, 29-Jan-2021.) $)
umgr2adedgspth $p |- ( ph -> F ( SPaths ` G ) P ) $=
( cvtx cfv wne wa wcel w3a cumgr cpr 3anass sylanbrc umgr2adedgwlklem syl
- simprd simpld wss ssid syl5sseqr jca eqid wi fveq2 eqcoms eqeq1d eqtr2 ex
+ simprd simpld wss ssid sseqtrrid jca eqid wi fveq2 eqcoms eqeq1d eqtr2 ex
syl6bi com13 eqcom prcom eqeq2i bitri umgrpredgv anim12d preqr1g eqneqall
wceq sylc syl6ci syl5bi syld neqne pm2.61d1 2spthd ) ABCDEGHIJKHUAUBZONAB
CUCCDUCUDZBWDUEZCWDUEZDWDUEZUFZAHUGUEZBCUHZFUEZCDUHZFUEZUFZWEWIUDAWJWLWNU
@@ -417705,7 +417719,7 @@ edge remains odd if it was odd before (regarding the subgraphs induced
( # ` x ) <_ 2 } ) $=
( cvv cword wcel cconcat co wceq chash cfv c2 crab wf w3a cv cpw cc0 cfzo
cle wbr c0 csn cdif konigsbergssiedgwpr wss prprrab wi 2re eqlei2 ss2rabi
- wrdf a1i eqsstr3i fss mpan2 iswrdb sylibr 3syl ) BJKZLCVFLDBCMNOUABAUBZPQ
+ wrdf a1i eqsstrri fss mpan2 iswrdb sylibr 3syl ) BJKZLCVFLDBCMNOUABAUBZPQ
ZROZAFUCZSZKLUDBPQUENZVKBTZBVHRUFUGZAVJUHUIUJZSZKLZABCDEFGHIUKVKBURVMVLVP
BTZVQVMVKVPULVRVKVIAVOSVPAFUMVIVNAVOVIVNUNVGVOLRVHUOUPUSUQUTVLVKVPBVAVBVP
BVCVDVE $.
@@ -424250,7 +424264,7 @@ Below is the final Metamath proof (which reorders some steps).
( F |` B ) = { <. 2 , 6 >. } ) $=
( c2 c6 cop c3 c9 cpr wceq c1 cres csn cun syl6eq c0 2re elexi wcel gtneii
cr simpl df-pr reseq1d resundir wrel cdm wss relsnop dmsnopss snsspr2 simpr
- wa 6re syl5sseqr syl5ss relssres sylancr 1re 1lt3 2lt3 nelpri eleq2d mtbiri
+ wa 6re sseqtrrid syl5ss relssres sylancr 1re 1lt3 2lt3 nelpri eleq2d mtbiri
wn ressnop0 syl uneq12d un0 eqtrd ) BCDEZFGEZHZIZAJCHZIZULZBAKZVJLZAKZVKLZA
KZMZVRVPVQVRVTMZAKWBVPBWCAVPBVLWCVMVOUAVJVKUBNUCVRVTAUDNVPWBVROMVRVPVSVRWAO
VPVRUEVRUFZAUGVSVRICDCTPQDTUMQUHVPWDCLZACDUIVPVNWEAJCUJVMVOUKZUNUOVRAUPUQVP
@@ -428096,7 +428110,7 @@ which is an Abelian group (i.e. the vectors, with the operation of
sspg $p |- ( ( U e. NrmCVec /\ W e. H ) -> F = ( G |` ( Y X. Y ) ) ) $=
( vx vy wcel wa wceq wfn wss cfv eqid syl cnv cxp cres cv co wral w3a cba
wfun nvgf ffund funres adantr sspnv ffnd fnresdm cns cnmcv isssp simplbda
- wf simp1d ssres eqsstr3d 3jca oprssov sylan eqcomd ralrimivva jctil sspba
+ wf simp1d ssres eqsstrrd 3jca oprssov sylan eqcomd ralrimivva jctil sspba
wb xpss12 syl2anc fnssres eqfnov mpbird ) AUAMZEDMZNZBCFFUBZUCZOZWAWAOZKU
DZLUDZBUEZWEWFWBUEZOZLFUFKFUFZNZVTWJWDVTWIKLFFVTWEFMWFFMNZNWHWGVTWBUIZBWA
PZBWBQZUGWLWHWGOVTWMWNWOVRWMVSVRCUIWMVRAUHRZWPUBZWPCACWPWPSZHUJZUKWACULTU
@@ -428128,7 +428142,7 @@ which is an Abelian group (i.e. the vectors, with the operation of
ssps $p |- ( ( U e. NrmCVec /\ W e. H ) -> R = ( S |` ( CC X. Y ) ) ) $=
( vx vy wcel wa cc wceq wss cfv eqid syl cnv cxp cres cv co wral wfun wfn
w3a cba nvsf ffund funres adantr wf sspnv ffnd fnresdm cpv cnmcv simplbda
- isssp simp2d ssres eqsstr3d 3jca oprssov sylan eqcomd ralrimivva jctil wb
+ isssp simp2d ssres eqsstrrd 3jca oprssov sylan eqcomd ralrimivva jctil wb
ssid sspba xpss12 sylancr fnssres syl2anc eqfnov mpbird ) CUAMZEDMZNZABOF
UBZUCZPZWDWDPZKUDZLUDZAUEZWHWIWEUEZPZLFUFKOUFZNZWCWMWGWCWLKLOFWCWHOMWIFMN
ZNWKWJWCWEUGZAWDUHZAWEQZUIWOWKWJPWCWPWQWRWAWPWBWABUGWPWAOCUJRZUBZWSBBCWSW
@@ -428270,7 +428284,7 @@ which is an Abelian group (i.e. the vectors, with the operation of
sspn $p |- ( ( U e. NrmCVec /\ W e. H ) -> M = ( N |` Y ) ) $=
( vx cnv wcel wa cr syl cfv wfn wss eqid cres wf sspnv nvf ffnd cba sspba
adantr fnssres syl2anc cv wfun cdm wceq ffund funres ad2antrr fnresdm cpv
- cns w3a isssp simplbda simp3d ssres eqsstr3d eleq2d biimpar sylan funssfv
+ cns w3a isssp simplbda simp3d ssres eqsstrrd eleq2d biimpar sylan funssfv
fdmd syl3anc eqcomd eqfnfvd ) ALMZEBMZNZKFCDFUAZVQFOCVQELMZFOCUBABEJUCZEC
FGIUDZPUEZVQDAUFQZRZFWCSVRFRVOWDVPVOWCODADWCWCTZHUDZUEUHABEWCFWEGJUGWCFDU
IUJVQKUKZFMZNZWGVRQZWGCQZWIVRULZCVRSZWGCUMZMZWJWKUNVOWLVPWHVODULWLVOWCODW
@@ -442844,7 +442858,7 @@ to a member of the subspace (Definition of complete subspace in [Beran]
$( A subset of Hilbert space is included in its span. (Contributed by NM,
2-Jun-2004.) (New usage is discouraged.) $)
spanss2 $p |- ( A C_ ~H -> A C_ ( span ` A ) ) $=
- ( vx chba wss cv csh crab cint cspn cfv ssintub spanval syl5sseqr ) ACDAB
+ ( vx chba wss cv csh crab cint cspn cfv ssintub spanval sseqtrrid ) ACDAB
EDBFGHAAIJBAFKBALM $.
$( The union of a set of subspaces is smaller than its supremum.
@@ -445221,7 +445235,7 @@ equals the join of their closures (double orthocomplements).
pjoml4i $p |- ( A vH ( B i^i ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) ) =
( A vH B ) $=
( cort cfv chj cin wss inss1 choccli chjcli chincli chlej2i ax-mp chdmm1i
- co chub1i ineq1i incom eqtri oveq2i inss2 pjoml2i eqtr3i chlej1i eqsstr3i
+ co chub1i ineq1i incom eqtri oveq2i inss2 pjoml2i eqtr3i chlej1i eqsstrri
wceq chlubii mp2an eqssi ) ABAEFZBEFZGQZHZGQZABGQZUOBIUPUQIBUNJUOBABUNDUL
UMACKBDKLMZDCNOAUPIBUPIUQUPIAUOCURRBABHZUOGQZUPUSUSEFZBHZGQZUTBVBUOUSGVBU
NBHUOVAUNBABCDPSUNBTUAUBUSBIVCBUHABUCUSBABCDMZDUDOUEUSAIUTUPIABJUSAUOVDCU
@@ -446264,7 +446278,7 @@ Note that the (countable) Axiom of Choice is used for this proof via
vH ( ( ( C vH F ) i^i ( D vH G ) ) i^i
( ( ( C vH R ) i^i ( D vH S ) ) vH
( ( F vH R ) i^i ( G vH S ) ) ) ) ) ) ) ) ) $=
- ( chj co cin cph cort cfv wss osumi ax-mp ineq12i chshii 5oalem7 eqsstr3i
+ ( chj co cin cph cort cfv wss osumi ax-mp ineq12i chshii 5oalem7 eqsstrri
shscli shincli shsleji chsleji ss2in mp2an shjshcli shlej2i shlej1i sstri
wceq chjcli chincli shjcli sslin ) ABUAUBZCDUAUBZUCZGHUAUBZEFUAUBZUCZUCZB
ACACUDUBZBDUDUBZUCZAEUDUBZBFUDUBZUCZCEUDUBZDFUDUBZUCZUDUBZUCZAGUDUBZBHUDU
@@ -446391,7 +446405,7 @@ Note that the (countable) Axiom of Choice is used for this proof via
(New usage is discouraged.) $)
3oai $p |- ( ( B vH R ) i^i ( C vH S ) ) C_
( B vH ( R i^i ( S vH ( ( B vH C ) i^i ( R vH S ) ) ) ) ) $=
- ( chj co cin cph cort cfv cch choccli chjcli chincli 3oalem5 eqsstr3i
+ ( chj co cin cph cort cfv cch choccli chjcli chincli 3oalem5 eqsstrri
eqeltri 3oalem3 3oalem6 sstri ) BDKLCEKLMBDNLCENLMZBDEBCKLDEKLMKLMKLZABCD
EFGHIJUAUGBDEBCNLDENLMNLMNLUHBCDEGHDBOPZBAKLZMQIUIUJBGRBAGFSTUCECOPZCAKLZ
MQJUKULCHRCAHFSTUCUDABCDEFGHIJUEUFUB $.
@@ -451352,7 +451366,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240.
$( Every bounded linear Hilbert space operator has an adjoint. (Contributed
by NM, 19-Feb-2006.) (New usage is discouraged.) $)
bdopssadj $p |- BndLinOp C_ dom adjh $=
- ( cbo clo ccop cin cado cdm lncnbd cnlnssadj eqsstr3i ) ABCDEFGHI $.
+ ( cbo clo ccop cin cado cdm lncnbd cnlnssadj eqsstrri ) ABCDEFGHI $.
$( Every bounded linear Hilbert space operator has an adjoint. (Contributed
by NM, 22-Feb-2006.) (New usage is discouraged.) $)
@@ -454635,7 +454649,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) =
( ( C vH D ) i^i B ) = ( ( C i^i B ) vH ( D i^i B ) ) ) $=
( wa cin wss chj co bicomi chjcli chlubi w3a wceq cch wcel cmd cdmd simpr
ssin anbi12i simpl 3pm3.2i dmdsl3 mpan syl3an chincli chub1i chlej1i mp1i
- wbr eqsstr3d chub2i sylib ssrind ssrin syl6ss adantr inss2 mpbir2an mdsl3
+ wbr eqsstrrd chub2i sylib ssrind ssrin syl6ss adantr inss2 mpbir2an mdsl3
jca a1i sseqtrd 3expb sylan2b lediri eqssd ) ABUAUOZBAUBUOZIZACDJKZCDLMZA
BLMZKZIZIZVQBJZCBJZDBJZLMZVTVOACKZADKZIZCVRKZDVRKZIZIWBWEKZVPWHVSWKWHVPAC
DUDNWKVSCDVRGHABEFOPNUEVOWHWKWLVOWHWKQZWBWEALMZBJZWEWMVQWNBWMCWNKZDWNKZIV
@@ -454656,7 +454670,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) =
( wa cin wss chj co w3a wceq simpr cch wcel chincli 3pm3.2i cmd cdmd ssin
wbr lejdiri bicomi chlubi anbi12i chub2i chlej1i chjcomi syl6sseq ssinss1
a1i ssini syl adantr chjcli dmdsl3 mpan syl3an inss1 ssrin ax-mp syl5sseq
- simpl mdsl3 inss2 ssind eqsstr3d 3expb sylan2b eqssd ) ABUAUDZBAUBUDZIZAB
+ simpl mdsl3 inss2 ssind eqsstrrd 3expb sylan2b eqssd ) ABUAUDZBAUBUDZIZAB
JZCDJZKZCDLMBKZIZIZVRALMZCALMZDALMZJZWCWFKWBCDAGHEUEUNWAVPVQCKZVQDKZIZCBK
ZDBKZIZIWFWCKZVSWIVTWLWIVSVQCDUCUFWLVTCDBGHFUGUFUHVPWIWLWMVPWIWLNZWFWFBJZ
ALMZWCVPVOWIAWFKZWLWFABLMZKZWPWFOZVNVOPWQWIAWDWEACEGUIADEHUIUOUNWJWSWKWJW
@@ -455132,7 +455146,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) =
its generating vector. (Contributed by NM, 24-Nov-2004.)
(New usage is discouraged.) $)
sh1dle $p |- ( ( A e. SH /\ B e. A ) -> ( _|_ ` ( _|_ ` { B } ) ) C_ A ) $=
- ( csh wcel wa csn cort cfv cspn chba wceq shel spansn syl spansnss eqsstr3d
+ ( csh wcel wa csn cort cfv cspn chba wceq shel spansn syl spansnss eqsstrrd
) ACDBADEZBFZGHGHZRIHZAQBJDTSKBALBMNABOP $.
$( A 1-dimensional subspace is less than or equal to any member of ` CH `
@@ -455318,7 +455332,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) =
( vy wss cat crab cuni cspn cfv wcel c0v wceq chba csh spanid 3syl adantr
cv mpan eleq1 wne wa sheli spansnsh spansna sylan spansnss sseq1 sylanbrc
csn elrab elssuni cch atssch chsssh sstri rabss2 uniss mp2b unimax shssii
- ax-mp eqsstri spanss eqsstr3d spansnid syl sseldd spancl sh0 a1i pm2.61ne
+ ax-mp eqsstri spanss eqsstrrd spansnid syl sseldd spancl sh0 a1i pm2.61ne
ssriv mp2an fveq2i eqtri sseqtri eqssi ) BASZBEZAFGZHZIJZDBWDDSZBKZWEWDKL
WDKZWELWELWDUAWFWELUBZUCZWEUKIJZWDWEWIWJWJIJZWDWFWKWJMZWHWFWENKZWJOKWLWEB
CUDZWEUEWJPQRWIWJWBKZWJWCEZWKWDEZWIWJFKZWJBEZWOWFWMWHWRWNWEUFUGWFWSWHBOKZ
@@ -456137,7 +456151,7 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130,
mdsymi $p |- ( A MH B <-> B MH A ) $=
( vx cmd wbr wb cort cfv c0h wne cdmd choccli cch wcel mddmd chba mp3an12
wa cv chj co eqid mdsymlem8 mp2an 3bitr4g wceq chssii fveq2 pjococi choc0
- wss 3eqtr3g syl5sseqr ssmd1 ssmd2 jca pm5.1 3syl pm2.61iine ) ABFGZBAFGZH
+ wss 3eqtr3g sseqtrrid ssmd1 ssmd2 jca pm5.1 3syl pm2.61iine ) ABFGZBAFGZH
ZBIJZAIJZKKVEKLVFKLTVFVEMGZVEVFMGZVBVCVEVFVEEUAUBUCZEBDNACNVIUDUEAOPZBOPZ
VBVGHCDABQUFVKVJVCVHHDCBAQUFUGVEKUHZABUMZVBVCTZVDVLRABACUIVLVEIJKIJZBRVEK
IUJBDUKULUNUOVMVBVCVJVKVMVBCDABUPSVJVKVMVCCDABUQSURVBVCUSZUTVFKUHZBAUMZVN
@@ -456180,7 +456194,7 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130,
eqeq1d syl6bb 3adant1 bitrd 3com23 syl3an 3expa eleq1 syl6bir mpbid simpr
imp jca exp31 reximdvai r19.42v syl6ib reximdva ancom bitri anass rexbii2
anbi1i syl6ibr chshii shincl sylancl ad2antrr shsel sylibrd expimpd ssrdv
- syl5bi adantl eqsstr3d chincl mpan2 chslej ad2antrl sstrd ralrimiv dmdbr2
+ syl5bi adantl eqsstrrd chincl mpan2 chslej ad2antrl sstrd ralrimiv dmdbr2
exp32 mp2an sylibr ) ABUAIZABUDIZJZBEKZLZXRXPMZXRAMZBUDIZLZNZEOUBZABUCUEZ
XQYDEOXQXROPZXSYCXQYGXSQZQZXTYABUAIZYBYIXTXRXOMZYJXQYKXTJYHXOXPXRUFUGYHYK
YJLXQYHFYKYJFKZYKPFEUHZYLXOPZQYHYLYJPZYLXRXOUIYHYMYNYOYNYLGKZHKZUJIZJZHBR
@@ -457668,7 +457682,7 @@ Class abstractions (a.k.a. class builders)
( wcel c2o cen wbr csn wceq cuni c1o ccrd cfv cfn 2onn adantl csuc syl6eq
vx wa sylib cdif cv wex com nnfi ax-mp enfi mpbiri diffi syl ensymd simpl
cardidd dif1card syl2anc cardennn mpan2 df-2o eqtr3d suc11reg breqtrd en1
- simpr unieqd vex unisn wss difssd eqsstr3d vsnid sylancl eqeltrd exlimddv
+ simpr unieqd vex unisn wss difssd eqsstrrd vsnid sylancl eqeltrd exlimddv
ssel2 ) BACZADEFZSZABGZUAZRUBZGZHZVSIZACRVQVSJEFWBRUCVQVSVSKLZJEVQWDVSVQV
SMVQAMCZVSMCVPWEVOVPWEDMCZDUDCZWFNDUEUFADUGUHOZAVRUIUJUMUKVQWDPZJPZHWDJHV
QAKLZWIWJVQWEVOWKWIHWHVOVPULABUNUOVPWKWJHVOVPWKDWJVPWGWKDHNADUPUQURQOUSWD
@@ -460327,7 +460341,7 @@ its graph has a given second element (that is, function value).
csdm wo wn csn cv crn wfun w3a wex brdom2 wi c1 chash cfv cfz co wf1o nfv
cfn isfinite2 isfinite4 adantr bren 3adant3 cdif cmpt cin f1of adantl cxp
sylib fconstmpt eqcomi wb simplr fconst2g mpbiri disjdif a1i fun syl21anc
- fz1ssnn undif mpbi feq2i 3adantl3 ssid wfo simpr forn syl5sseqr orcd ssun
+ fz1ssnn undif mpbi feq2i 3adantl3 ssid wfo simpr forn sseqtrrid orcd ssun
f1ofo rnun syl6sseqr dff1o3 simprbi cnvun reseq1i resundir dff1o4 fnresdm
eqtri wfn simpl3 cnveqi cnvxp incom disjsn biimpri syl5eqr syl5eq uneq12d
xpdisjres un0 syl6eq funeqd mpbird nnex difexg ax-mp mptex unex feq1 rneq
@@ -460646,7 +460660,7 @@ its graph has a given second element (that is, function value).
cin syl3anc sseqin2 3sstr4d simpl inundif fneq2i sylibr vex a1i fnsuppres
ffs2 inindif syl121anc mpbid uneq12d eqtrd ad2antrl fconst6g disjdif fun2
jca syl21anc feq12d biimpar cfv fveq1d fconstg ad3antlr syl112anc fvconst
- ffnd fvun2 3eqtrd suppss eqsstr3d reseq1d res0 eqtr4i reseq2i fresaunres1
+ ffnd fvun2 3eqtrd suppss eqsstrrd reseq1d res0 eqtr4i reseq2i fresaunres1
3eqtr4ri jca31 impbida syl5bb f1od ) AFLZBGLZCAUCZUDZIBLZMZDUAHBCUEUFZDUI
ZCNZUAUIZACUGZIUHZUJZUKZEOOKUUHHLZUUIOLUUFUUHCHULUMUUDUUJUUGLZUUNOLZUUEUU
DUUPMUUPUUMOLZUUQUUDUUPUNUUDUURUUPUUDUUKOLZUULOLUURUUAUUBUUSUUCACFUOUPIVA
@@ -463918,7 +463932,7 @@ real number multiplication operation (this has to be defined in the main
xrge00 $p |- 0 = ( 0g ` ( RR*s |`s ( 0 [,] +oo ) ) ) $=
( cxrs cxr cmnf cdif cress co cmnd wcel cc0 cpnf cicc wss cfv wceq ax-mp wn
wbr mpbi cin cvv csn c0g eqid xrs1mnd ccmn xrge0cmn cmnmnd cle wa mnflt0 wb
- clt mnfxr 0xr xrltnle mp2an intnan elxrge0 mtbir iccssxr eqsstr3i 0e0iccpnf
+ clt mnfxr 0xr xrltnle mp2an intnan elxrge0 mtbir iccssxr eqsstrri 0e0iccpnf
difsn ssdif cbs difss df-ss xrex difexg xrsbas ressbas eqtr3i ovex ressress
xrs10 dfss incom eqtr2i oveq2i submnd0 mp4an ) ABCUAZDZEFZGHAIJKFZEFZGHZWEW
CLZIWEHIWFUBMNWDWDUCZUDWFUEHWGUFWFUGOWEWEWBDZWCCWEHZPWJWENWKCBHZICUHQZUIWMW
@@ -465845,7 +465859,7 @@ Formula in property (b) of [Lang] p. 32. (Contributed by Thierry
-> ( W |`s A ) e. Archi ) $=
( vx vy vn wcel wa cfv co cv wbr cn wi wral cmnd wb eqid syl adantl wceq
ctos carchi csubmnd c0g cplt cmg cple wrex submrcl isarchi2 sylan2 biimpa
- cress cbs an32s wss submbas submss eqsstr3d ssralv ralimdv subm0 ad2antrr
+ cress cbs an32s wss submbas submss eqsstrrd ssralv ralimdv subm0 ad2antrr
syld cdif ressle difeq1d pltfval submmnd 3eqtr4d eqidd breq123d ad3antrrr
cid cn0 simplll nnnn0d simpllr eleqtrrd submmulg syl3anc rexbidva imbi12d
simpr ralbidva sylibd mpd resstos syl2anc adantlr mpbird ) BUAFZBUBFZGZAB
@@ -469789,7 +469803,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
crg clinds clvec eqeltrd sralvec lveclmod lbsss clspn islbs4 islinds5
syl21anc rspcdva mpand imp suppss ssfid suppssdm sselda eleq1w anbi2d
fssdm breq1d chvarv iunfi xpfi fveq1d cbvmptv syl6eq suppovss feqmptd
- subrg0 feq1d ressbas2 eqsstr3d iuneq12d xpeq12d 3sstr3d suppssfifsupp
+ subrg0 feq1d ressbas2 eqsstrrd iuneq12d xpeq12d 3sstr3d suppssfifsupp
eqtrd fssd syl32anc breqtrd drgextgsum cmulr csubg subrgsubg subgsubm
csubmnd lmodvscl gsumsubm ressmulr sstrd sseqtr4d ringass syl13anc wb
fmpttd cplusg breq2d rmfsupp2 gsummulc1 3eqtr4rd w3a offval22 ovmpt4g
@@ -469898,7 +469912,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
sseqtrd eqtr3d fveq2d sseldd cplusg 3eqtr3rd 3eqtr3d eleqtrd syl21anc
cvv ovexd w3a fvexd wf wb simpr eqidd oveq1d ad2antrr oveqd mpteq2dva
oveq2d drgextgsum wne crab cfn mptexd fmpttd ovex rgenw mpteqb eqeq1d
- wn ax-mp sylib 3ad2ant1 csn subsubrg simpld clbs lbsss eqsstr3d sstrd
+ wn ax-mp sylib 3ad2ant1 csn subsubrg simpld clbs lbsss eqsstrrd sstrd
csra clss srasubrg cmulr simprd simprl ressabs oveq1i 3eqtr4g 3eqtr4d
syl5eq eleqtrrd simprr subrgmcl eqeltrrd ralrimivva syl12anc sseqtr4d
islss4 biimpar lbslinds sseldi resssca sraaddg eqeltrd lmodvscl 3expb
@@ -470070,7 +470084,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
anasss ad5antr simpllr srasca syl5eq fveq2d simp-4r clspn ad7antr oveq12d
eqtr3d clmod cgsu lveclmod fveq2 cmap wrex cmpt ffvelrnd eleqtrrd cbvmptv
cfsupp oveq1d oveq2d anbi12d ad8antr exlimddv cnzr dimval subsubrg oveq1i
- ressabs 3eqtr4g eqeltrd crn cen wf1 crg drngring ressbas2 eqsstr3d ringcl
+ ressabs 3eqtr4g eqeltrd crn cen wf1 crg drngring ressbas2 eqsstrrd ringcl
cmulr sstrd oveq2 oveq1 cplusg clinds lbslinds sseldi ad3antrrr wn islbs4
0nellinds nelne2 drgext0g neeqtrrd ovexd ovmpt4g drgextvsca simprr simprl
oveqd ovmpod adantllr adantl3r 3eqtr3d linds2eq ex f1opr sylanbrc hasheni
@@ -470408,7 +470422,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
wel oveqd cdr cfield fldextfld1 ccrg isfld fldextfld2 eqeltrrd drgextgsum
simplbi adantlr cvv cmnd crg drngring 3syl ringmnd ad4antr vex a1i elmapi
ad3antrrr wf adantl vsnid syl5eleqr ffvelrnd srasca eqtrd fveq2d eleqtrrd
- sseldd lbsss eqsstr3d snss sylibr srabase ringcl syl3anc oveq12d ressmulr
+ sseldd lbsss eqsstrrd snss sylibr srabase ringcl syl3anc oveq12d ressmulr
gsumsnd eqtr4d 3eqtr3d cur cinvr cui syl2an2r eqeltrd wrex lbslsp r19.29a
wn exlimddv simp-5l simprbi crngcom 3eqtr2d simpld unitinvinv cdif simprd
invrvald sralmod0 clinds lbslinds sseldi eqneltrd nelne2 eldifsn sylanbrc
@@ -471655,7 +471669,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
symgfv cn nnuz syl6eleq eluzfz2 eqtr3d 3eqtr3d ex con3d iffalsed ifeqda
cuz imp cvv simp2 adantr ovexd oveqi mpoeq3ia 3eqtr4d cmat cmdat oveq2d
cbs ovmpt4g mpoeq3dva matecld matbas2d ringidcl marrepval cmin csn cdif
- csubma submaval fzdif2 mpoeq12 difssd eqsstr3d submabas mdetcl ringlidm
+ csubma submaval fzdif2 mpoeq12 difssd eqsstrrd submabas mdetcl ringlidm
wss eqeltrd cmulr smadiadetr fveq1i eqeq12i sylibr submat1n submatminr1
fveq1d eqtr4d 3eqtrd ) APORQVCVDZJDVCZEHVCFHVCVEVDUAVCZTDVCZIVDUVMOPRVF
VCVDZMVCZIVDAUVKOPRVGSVNVDZGVHVDZVCVDZDVCZUVLARCVIZPUVQVIZOUVQVIZUVKUVT
@@ -472012,7 +472026,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
txomap $p |- ( ph -> ( H " A ) e. ( L tX M ) ) $=
( vc va vb vz cima ctx co wcel cxp wss wrex wral cfv wceq simp-6l simpllr
cv wa syl2anc simplr wfn cop opex fnmpoi a1i ctopon toponss xpss12 simprl
- syl fnfvima syl3anc simp-4r wf ffn 3syl fimaproj imass2 ad2antll eqsstr3d
+ syl fnfvima syl3anc simp-4r wf ffn 3syl fimaproj imass2 ad2antll eqsstrrd
3eltr3d xpeq1 eleq2d sseq1d anbi12d xpeq2 rspc2ev syl112anc wb eltx mpbid
r19.21bi adantlr adantr r19.29vva wfun mpofun fvelima mpan adantl r19.29a
ralrimiva mpbird ) AHDUJZKLUKULUMZUFVBZUGVBZUHVBZUNZUMZXNXIUOZVCZUHLUPUGK
@@ -473221,7 +473235,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
( vz cle cordt cfv crest co wceq wtru cxr wcel ax-mp wss cv wa wbr cnvtsr
ccnv cxp cin crn cdm lern df-rn eqtri ctsr letsr a1i crab wb brcnvg simpr
adantlr simplr syl2anc anbi12d ancom syl6bb rabbidva sseldi iccval ancoms
- simpl eqsstr3d eqsstrd adantl ordtrest2 mptru tsrps ordtcnv oveq1i eqtr2i
+ simpl eqsstrrd eqsstrd adantl ordtrest2 mptru tsrps ordtcnv oveq1i eqtr2i
cicc cps ) GUBZCCUCUDHIZVSHIZCJKZGHIZCJKVTWBLMBAFCVSNNGUEVSUFUGGUHUIVSUJO
ZMGUJOZWDUKGUAPULCNQMDULBRZCOZARZCOZSZWFFRZVSTZWKWHVSTZSZFNUMZCQMWJWOWHWK
GTZWKWFGTZSZFNUMZCWJWNWRFNWJWKNOZSZWNWQWPSWRXAWLWQWMWPWGWTWLWQUNWIWFWKCNG
@@ -473461,7 +473475,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
simplr bitr3i syl5bb posrasymb rexbii rexcom rexnal 3bitr3i sselda nelne2
necon3bbid r19.41v syl3anc mpbird pm5.17 rexbidva ex inex1 eqeltri fveq2d
cxp ordttopon prsdm syl5eleq syl5eqel ctg snex cbs fvexi ssfii ordtprsval
- bastg sstri syl5eq syl5sseqr syl unssad simplrl simplrr oran 3expa nrexdv
+ bastg sstri syl5eq sseqtrrid syl unssad simplrl simplrr oran 3expa nrexdv
anbi12i anandi exbii n0f df-rex necon1bbii ineq1d 0in syl6eq eldif ssconb
nfin vex sylanb anass1rs mpbi2and nfun ianor equcom syl6bb syl5bbr 3expia
pm5.32d orbi12i elun andi 3bitr4ri eldifsn bicomi eqrd sseqtr4d nconnsubb
@@ -473945,7 +473959,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
cicc cmpt wa simpl oveq2d simpr sseldi xmul01 syl eqtrd mpteq2dva 3eqtr4g
fconstmpt c0ex fconst2 sylibr cnconst syl22anc cres eqid oveq1 cbvmptv id
adantl xrmulc1cn cnrest sylancl resmpt ax-mp eqtr4i eqcomi oveq1i 3eltr3g
- letopuni crn ioorp ioossicc eqsstr3i ge0xmulcl syl2anc fmptd frnd cnrest2
+ letopuni crn ioorp ioossicc eqsstrri ge0xmulcl syl2anc fmptd frnd cnrest2
wb syl3anc mpbid oveq2i syl6eleqr eleq2s cico wo clt wbr 0xr pnfxr 0ltpnf
elicoelioo mp3an sylib mpjaodan ) ACJKZDEELMZNZCJOUAMZNZXQXSAXQEJOUQMZUBP
ZNZYDJYBNZYBJUCZDUEZXSYDXQEUFUDPZYBUGMZYCFYHQUBPNZYBQUHZYIYCNUIJOUJZYBYHQ
@@ -476982,7 +476996,7 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be
6-Jul-2017.) $)
esumdivc $p |- ( ph -> ( sum* k e. A B /e C ) = sum* k e. A ( B /e C ) ) $=
( cesum c1 cxdiv co cc0 cpnf wcel wceq syl3anc sseldi cxr cxmu cdiv rpred
- cico wne 1red rpne0d rexdiv cioo ioorp ioossico eqsstr3i rpreccld eqeltrd
+ cico wne 1red rpne0d rexdiv cioo ioorp ioossico eqsstrri rpreccld eqeltrd
cr crp esummulc1 cicc iccssxr wral ralrimiva esumcl syl2anc xdivrec cv wa
nfcv adantr esumeq2dv 3eqtr4d ) ABCEJZKDLMZUAMZBCVLUAMZEJVKDLMZBCDLMZEJAB
CVLEFGHAVLKDUBMZNOUDMZAKUOPDUOPZDNUEZVLVQQAUFADIUCZADIUGZKDUHRAUPVRVQUPNO
@@ -477854,7 +477868,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
( vx vo csiga wcel cuni wceq wa elex cpw wss cdif wral w3a elpwuni biimpa
cv ancom eqcom cfv crn fvssunirn sseli cvv com cdom wbr wi issiga 3imtr4i
3ad2antr1 syl6bi mpcom jca wex isrnsiga simprbi pweq sseq2d difeq1 eleq1d
- wb eleq1 ralbidv 3anbi12d anbi12d syl ibi exlimiv biimprd pwuni syl5sseqr
+ wb eleq1 ralbidv 3anbi12d anbi12d syl ibi exlimiv biimprd pwuni sseqtrrid
simprd jctild anim2d biimpar syl56 impcom impbii ) ABEUAZFZAEUBGZFZBAGZHZ
IWBWDWFWAWCAEBUCUDAUEFZWBWFAWAJWGWBABKZLZBAFZBCRZMZAFZCANZWKUFUGUHWKGAFUI
CAKNZOZIZWFCABUJZWIWNWJWFWOWJWIIWEBHZWIWJIWFWJWIWSABPQWIWJSBWETUKULUMUNUO
@@ -478137,7 +478151,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
$( A set is a subset of the sigma-algebra it generates. (Contributed by
Thierry Arnoux, 24-Jan-2017.) $)
sssigagen $p |- ( A e. V -> A C_ ( sigaGen ` A ) ) $=
- ( vs wcel cv wss cuni cfv crab cint csigagen ssintub sigagenval syl5sseqr
+ ( vs wcel cv wss cuni cfv crab cint csigagen ssintub sigagenval sseqtrrid
csiga ) ABDACEFCAGOHZIJAAKHCAPLABCMN $.
$}
@@ -479233,7 +479247,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
= sum* n e. N ( M ` ( A \ U_ k e. ( 1 ..^ n ) B ) ) ) $=
( cfv c1 wcel wa cn wss ciun cv cfzo co cdif cesum iundisjcnt fveq2d wral
cmeas com cdom wbr wdisj wceq crn cuni measbase syl adantr simpll fzossnn
- csiga simpr syl5sseqr cuz simplr eleqtrd elfzouz2 fzoss2 3syl sseqtr4d wo
+ csiga simpr sseqtrrid cuz simplr eleqtrd elfzouz2 fzoss2 3syl sseqtr4d wo
mpjaodan sselda wsb sbimi sban sbv clelsb3 anbi12i bitri csb sbsbc wb cvv
wsbc sbcel1g elv nfcv cbvcsb csbid eqtri eleq1i 3imtr3i syl2anc ralrimiva
3bitri sigaclfu2 difelsiga eqimss sseq1 mpbiri jaoi nnct ssct iundisj2cnt
@@ -480609,7 +480623,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
c1 2re a1i cc0 wne 2ne0 reexpclzd 2cnd expne0d redivcld 1red readdcld
simplr rexrd icossre syl2anc eqsstrd ex rexlimivv sylbi mprgbir sseli
elpwid xpss12 syl2an adantr ctx ctop cioo ctg eqeltri txtopi uniretop
- cfv retop unieqi eqtr4i txunii topopn dya2iocuni mp2b unissd eqsstr3d
+ cfv retop unieqi eqtr4i txunii topopn dya2iocuni mp2b unissd eqsstrrd
elpwi rexlimiva ax-mp eqssi ) DUBZUCZLLUDZXIXJMKNZXJMZKXHKXHXJUEXKXHO
XKCNZBNZUDZUFZBFUBZPCXQPXLCBXQXQXOXKDJXMXNCUHBUHUGUIXPXLCBXQXQXMXQOZX
NXQOZQZXPXLXTXPQXKXOXJXTXPUJXTXOXJMZXPXRXMLMXNLMYAXSXRXMLXQLUKZXMXQYB
@@ -481411,7 +481425,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
xrletri3 fveq2 nnex simpr sseldd ad2antrr cima cdif wdisj cnvimass 3syl
esumrnmpt2 c1 cin 3adant1r wfn cfn mpan2 ax-mp disjss1 syl6sseq sseqin2
sylib eqbrtrrd fssdm ffun difpreima fimacnv difeq1d uncom difun2 eqtr3i
- difeq1i eqsstri sspreima eqsstr3d fvimacnvi wf1o simpr3 fresf1o disjrdx
+ difeq1i eqsstri sspreima eqsstrrd fvimacnvi wf1o simpr3 fresf1o disjrdx
difss fvres disjeq2dv bitr3d mpbid biimpa syl21anc 3eqtr3rd ciun uniiun
disjss3 elpwiuncl syl5eqel cfz cbvesum syl6breq fz1ssnn fzfi fnfi resss
ffn fnssres rnfi rnss cbvdisj disjun0 sylbi sylc carsgclctunlem1 unissd
@@ -482322,7 +482336,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
wi cr crrh wf cioo ctg ctopn czlm cds cxp cres xpeq12i reseq12i eqtri
fveq2i eqid cdr crrext syl5eqel rrextdrg syl5eqelr cnrg rrextnrg cnlm
syl rrextnlm cchr wceq rrextchr syl5eqr ccusp rrextcusp cuss rrextust
- cmetu rrhf feq1i ffund rge0ssre fdmd syl5sseqr funfvima2 syl2anc sylc
+ cmetu rrhf feq1i ffund rge0ssre fdmd sseqtrrid funfvima2 syl2anc sylc
sylibr cuni cmeas csiga dmmeas csigagen fvexi a1i sgsiga sibfmbl frnd
mbfmf unieqi w3a cfn sylancl sstri eqeltrd unisg syl5eq tpsuni eqtr4d
mp1i sseqtrd ssdifd sselda eldifad simp2 eleq1 3anbi2d eleq1d imbi12d
@@ -488693,7 +488707,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
rpsqrtcn $p |- ( sqrt |` RR+ ) e. ( RR+ -cn-> RR+ ) $=
( vx csqrt crp cres ccncf co wcel wf cv cdm cc cr wceq sqrtf ax-mp wb mpbir
wss cc0 cpnf cfv wa wral rpssre ax-resscn sstri fdm sseqtr4i sseli rpsqrtcl
- rgen wfun ffun ffvresb cico cioo ioossico eqsstr3i resabs1 resqrtcn rescncf
+ rgen wfun ffun ffvresb cico cioo ioossico eqsstrri resabs1 resqrtcn rescncf
jca ioorp mp2 eqeltrri cncffvrn mp2an ) BCDZCCEFGZCCVHHZVJAIZBJZGZVKBUACGZU
BZACUCZVOACVKCGVMVNCVLVKCKVLCLKUDUEUFZKKBHZVLKMNKKBUGOUHUIVKUJVBUKBULZVJVPP
VRVSNKKBUMOACCBUNOQCKRVHCLEFZGVIVJPVQBSTUOFZDZCDZVHVTCWARZWCVHMCSTUPFWAVCST
@@ -493425,7 +493439,7 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a
$( First-order logic and set theory. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.) (New usage is discouraged.) $)
bnj1241 $p |- ( ( ph /\ ps ) -> C C_ B ) $=
- ( wa wceq eqcomd adantl wss adantr eqsstr3d ) ABHECDBCEIABECGJKACDLBFMN
+ ( wa wceq eqcomd adantl wss adantr eqsstrrd ) ABHECDBCEIABECGJKACDLBFMN
$.
$}
@@ -495586,7 +495600,7 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a
w-bnj15 wfn csuc wral w3a csn cdif cab c-bnj18 wss weu wne 1onn 1n0 ne0ii
eldifsn mpbir2an biid eqid bnj852 r19.2z sylancr euex bnj31 rexcom4 sylib
bnj1198 simp2 reximi sylbi df-rex 19.41v simprbi cdm bnj900 fveq2 ssiun2s
- abid ssiun2 bnj882 syl6sseqr sstrd eqsstr3d exlimiv ) ABUBCAHIZDJZWGEJZUC
+ abid ssiun2 bnj882 syl6sseqr sstrd eqsstrrd exlimiv ) ABUBCAHIZDJZWGEJZUC
ZKWGLZABCMZNZFJZUDZWHHWNWGLGWMWGLZABGJMONUAFPUEZUFZEPKUGUHZQZDUIZHZDRWKAB
CUJZUKZWFWSDXAWFWQDRZEWRQWSDRWFWQDULZXDEWRWFWRKUMXEEWRUEXEEWRQSWRSWRHSPHS
KUMUNUOSPKUQURUPWLWPGAWRBDFECWLUSZWPUSZWRUTZVAXEEWRVBVCWQDVDVEWQEDWRVFVGW
@@ -500943,7 +500957,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by
f1osn sneqd fzsplit cz 1p1e2 oveq1i uneq12i syl6req cin snssd incom
fzsn wn cle clt 1lt2 1re 2re ltnlei mpbi elfzle1 disjsn mpbir eqtri
mto uneqdifeq mpbid reseq2 f1oeq2 f1oeq3 3bitrd fzp1ss subfacp1lem4
- eqsstr3i fveq1d r19.21bi sylan2 eleq2i eldifsn subfacp1lem2b fvresi
+ eqsstrri fveq1d r19.21bi sylan2 eleq2i eldifsn subfacp1lem2b fvresi
sseli bitri adantl sylan2br adantlr expr pm2.61dne syl2an ralrimiva
ffvelrn cpr difexg fex prex unex coex resex fvres sylan9eq ralbidva
f1oun mpanr12 sylancr bitrd biimpa syldan wo eleqtrrd nelne2 simp2d
@@ -503135,7 +503149,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by
cvmscld $p |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) ->
A e. ( Clsd ` ( C |`t ( `' F " U ) ) ) ) $=
( vx wcel cuni wss wceq syl2anc cin c0 ccvm co cfv w3a ccnv cima csn cdif
- crest ccld ctop cvmtop1 3ad2ant1 cvmsuni 3ad2ant2 cvmsss eqsstr3d restuni
+ crest ccld ctop cvmtop1 3ad2ant1 cvmsuni 3ad2ant2 cvmsss eqsstrrd restuni
unissd eqid difeq1d cun unisng 3ad2ant3 uneq2d uniun undif1 simp3 ssequn2
snssd sylib syl5eq unieqd eqtrd syl5eqr eqtr3d difss unissi syl5sseq ciun
wb cv uniiun ineq2i incom iunin2 3eqtr4i wa wne eldifsn wn nesym cvmsdisj
@@ -503816,7 +503830,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by
syl6eleq fzsn ax-mp 1ex syl6eqr eluzfz1 1m1e0 oveq1i nnne0d div0d a1d
syl5eq elnnuz biimpi peano2fzr ex imim1d simplbi elfznn adantr fveq2i
cioo crn ctg ssun1 nnnn0d nn0ge0d nngt0d divge0 syl22anc ltp1d ltdiv1
- syl112anc ltled w3a elicc2 mpbir3and iccsplit syl5sseqr unieqi eqtr4i
+ syl112anc ltled w3a elicc2 mpbir3and iccsplit sseqtrrid unieqi eqtr4i
uniretop restcldi ssun2 eqeltri restuni cin simprbi cnf mpbird ax-1cn
retop pncan sylancl cop wfun cdm ffund ssiun2s peano2rem ltm1d ubicc2
fdmd eleqtrrd funssfv fveq2d fveq12d cvmliftlem9 3eqtr2d opeq2d sneqd
@@ -505886,7 +505900,7 @@ codes is an increasing chain (with respect to inclusion). (Contributed
( vb com wcel wa wss cfv wi cv wceq fveq2 sseq2d imbi2d va vx vu vv vy vi
vk vz csuc weq ssidd a1i pm2.27 adantl simpr c1st cgna cmap c2nd cin cdif
co wrex cgol cop csn cres wral crab wo copab ssun1 simpl simplll satfvsuc
- cun syl2an23an syl5sseqr adantr sstrd ex syld com23 findsg impcom ) AJKBJ
+ cun syl2an23an sseqtrrid adantr sstrd ex syld com23 findsg impcom ) AJKBJ
KZLZEFKZDGKZLZBAMZBCNZACNZMZOWGWKWJWNWGWKWJWNOZWJWLIPZCNZMZOWJWLWLMZOZWJW
LUAPZCNZMZOZWJWLXAUIZCNZMZOWOIUAABWPBQZWRWSWJXHWQWLWLWPBCRSTIUAUJZWRXCWJX
IWQXBWLWPXACRSTWPXEQZWRXGWJXJWQXFWLWPXECRSTWPAQZWRWNWJXKWQWMWLWPACRSTWTWF
@@ -506633,7 +506647,7 @@ codes is an increasing chain (with respect to inclusion). (Contributed
height ` N + 1 ` . (Contributed by AV, 20-Oct-2023.) $)
fmlasssuc $p |- ( N e. _om -> ( Fmla ` N ) C_ ( Fmla ` suc N ) ) $=
( vx vu vv vi com wcel cfmla cfv cv cgna co wceq wrex cgol cab csuc ssun1
- wo cun fmlasuc syl5sseqr ) AFGAHIZBJZCJZDJKLMDUCNUDUEEJOMEFNSCUCNBPZTUCAQ
+ wo cun fmlasuc sseqtrrid ) AFGAHIZBJZCJZDJKLMDUCNUDUEEJOMEFNSCUCNBPZTUCAQ
HIUCUFRBDCEAUAUB $.
$}
@@ -507340,7 +507354,7 @@ codes is an increasing chain (with respect to inclusion). (Contributed
-> ( ( M Sat E ) ` _om ) : ( Fmla ` _om ) --> ~P ( M ^m _om ) ) $=
( vx vy wcel wa com cfmla cfv co wf cv ciun wss wral wbr adantl wb cpw wo
cmap csat satff 3expa csdm cen w3o entric wpss nnsdomo pm3.22 anim2i eqid
- pssss satfsschain imp syl2an orcd ex sylbid weq ssid fveq2 syl5sseqr olcd
+ pssss satfsschain imp syl2an orcd ex sylbid weq ssid fveq2 sseqtrrid olcd
nneneq syl6bi ancoms impel 3jaod mpd expr ralrimiv jca ralrimiva fvex syl
fiun satom wceq fmla a1i feq12d mpbird ) BCGZADGZHZIJKZBIUCLUAZIBAUDLZKZM
EIENZJKZOZWKEIWNWLKZOZMZWIWOWKWQMZWQFNZWLKZPZXBWQPZUBZFIQZHZEIQWSWIXGEIWI
@@ -516343,7 +516357,7 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $=
ndmfv 3orrot incom word nodmord ordirr disjsn xpdisj1 eqtr3d resundir un0
uneq2d eqcomi 3eqtr4g wrel resdm sssucid resabs1 mp1i 3brtr4d elexi prid2
funrel noextend sucelon sltres soasym df-suc reseq2i resundi eqtri rabbii
- dmres necom inteqi necomd syl5eqss eqsstr3d df-ss biimpar ralrimiv funres
+ dmres necom inteqi necomd syl5eqss eqsstrrd df-ss biimpar ralrimiv funres
nesym eqfunfv mpbir2and wfn funfn 1oex prid1 neeq1d 3brtr3d brtp ecase23d
3bitri simprd neeq1 con4i fnressn opeq2d sneqd uneq12d eqnbrtrd jaodan
mpdan ) CBHZCAUSIJKABUBZLZBMUCZBUDHZDMHZUEZEUSZDIJZEBUBZUEZCDIJZDCUFZUGZU
@@ -516603,7 +516617,7 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $=
noetalem4 $p |- ( ( ( A C_ No /\ A e. _V ) /\ ( B C_ No /\ B e. _V ) ) ->
( bday ` Z ) C_ suc U. ( bday " ( A u. B ) ) ) $=
( csur wss wcel cbday cun wceq ax-mp eqtri con0 cvv wa cdm cima nosupno
- cuni csuc cfv bdayval syl nosupbday eqsstr3d adantr unss1 simpll simplr
+ cuni csuc cfv bdayval syl nosupbday eqsstrrd adantr unss1 simpll simplr
simprr noetalem1 syl3anc cdif c1o csn cxp dmeqi dmun wne 1oex snnz dmxp
c0 uneq2i undif2 syl6eq imaundi unieqi uniun suceq word crn imassrn wfo
bdayfo forn sseqtri ssorduni ordsucun mp2an a1i 3sstr4d ) ELMZEUANZUBZF
@@ -522229,7 +522243,7 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and
( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) ) ->
( A Line B ) = { x e. ( EE ` N ) | x Colinear <. A , B >. } ) $=
( cn wcel cee cfv wne w3a wa cline2 co cv cop ccolin wbr cab cin crab wss
- fvline wceq liness eqsstr3d df-ss sylib eqtr4d dfrab2 syl6eqr ) DEFBDGHZF
+ fvline wceq liness eqsstrrd df-ss sylib eqtr4d dfrab2 syl6eqr ) DEFBDGHZF
CUKFBCIJKZBCLMZANBCOPQZARZUKSZUNAUKTULUMUOUPABCDUBZULUOUKUAUPUOUCULUOUMUK
UQBCDUDUEUOUKUFUGUHUNAUKUIUJ $.
$}
@@ -523546,7 +523560,7 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and
` A ) ) = A <-> E. c e. ( Clsd ` J ) A = ( ( int ` J ) ` c ) ) ) $=
( ctop wcel wss wa ccl cfv wceq cv ccld wrex clscld syl2anc ntrss syl3anc
cnt eqcom biimpi fveq2 rspceeqv syl2an cldrcl ntrss2 clsss2 ntridm ntrss3
- ex cldss mpdan clsss3 sscls eqsstr3d eqssd adantl id syl5ibrcom rexlimdva
+ ex cldss mpdan clsss3 sscls eqsstrrd eqssd adantl id syl5ibrcom rexlimdva
2fveq3 eqeq12d impbid ) BFGZACHIZABJKZKZBTKZKZALZADMZVIKZLZDBNKZOZVFVKVPV
FVHVOGAVJLZVPVKABCEPVKVQVJAUAUBDVHVOVMVJAVLVHVIUCUDUEUKVFVNVKDVOVFVLVOGZI
VKVNVMVGKZVIKZVMLZVRWAVFVRVTVMVRVEVLCHZVSVLHZVTVMHVLBUFZVLBCEULZVRVMVLHZW
@@ -524476,7 +524490,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $=
filnetlem3 $p |- ( H = U. U. D /\
( F e. ( Fil ` X ) -> ( H C_ ( F X. X ) /\ D e. DirRel ) ) ) $=
( vv vw vz cfv wcel wss wa cv c1st wbr cvv vu cuni wceq cfil cxp wi cdm
- cdir crn cun cid cres dmresi filnetlem2 dmss ax-mp eqsstr3i ssun1 sstri
+ cdir crn cun cid cres dmresi filnetlem2 dmss ax-mp eqsstrri ssun1 sstri
simpli dmrnssfld simpri uniss mp2b unixpss unidm sseqtri eqssi csn ciun
wral filelss xpss2 syl ralrimiva iunxpconst syl6sseq syl5eqss wrel ccom
ss2iun ccnv a1i relopabi jctil wex cin c0 wne simpl adantr simprl xp1st
@@ -524528,7 +524542,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $=
dfss3 rexiunxp 3bitri fileln0 adantlr r19.9rzv rspcv adantl sstr2 com12
ssid mpii ralrimivw impbid1 bitr3d syl5bb 3bitrd pm5.32da filn0 wo snnz
jctil neanior sylnibr ralrimiva r19.2z rexnal sseq1i 3bitr3i necon3abii
- xpeq0 iunss sylibr cid dmresi filnetlem2 eqsstr3i dmxpid sseqtri tailfb
+ xpeq0 iunss sylibr cid dmresi filnetlem2 eqsstrri dmxpid sseqtri tailfb
eqssi elfm filfbas elfg eqrdv fgfil eqtr2d jca feq1 oveq2 fveq1d spcegv
3bitr4d sylc dmeq fveq2 rneqd fveq2d exbidv rspcev ) FHUDMZNZCUENZCUFZH
DUGZVBZFCUHMZUIZHUYHUJVCZMZOZPZDUKZIUGZUFZHUYHVBZFUYQUHMZUIZUYLMZOZPZDU
@@ -527942,9 +527956,9 @@ Utility lemmas or strengthenings of theorems in the main part (biconditional
bj-mpgs.min $e |- ph $.
bj-mpgs.maj $e |- ( ( ph /\ A. x ph ) -> ps ) $.
$( From a closed form theorem (the major premise) with an antecedent in the
- "strong necessity" modality form (in the language of modal logic),
- deduce the inference ` |- ph => |- ps ` . Strong necessity is stronger
- than necessity, and equivalent to it when ~ sp (modal T) is available.
+ "strong necessity" modality (in the language of modal logic), deduce the
+ inference ` |- ph => |- ps ` . Strong necessity is stronger than
+ necessity, and equivalent to it when ~ sp (modal T) is available.
Therefore, this theorem is stronger than ~ mpg when ~ sp is not
available. (Contributed by BJ, 1-Nov-2023.) $)
bj-mpgs $p |- ps $=
@@ -528378,12 +528392,6 @@ Utility lemmas or strengthenings of theorems in the main part (biconditional
) BCDZAEZBFZQAGZBHZSUABFZUAQAUAGGZBHSUBGABCIUCRUABQAUAJKLTBMNABCOP $.
$}
- $( See ~ sbft . This proof is from Tarski's FOL together with ~ sp (and its
- dual). (Contributed by BJ, 22-Dec-2020.) $)
- bj-ssbft $p |- ( F/ x ph -> ( [ t / x ] ph <-> ph ) ) $=
- ( wnf wsb wex wal spsbe wi df-nf biimpi sp syl56 19.8a stdpc4 impbid ) ABDZ
- ABCEZARABFZQABGZAABCHQSTIABJKZABLMASQTRABNUAABCOMP $.
-
${
$d y x $. $d y ph $.
$( A special case of ~ sbequ2 . (Contributed by BJ, 22-Dec-2020.) $)
@@ -531275,10 +531283,10 @@ FOL part ( ~ bj-ru0 ) and then two versions ( ~ bj-ru1 and ~ bj-ru ).
from which the theorem follows. QED
2. Suppose that S also contains (the FOL version of) modal logic KB and
- commutation of quantifiers ~ alcom and ~ excom (possibly weakend by a DV
+ commutation of quantifiers ~ alcom and ~ excom (possibly weakened by a DV
condition on the quantifying variables), and that S can be axiomatized
such that the only axioms with a DV condition involving a formula variable
- are among ~ ax-5 , ~ ax5e , ax5ea . If the scheme
+ are among ~ ax-5 , ~ ax5e , ~ ax5ea . If the scheme
(PHI_1 ` & ... & ` PHI_n ` => ` PHI_0, DV)
@@ -531320,9 +531328,9 @@ variables all in OC(PHI) ` \ { ph } ` , so ` ( F// x ` PHI,
` \ { ph } } ) ` by ~ a1i . For the induction step, PHI is either an
implication, a negation, a conjunction, a disjunction, a biconditional, a
universal or an existential quantification of formulas where ` x ` does
- not occur. We use respectively ~ bj-nnfim , ~ bj-nnfnt , nnfant, nnfort,
- nnfbit, ~ bj-nnfalt , ~ bj-nnfext . For instance, in the implication
- case, if we have by induction hypothesis
+ not occur. We use respectively ~ bj-nnfim , ~ bj-nnfnt , ~ bj-nnfant ,
+ ~ bj-nnfort , nnfbit, ~ bj-nnfalt , ~ bj-nnfext . For instance, in the
+ implication case, if we have by induction hypothesis
` ( ( A. x `_1 ` ... A. x `_m ` F// x ph -> F// x ` PHI),
` { { x , a } | a e. ` OC(PHI) ` \ { ph } } ) ` and ` ( ( A. y `_1
@@ -531442,16 +531450,18 @@ quantifiers appear (since they typically require ~ ax-10 to work with),
( wnnf wex wal wi wnf bj-nnfea df-nf sylibr ) ABCABDABEFABGABHABIJ $.
$( A variable is nonfree in a formula if and only if it is nonfree in its
- negation. The foward implication is intuitionistically valid.
+ negation. The foward implication is intuitionistically valid (and that
+ direction is sufficient for the purpose of recursively proving that some
+ formulas have a given variable not free in them, like ~ bj-nnfim ).
Intuitionistically, ` |- ( F// x -. ph <-> F// x -. -. ph ) ` . See
~ nfnt . (Contributed by BJ, 28-Jul-2023.) $)
bj-nnfnt $p |- ( F// x ph <-> F// x -. ph ) $=
( wex wi wal wa wn wnnf eximal alimex anbi12ci df-bj-nnf 3bitr4i ) ABCADZAA
BEDZFAGZBCPDZPPBEDZFABHPBHNROQAABIAABJKABLPBLM $.
- $( A variable is nonfree in a theorem. Note that the antecedent is in the
- form of what is called "strong necessity" in modal logic. (Contributed by
- BJ, 28-Jul-2023.) $)
+ $( A variable is nonfree in a theorem. The antecedent is in the "strong
+ necessity" modality of modal logic in order not to require ~ sp (modal T).
+ (Contributed by BJ, 28-Jul-2023.) $)
bj-nnftht $p |- ( ( ph /\ A. x ph ) -> F// x ph ) $=
( wal wa wex wi wnnf ax-1 anim12i df-bj-nnf sylibr ) AABCZDABEZAFZALFZDABGA
NLOAMHLAHIABJK $.
@@ -531493,20 +531503,72 @@ quantifiers appear (since they typically require ~ ax-10 to work with),
wi syl6 ) ACDBCDEZABOZCFZRORRCGZORCDSACGBCFOQRABCHABCIJQRACFBCGOTABCKABCLPR
CMN $.
+ $( Nonfreeness in both conjuncts implies nonfreeness in the conjunction.
+ (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof
+ using the definition of conjunction in terms of implication and negation,
+ so using ~ bj-nnfim , ~ bj-nnfnt and ~ bj-nnfbi , but we want a proof
+ valid in intuitionistic logic. (Proof modification is discouraged.) $)
+ bj-nnfant $p |- ( ( F// x ph /\ F// x ps ) -> F// x ( ph /\ ps ) ) $=
+ ( wnnf wa wex wi wal df-bj-nnf 19.40 prth syl5 id alanimi syl6 anim12i an4s
+ syl2anb sylibr ) ACDZBCDZEABEZCFZUBGZUBUBCHZGZEZUBCDTACFZAGZAACHZGZEBCFZBGZ
+ BBCHZGZEUGUAACIBCIUIUMUKUOUGUIUMEZUDUKUOEZUFUCUHULEUPUBABCJUHAULBKLUQUBUJUN
+ EUEAUJBUNKABUBCUBMNOPQRUBCIS $.
+
+ ${
+ bj-nnfand.1 $e |- ( ph -> F// x ps ) $.
+ bj-nnfand.2 $e |- ( ph -> F// x ch ) $.
+ $( Nonfreeness in both conjuncts implies nonfreeness in the conjunction.
+ Deduction form. Note: compared with the proof of ~ bj-nnfant , it has a
+ few more essential steps (this difference would be reduced by adding
+ deduction forms of ~ bj-nnfe and ~ bj-nnfa ) and has fewer total steps
+ (since there are fewer intermediate formulas to build) and is slightly
+ easier to understand. This statement is of intermediate complexity: for
+ simpler statements, closed-style proofs will generally be shorter than
+ deduction-style proofs, while for more complex statements, the opposite
+ will be true (and deduction-style proofs will generally be easier to
+ understand). (Contributed by BJ, 19-Nov-2023.)
+ (Proof modification is discouraged.) $)
+ bj-nnfand $p |- ( ph -> F// x ( ps /\ ch ) ) $=
+ ( wa wex wi wal wnnf 19.40 bj-nnfe syl anim12d syl5 bj-nnfa alanimi syl6
+ id df-bj-nnf sylanbrc ) ABCGZDHZUCIUCUCDJZIUCDKUDBDHZCDHZGAUCBCDLAUFBUGCA
+ BDKZUFBIEBDMNACDKZUGCIFCDMNOPAUCBDJZCDJZGUEABUJCUKAUHBUJIEBDQNAUICUKIFCDQ
+ NOBCUCDUCTRSUCDUAUB $.
+ $}
+
+ $( Nonfreeness in both disjuncts implies nonfreeness in the disjunction.
+ (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof
+ using the definition of disjunction in terms of implication and negation,
+ so using ~ bj-nnfim , ~ bj-nnfnt and ~ bj-nnfbi , but we want a proof
+ valid in intuitionistic logic. (Proof modification is discouraged.) $)
+ bj-nnfort $p |- ( ( F// x ph /\ F// x ps ) -> F// x ( ph \/ ps ) ) $=
+ ( wnnf wa wo wex wi wal df-bj-nnf 19.43 pm3.48 syl5bi 19.33 anim12i syl2anb
+ syl6 an4s sylibr ) ACDZBCDZEABFZCGZUBHZUBUBCIZHZEZUBCDTACGZAHZAACIZHZEBCGZB
+ HZBBCIZHZEUGUAACJBCJUIUMUKUOUGUIUMEZUDUKUOEZUFUCUHULFUPUBABCKUHAULBLMUQUBUJ
+ UNFUEAUJBUNLABCNQORPUBCJS $.
+
$( If two formulas are equivalent for all ` x ` , then nonfreeness of ` x `
in one of them is equivalent to nonfreeness in the other. Compare
~ nfbiit . From this and ~ bj-nnfim and ~ bj-nnfnt , one can prove
analogous nonfreeness conservation results for other propositional
- operators.
-
- Note: The antecedent is in the "strong necessity" modality of modal logic
- (see also ~ bj-nnftht ) in order not to require ~ sp (modal T).
+ operators. The antecedent is in the "strong necessity" modality of modal
+ logic (see also ~ bj-nnftht ) in order not to require ~ sp (modal T).
(Contributed by BJ, 27-Aug-2023.) $)
bj-nnfbi $p |- ( ( ( ph <-> ps ) /\ A. x ( ph <-> ps ) ) ->
( F// x ph <-> F// x ps ) ) $=
( wb wal wa wex wi wnnf bj-hbyfrbi bj-hbxfrbi anbi12d df-bj-nnf 3bitr4g ) A
BDZOCEFZACGAHZAACEHZFBCGBHZBBCEHZFACIBCIPQSRTABCJABCKLACMBCMN $.
+ ${
+ bj-nnfbii.1 $e |- ( ph <-> ps ) $.
+ $( If two formulas are equivalent for all ` x ` , then nonfreeness of ` x `
+ in one of them is equivalent to nonfreeness in the other. Compare
+ ~ nfbii . From this and ~ bj-nnfim and ~ bj-nnfnt , one can prove
+ analogous nonfreeness conservation results for other propositional
+ operators. (Contributed by BJ, 18-Nov-2023.) $)
+ bj-nnfbii $p |- ( F// x ph <-> F// x ps ) $=
+ ( wb wnnf bj-nnfbi bj-mpgs ) ABEACFBCFECDABCGH $.
+ $}
+
${ $d x ph $.
$( A non-occurring variable is nonfree in a formula. (Contributed by BJ,
28-Jul-2023.) $)
@@ -531612,6 +531674,12 @@ quantifiers appear (since they typically require ~ ax-10 to work with),
$.
$}
+ $( Version of ~ sbft using ` F// ` , proved from core axioms. (Contributed
+ by BJ, 19-Nov-2023.) $)
+ bj-sbft $p |- ( F// x ph -> ( [ t / x ] ph <-> ph ) ) $=
+ ( wnnf wsb wex spsbe bj-nnfe syl5 wal bj-nnfa stdpc4 syl6 impbid ) ABDZABCE
+ ZAPABFOAABCGABHIOAABJPABKABCLMN $.
+
$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
@@ -532266,6 +532334,15 @@ sethood hypotheses (compare ~ opth ). (Contributed by BJ, 6-Oct-2018.) $)
ZBCKZLCMDNABOPZCQZACQZLDERBUCUDFGSUFUGUECHTACITSABCDEUAUB $.
$}
+ $( Join of ~ elsng and ~ elsn2g . (Contributed by BJ, 18-Nov-2023.) $)
+ bj-elsn12g $p |- ( ( A e. V \/ B e. W ) -> ( A e. { B } <-> A = B ) ) $=
+ ( wcel csn wceq wb elsng elsn2g jaoi ) ACEABFEABGHBDEABCIABDJK $.
+
+ $( Biconditional version of ~ elsng . (Contributed by BJ, 18-Nov-2023.) $)
+ bj-elsnb $p |- ( A e. { B } <-> ( A e. _V /\ A = B ) ) $=
+ ( cvv wcel csn wceq wb wi wa elsng elelb mpbi ) ACDZABEZDZABFZGHOMPIGABCJPA
+ NKL $.
+
${
$d y f A $.
$( Remove hypothesis from ~ pwcfsdom . Illustration of how to remove a
@@ -535458,7 +535535,7 @@ Complex numbers (supplements)
( vx vy vz con0 wcel cfv wss wi wceq c0 cvv fveq2 crdg cv wa wral nfsbc1v
wsbc 0ex rzal sbceq1a mpbid vtoclef csuc wo vex elsuc csb cun ssun1 csbex
fvex unex nfcv cmpt nfmpt1 nfcxfr nfrdg nffv nfcsb1 nfun ax-mp id csbeq1a
- rdgeq1 uneq12d rdgsucmptf mpan2 syl5sseqr sstr2 syl5com imim2d imp sseq1d
+ rdgeq1 uneq12d rdgsucmptf mpan2 sseqtrrid sstr2 syl5com imim2d imp sseq1d
syl5ibrcom adantr jaod syl5bi ralimdv2 cab df-sbc sucex sseq2d raleqbi1dv
cbvabv elab2 bitri syl6ibr wlim ciun ssiun2 adantl rdglim2a mpan sseqtr4d
ex ralrimiva eleq2i abid sylibr a1d tfindes rsp syl eleq1 sseq12d imbi12d
@@ -536214,7 +536291,7 @@ Complex numbers (supplements)
syldan sylan2 adantlrr isfinite adantrr ad2antrr mpd eqeq2d sylibr df-rex
ex unieq a1i eximdv exlimdv sylan2b adantlr sseld wfn f1fn fniunfv cldopn
sseqtrd ancomd eqcomd eqimss ssun4 uniun ssun3 uncom undif1 eqtri ssequn2
- biimpi syl5eq eqsstr3d sylanr1 sylanr2 f1f frn topopn snssd uniss ancom1s
+ biimpi syl5eq eqsstrrd sylanr1 sylanr2 f1f frn topopn snssd uniss ancom1s
eqssd mpand impr wf1o f1f1orn f1oen3g sylancr enen1 endom snfi mpbi unctb
difopn sdomdom sylancl syl6bi impcom adantll elpw2g biimprd imbi2i ralbii
ad2ant2lr simprbi cbvrexv breq1 anbi12d pweq ineq1d rexeqdv rspccv mp2and
@@ -536449,13 +536526,13 @@ the following slightly extended version (related to ~ pm2.65 ):
$}
${
- wl-luk-syl5.1 $e |- ( ph -> ps ) $.
- wl-luk-syl5.2 $e |- ( ch -> ( ps -> th ) ) $.
+ wl-luk-imtrid.1 $e |- ( ph -> ps ) $.
+ wl-luk-imtrid.2 $e |- ( ch -> ( ps -> th ) ) $.
$( A syllogism rule of inference. The first premise is used to replace the
second antecedent of the second premise. Copy of ~ syl5 with a
different proof. (Contributed by Wolf Lammen, 17-Dec-2018.)
(New usage is discouraged.) (Proof modification is discouraged.) $)
- wl-luk-syl5 $p |- ( ch -> ( ph -> th ) ) $=
+ wl-luk-imtrid $p |- ( ch -> ( ph -> th ) ) $=
( wi wl-luk-imim1i wl-luk-syl ) CBDGADGFABDEHI $.
$}
@@ -536474,7 +536551,7 @@ the following slightly extended version (related to ~ pm2.65 ):
by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-con4i $p |- ( ps -> ph ) $=
- ( wn ax-luk3 wl-luk-syl5 wl-luk-pm2.18d ) BAADBDBACBAEFG $.
+ ( wn ax-luk3 wl-luk-imtrid wl-luk-pm2.18d ) BAADBDBACBAEFG $.
$}
${
@@ -536502,7 +536579,7 @@ the following slightly extended version (related to ~ pm2.65 ):
(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-mpi $p |- ( ph -> ch ) $=
- ( wn wl-luk-a1i wl-luk-syl5 wl-luk-pm2.18d ) ACCFZBACBJDGEHI $.
+ ( wn wl-luk-a1i wl-luk-imtrid wl-luk-pm2.18d ) ACCFZBACBJDGEHI $.
$}
${
@@ -536515,13 +536592,13 @@ the following slightly extended version (related to ~ pm2.65 ):
$}
${
- wl-luk-syl6.1 $e |- ( ph -> ( ps -> ch ) ) $.
- wl-luk-syl6.2 $e |- ( ch -> th ) $.
+ wl-luk-imtrdi.1 $e |- ( ph -> ( ps -> ch ) ) $.
+ wl-luk-imtrdi.2 $e |- ( ch -> th ) $.
$( A syllogism rule of inference. The second premise is used to replace
the consequent of the first premise. Copy of ~ syl6 with a different
proof. (Contributed by Wolf Lammen, 17-Dec-2018.)
(New usage is discouraged.) (Proof modification is discouraged.) $)
- wl-luk-syl6 $p |- ( ph -> ( ps -> th ) ) $=
+ wl-luk-imtrdi $p |- ( ph -> ( ps -> th ) ) $=
( wi wl-luk-imim2i wl-luk-syl ) ABCGBDGECDBFHI $.
$}
@@ -536529,8 +536606,8 @@ the following slightly extended version (related to ~ pm2.65 ):
17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-ax3 $p |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) $=
- ( wn wi ax-luk3 ax-luk1 wl-luk-syl5 ax-luk2 wl-luk-syl6 ) ACZBCZDZBJADZABKA
- DLMBAEJKAFGAHI $.
+ ( wn wi ax-luk3 ax-luk1 wl-luk-imtrid ax-luk2 wl-luk-imtrdi ) ACZBCZDZBJADZ
+ ABKADLMBAEJKAFGAHI $.
$( ~ ax-1 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen,
17-Dec-2018.) (New usage is discouraged.)
@@ -536544,8 +536621,8 @@ the following slightly extended version (related to ~ pm2.65 ):
17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-pm2.27 $p |- ( ph -> ( ( ph -> ps ) -> ps ) ) $=
- ( wi wn wl-luk-ax1 ax-luk1 wl-luk-syl ax-luk2 wl-luk-syl6 ) AABCZBDZBCZBAKA
- CJLCAKEKABFGBHI $.
+ ( wi wn wl-luk-ax1 ax-luk1 wl-luk-syl ax-luk2 wl-luk-imtrdi ) AABCZBDZBCZBA
+ KACJLCAKEKABFGBHI $.
${
wl-luk-com12.1 $e |- ( ph -> ( ps -> ch ) ) $.
@@ -536554,7 +536631,7 @@ the following slightly extended version (related to ~ pm2.65 ):
17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-com12 $p |- ( ps -> ( ph -> ch ) ) $=
- ( wi wl-luk-pm2.27 wl-luk-syl5 ) ABCEBCDBCFG $.
+ ( wi wl-luk-pm2.27 wl-luk-imtrid ) ABCEBCDBCFG $.
$}
$( From a wff and its negation, anything follows. Theorem *2.21 of
@@ -536571,7 +536648,7 @@ the following slightly extended version (related to ~ pm2.65 ):
(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-con1i $p |- ( -. ps -> ph ) $=
- ( wn wl-luk-pm2.21 wl-luk-syl5 wl-luk-pm2.18d ) BDZAADBHACBAEFG $.
+ ( wn wl-luk-pm2.21 wl-luk-imtrid wl-luk-pm2.18d ) BDZAADBHACBAEFG $.
$}
${
@@ -536581,8 +536658,8 @@ the following slightly extended version (related to ~ pm2.65 ):
different proof. (Contributed by Wolf Lammen, 17-Dec-2018.)
(New usage is discouraged.) (Proof modification is discouraged.) $)
wl-luk-ja $p |- ( ( ph -> ps ) -> ch ) $=
- ( wi wn wl-luk-con1i wl-luk-imim2i wl-luk-syl5 wl-luk-pm2.18d ) ABFZCCGAL
- CACDHBCAEIJK $.
+ ( wi wn wl-luk-con1i wl-luk-imim2i wl-luk-imtrid wl-luk-pm2.18d ) ABFZCCG
+ ALCACDHBCAEIJK $.
$}
$( A closed form of syllogism (see ~ syl ). Theorem *2.05 of
@@ -536631,7 +536708,7 @@ In classical logic (our logic) this is always true. In intuitionistic
by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-pm2.04 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) $=
- ( wi wl-luk-ax1 wl-luk-ax2 wl-luk-syl5 ) BABDABCDDACDBAEABCFG $.
+ ( wi wl-luk-ax1 wl-luk-ax2 wl-luk-imtrid ) BABDABCDDACDBAEABCFG $.
$(
@@ -536715,7 +536792,7 @@ indicating the (minimal) length of the longest chain involved.
wl-impchain-mp-2.a $e |- ( th -> ( ch -> ps ) ) $.
$( Use to replace the consequent. $)
wl-impchain-mp-2.b $e |- ( ps -> ph ) $.
- $( This theorem is in fact a copy of ~ wl-luk-syl6 , and repeated here to
+ $( This theorem is in fact a copy of ~ wl-luk-imtrdi , and repeated here to
demonstrate a recursive proof scheme. The number '2' in the theorem
name indicates that a chain of length 2 is modified. (Contributed by
Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.)
@@ -540760,7 +540837,7 @@ curry M LIndF ( R freeLMod I ) ) ) $=
unex coex fz1ssfz0 syl6ss fveq2 breq2d oveq12d simprbi elfznn0 nn0red
cc f1ofn df-pr indi 3eqtr3g elfzuz3 reldisj crn imassrn fnima elfzelz
ltled cn0 nn0p1nn elfz syl3anc 1red ltle imp leadd1dd peano2zd imass2
- eqsstr3d eqssd wfo f1ofo forn fnresi eqtri mp2an nnuz syl6eleq ssdifd
+ eqsstrrd eqssd wfo f1ofo forn fnresi eqtri mp2an nnuz syl6eleq ssdifd
fzss1 imaundi eqtr3d iftrue vex weq oveq2 oveq1 3eqtr4d lenlt biimpar
oveq1d lep1d letrd eluzp1p1 ltadd1dd elfzle1 syldan iffalse pm2.61dan
mpteq2dva eqtr4d op1std op2ndd id imaeq1 oveqan12d sylan9eqr sylanbrc
@@ -544069,7 +544146,7 @@ curry M LIndF ( R freeLMod I ) ) ) $=
sstr mtbird syl5bi imp cxp cmap cabs mp3an3 cico icossxr peano2re c2 cdiv
1rp mpani resubcl posdif elrpd rphalfcld rehalfcld jctl ltsubrpd retopbas
biimpd ctop bastg mblfinlem2 indif2 df-ss syl5eq opncld incld simpr simpl
- difeq1d ssdif2d dfin4 inss2 eqsstr3i anbi1d eqcoms biantrud bitr4d rspcev
+ difeq1d ssdif2d dfin4 inss2 eqsstrri anbi1d eqcoms biantrud bitr4d rspcev
sseq1 fveq2 an12s adantrrr adantlr simp-6r difdif2 fveq2i unssi ad3antrrr
cun ssdifss ssinss1 ovolun syl22anc sylancl eleq1w vtoclga subaddd biimpi
ssdif oveq2d ad2antll eqtr3d breqtrd ltsub23d lelttrd ad4antr ssun2 incom
@@ -545426,7 +545503,7 @@ curry M LIndF ( R freeLMod I ) ) ) $=
0le0 iftrue 3brtr4d ex iffalse ifbi ifan eqtri breq2d reex eqidd ad2antrr
cvv cima csup crn imassrn cico frnd icossxr syl6ss syl5ss supxrcl wne cxp
c0 ltrelxr ssbri brxp ioon0 r19.2z wfn ffnd ioossre qbtwnxr anim1i anasss
- crab qre rpxr syl2an cicc ioorp ioossicc eqsstr3i sseli cmul covol ioombl
+ crab qre rpxr syl2an cicc ioorp ioossicc eqsstrri sseli cmul covol ioombl
cin cdm elioore xrltnle biimpar xrre2 syl32anc mnfxr mnfle xrlelttrd xrre
mblvol ltsubposd eliooord ltaddpos2d lttrd ovolioo syl5eq eqeltrd mulgt0d
iooin ifbid mpteq2dv rpge0 sylanbrc itg2const eqtrd fvoveq1 imbrov2fvoveq
@@ -546697,7 +546774,7 @@ curry M LIndF ( R freeLMod I ) ) ) $=
cico reex off abstrid ralrimiva ovexd eqidd fvexd absmul oveq1i mulid2d
absi eqtr2d offval2 ofrfval2 itg2le syl3anc ccom cofmpt resubcl crn csn
absf iunin2 imaiun iunid imaeq2i eqtr3i ineq2i cnvimass cnvimarndm fdmd
- dmmpti sseqtri syl5sseqr df-ss sylib frnd sselda ad2antrr 2thd ltaddsub
+ dmmpti sseqtri sseqtrrid df-ss sylib frnd sselda ad2antrr 2thd ltaddsub
sylan syl3an3 3comr 3expa readdcl adantll elioopnf rexr 3bitr4rd bibi1d
oveq2 syl5ibrcom pm5.32rd adantllr syldan rabbidv imaeq1d vex mptiniseg
cnveqd 3eqtr4d eqtr3d cfn i1frn mbfss mbfima i1fima ralrimivw finiunmbl
@@ -549475,7 +549552,7 @@ curry M LIndF ( R freeLMod I ) ) ) $=
cv cbl ciun cpw cfn wrex cxp cres metres2 syl5eqel istotbnd3 baib simpllr
sspwb sylib ssrind simprl sseldd simprr wel cxmet metxmet ad4antr simplbi
wb elfpw adantl sselda simp-4r sseqin2 eleqtrrd rpxrd blres syl3anc inss1
- syl6eqss ralrimiva ss2iun adantrr eqsstr3d ex reximdv2 ralimdva sylbid c2
+ syl6eqss ralrimiva ss2iun adantrr eqsstrrd ex reximdv2 ralimdva sylbid c2
cxr jca cdiv wi simpr rphalfcld oveq2 iuneq2d sseq2d rexbidv rspcv wne wf
crab simprbi ad2antrl ssrab2 ssfi sylancl oveq1 ineq1d incom syl6eq dfin5
wex weq neeq1d rabn0 syl6bb rgen cima mpbird wfn elpreima imaeq2d eleq12d
@@ -549844,7 +549921,7 @@ counterexample is the discrete extended metric (assigning distinct
necon3i wor w3a ltso fisupcl mpan sseldd cxmet cmin metxmet ad2antrr 1red
fimaxre2 syl2anc cvv elrnmpt1 mpan2 adantl suprub syl31anc leaddsub mpbid
wb blss2 syl33anc ralrimiva nfcv nfmpt1 nfrn nfsup nfov nfss oveq1 sseq1d
- nfv cbvral sylibr iunss eqsstr3d cxr rpxrd blssm eqssd rspceeqv ralrimdva
+ nfv cbvral sylibr iunss eqsstrrd cxr rpxrd blssm eqssd rspceeqv ralrimdva
rexlimdvaa isbnd baib sylibrd sylc ) ABUBGHZABUCGHZCDIZCIZJAUDGZKZUEZBLZD
BUFUGUHZUIZABUJGHZABUKJMHUULCUUNUUOEIZUUPKZUEZBLZDUUTUIZEMNZUVAULUULUUMUV
HCDABEUMUOUVGUVAEJMUVCJLZUVFUUSDUUTUVIUVEUURBUVICUUNUVDUUQUVCJUUOUUPUPUQU
@@ -550516,7 +550593,7 @@ counterexample is the discrete extended metric (assigning distinct
selpw anbi1i csn cun ffn 0z ssv int0 sseqtr4i ssun1 anim2i imim1i ssun2
ralsn sscls sseq2 syl5ibrcom anim12i intunsn syl6sseqr rexlimdvv reeanv
ssin ss2in cbvrexv 3imtr3g expd findcard2 com12 impr ffnd eqeltri mpan2
- sylcom 1z uzn0 wfun fnfun fndm syl5sseqr funfvima2 necomd neneqd nrexdv
+ sylcom 1z uzn0 wfun fnfun fndm sseqtrrid funfvima2 necomd neneqd nrexdv
ne0i 0ex zex pwex frn ssexi abrexex elfi sylnibr cmptop cmpfi ibi fveq2
notbid neeq1d syl6bb rspccv syl3c wrel lmrel r19.23v albii eleq2 ralbii
ceqsalv ralcom4 bitr3i elintab rspceeqv elab mpbir intss1 clsss3 sselda
@@ -550619,7 +550696,7 @@ counterexample is the discrete extended metric (assigning distinct
cdm expr c1 nnuz 1zzd simpl iscmet3 mpbird cbl co cab cpw cin wss metxmet
cxmet cxr id rpxr blopn syl3an 3com23 eleq1a syl rexlimdva adantlr abssdv
3expa ad2antrr mopnuni blcntr syl3an1 elabrex adantl elunii syl2anc nfre1
- ovex nfcv nfab nfuni dfss3f sylibr eqsstr3d unissd eqssd eqid cmpcov elin
+ ovex nfcv nfab nfuni dfss3f sylibr eqsstrrd unissd eqssd eqid cmpcov elin
syl3anc ancom bitri anbi1i anass rexbii2 sylib eqcom eqeq1d anbi1d syl5bb
syl5rbb elpwi ssabral anim2i syl6bi reximdv mpd istotbnd sylanbrc jca ) A
CUBIJZBUCJZKZACUDIJZACUEIJZYFYGUFCELZUGZYIBUAIUNJZMZEAUHIZNYFYLEYMYFYIYMJ
@@ -550963,7 +551040,7 @@ the next (since the empty set has a finite subcover, the
cc0 wb elrp 1re lediv1 mp3an12 sylbi mpbii fveq2 opeq12d opex fvmpt
3re op2nd breqtrrd ssbl syl221anc oveq1 ovmpo oveq1d eqtr3d 1st2nd2
op1st df-ov syl6reqr ctop mopntop blssm mopnuni eqid simprr blsscls
- ccl sscls syl23anc eqsstr3d rpre ccom clm heiborlem6 caublcls mpdan
+ ccl sscls syl23anc eqsstrrd rpre ccom clm heiborlem6 caublcls mpdan
3expia imp blhalf syl22anc sstr2 cpw cin csn unisng snssd snex elpw
biimpar sylibr snfi a1i elind unieq rspcev sylan syldan sseq1 elab2
notbid con2bii sylib ex syld mt2d rexlimddv nrexdv pm2.21dd ) AUBCV
@@ -551094,7 +551171,7 @@ the next (since the empty set has a finite subcover, the
iuneq2 syl5eq syl simp2r 3eqtr2rd iuneq1 rspceeqv 3expia adantrrr exlimdv
fofi mpd rexlimdvaa syld ralrimdva pwex inex1 com eqid simpl weq iuneq2dv
oveq2d nn0ennn nnenom entri axcc4 syl6 elpwi cab copab pweqd ineq1d feq3d
- cmpo wn biimpar adantrr cbviunv id syl5sseqr fss syl2anr ffvelrnda elpwid
+ cmpo wn biimpar adantrr cbviunv id sseqtrrid fss syl2anr ffvelrnda elpwid
inss1 sselda ovex ovmpo biimprd ralimdva impr fveq2 iuneq1d eqtrd cbvralv
simplr heiborlem10 exp32 syl5 ralrimiv ex imp iscmp sylanbrc jca impbii )
ACUBUCKZBUFKZLACUGUCKZACUHUCKZLZABCDUIUWIUWEUWFUWGUWEUWHACUJZUKUWIBULKZBU
@@ -553339,7 +553416,7 @@ the next (since the empty set has a finite subcover, the
/\ B e. ( X \ { Z } ) ) -> ( A H B ) e. ( X \ { Z } ) ) $=
( wcel cxp wa co wceq adantl cdm wss eleq2d cdrng crngo csn cdif cres cgr
isdrngo1 ovres crn wi eqid grpocl 3expib grporndm difss xpss12 mp2an fdmd
- rngosm syl5sseqr ssdmres sylib adantr dmeqd dmxpid syl6eq anbi12d 3imtr3d
+ rngosm sseqtrrid ssdmres sylib adantr dmeqd dmxpid syl6eq anbi12d 3imtr3d
eqtrd imp eqeltrrd 3impb syl3an1b ) CUALCUBLZEFGUCZUDZVPMZUEZUFLZNZAVPLZB
VPLZABEOZVPLZCDEFGHIJKUGVTWAWBWDVTWAWBNZNABVROZWCVPWEWFWCPVTABVPVPEUHQVTW
EWFVPLZVTAVRUIZLZBWHLZNZWFWHLZWEWGVSWKWLUJVNVSWIWJWLABVRWHWHUKULUMQVTWIWA
@@ -553356,7 +553433,7 @@ the next (since the empty set has a finite subcover, the
A. x e. ( X \ { Z } ) E. y e. ( X \ { Z } ) ( y H x ) = U ) ) ) $=
( vz wcel wa co wceq wrex adantr vu vv vw cdrng csn cdif cxp cres cgr wne
crngo cv wral isdrngo1 dvrunz sylbir crn cdm grporndm adantl difss xpss12
- wss mp2an rngosm syl5sseqr ssdmres sylib dmeqd dmxpid syl6eq eqtrd eleq2d
+ wss mp2an rngosm sseqtrrid ssdmres sylib dmeqd dmxpid syl6eq eqtrd eleq2d
fdmd biimpar cgn cfv grpoinvcl adantll cgi grpolinv cmagm cexid cin cmndo
eqid rngomndo mndomgmid syl sseqtri rngorn1eq syl5sseq c1st rneqi rngo1cl
eqtri eldifsn sylanbrc grpomndo mndoismgmOLD syl31anc oveq1 eqeq1d rspcev
@@ -554856,7 +554933,7 @@ the next (since the empty set has a finite subcover, the
Madsen, 10-Jun-2010.) $)
igenss $p |- ( ( R e. RingOps /\ S C_ X ) -> S C_ ( R IdlGen S ) ) $=
( vj crngo wcel wss wa cv cidl cfv crab cint cigen co ssintub igenval
- syl5sseqr ) AHIBDJKBGLJGAMNZOPBABQRGBUBSABGCDEFTUA $.
+ sseqtrrid ) AHIBDJKBGLJGAMNZOPBABQRGBUBSABGCDEFTUA $.
$( The ideal generated by a set is an ideal. (Contributed by Jeff Madsen,
10-Jun-2010.) $)
@@ -563630,7 +563707,7 @@ inference form (e.g.,
(Contributed by NM, 15-Jan-2015.) $)
lsmsat $p |- ( ph -> E. p e. A ( p C_ T /\ Q C_ ( p .(+) U ) ) ) $=
( vr vy vz vq cv csn clspn cfv wceq cbs cdif wrex wss co wa wcel clmod wb
- eqid islsat syl mpbid cplusg simp3 3ad2ant1 eqsstr3d lsmcl syl3anc eldifi
+ eqid islsat syl mpbid cplusg simp3 3ad2ant1 eqsstrrd lsmcl syl3anc eldifi
w3a 3ad2ant2 lspsnel5 mpbird lsssssubg sseldd lsmelval syl2anc wne lssne0
csubg wi adantr simpr2 lssel simpr3 lspsnel5a simpl3 simpr1 oveq1d simp2r
lsatlspsn2 lmod0vlid 3eqtrd sneqd fveq2d eqsstrd lsmub2 sstrd sseq1 oveq1
@@ -563763,7 +563840,7 @@ inference form (e.g.,
( vy wcel wa cv wss crab cuni cfv wceq syl2anc syl3anc clmod eleq1 simplr
c0g wne csn simplll cbs simpllr lssel lspsncl lspid lsatlss adantr rabss2
eqid uniss 3syl unimax lssss eqsstrd adantl ad2antrr lsatlspsn2 lspsnel5a
- sstrd simpr sseq1 elrab sylanbrc elssuni syl lspss eqsstr3d sseldd simpll
+ sstrd simpr sseq1 elrab sylanbrc elssuni syl lspss eqsstrrd sseldd simpll
lspsnid lspcl syldan lss0cl pm2.61ne ex ssrdv simpl fveq2d eqtrd sseqtrd
eqssd ) FUAKZDCKZLZDAMZDNZABOZPZEQZWKJDWPWKJMZDKZWQWPKZWKWRLZWSFUDQZWPKZW
QXAWQXAWPUBWTWQXAUEZLZWQUFEQZWPWQXDXEXEEQZWPXDWIXECKZXFXERWIWJWRXCUGZXDWI
@@ -566826,7 +566903,7 @@ Functionals and kernels of a left vector space (or module)
<-> ( G =/= .0. /\ H = .0. ) ) ) $=
( cfv wne wceq wa simpr wpss cbs wss df-pss eqid clvec clmod lveclmod syl
wcel lkrssv adantr psssstrd pssned sylan2br clsh simplr ad3antrrr simpllr
- wn ad2antrr eqsstr3d eqssd wb lkrshp4 necon1bbid mpbird pm2.21dd lkrshpor
+ wn ad2antrr eqsstrrd eqssd wb lkrshp4 necon1bbid mpbird pm2.21dd lkrshpor
wo mpjaodan lshpcmp mpbid necon3ad impr jca simprr eqcomd sseqtrd neeqtrd
ex simprl impbida syl5bb lkr0f2 necon3bid anbi12d bitrd ) ADFPZEFPZUAZWIG
UBPZQZWJWLRZSZDHQZEHRZSWKWIWJUCZWIWJQZSZAWOWIWJUDZAWTWOAWTSZWMWNWTAWKWMXA
@@ -576865,7 +576942,7 @@ Cambridge University Press (1988). Unlike them, we do not assume that
( vk wcel cv wral cfv catm cvv wss co wbr wi wa cab wceq elex cpsubsp cjn
fveq2 syl6eqr sseq2d oveqd breq2d breqd imbi1d raleqbidv 2ralbidv anbi12d
cple bitrd abbidv df-psubsp cpw fvexi pwex selpw anbi1i abbii ssab2 ssexi
- eqsstr3i fvmpt syl5eq syl ) EBPEUAPZCGQZAUBZHQZJQZIQZDUCZFUDZWAVSPZUEZHAR
+ eqsstrri fvmpt syl5eq syl ) EBPEUAPZCGQZAUBZHQZJQZIQZDUCZFUDZWAVSPZUEZHAR
ZIVSRJVSRZUFZGUGZUHEBUIVRCEUJSWKNOEVSOQZTSZUBZWAWBWCWLUKSZUCZWLVBSZUDZWFU
EZHWMRZIVSRJVSRZUFZGUGWKUAUJWLEUHZXBWJGXCWNVTXAWIXCWMAVSXCWMETSAWLETULMUM
ZUNXCWTWHJIVSVSXCWSWGHWMAXDXCWRWEWFXCWRWAWDWQUDWEXCWPWDWAWQXCWODWBWCXCWOE
@@ -577964,7 +578041,7 @@ their common point (atom). (Contributed by NM, 2-May-2013.)
paddunssN $p |- ( ( K e. B /\ X C_ A /\ Y C_ A )
-> ( X u. Y ) C_ ( X .+ Y ) ) $=
( vp vq vr wcel wss w3a cun cv cfv co wrex eqid cjn cple wbr crab paddval
- ssun1 syl5sseqr ) DBLEAMFAMNEFOZIPJPKPDUAQZRDUBQZUCKFSJESIAUDZOUHEFCRUHUK
+ ssun1 sseqtrrid ) DBLEAMFAMNEFOZIPJPKPDUAQZRDUBQZUCKFSJESIAUDZOUHEFCRUHUK
UFABCUIDUJEFKJIUJTUITGHUEUG $.
$( Member of projective subspace sum with at least one empty set.
@@ -578028,14 +578105,14 @@ their common point (atom). (Contributed by NM, 2-May-2013.)
analog.) (Contributed by NM, 3-Jan-2012.) $)
sspadd1 $p |- ( ( K e. B /\ X C_ A /\ Y C_ A ) -> X C_ ( X .+ Y ) ) $=
( vp vq vr wcel wss cun cv cfv co wrex ssun1 eqid w3a cjn cple crab sstri
- wbr paddval syl5sseqr ) DBLEAMFAMUAEFNZIOJOKODUBPZQDUCPZUFKFRJERIAUDZNZEE
+ wbr paddval sseqtrrid ) DBLEAMFAMUAEFNZIOJOKODUBPZQDUCPZUFKFRJERIAUDZNZEE
FCQEUIUMEFSUIULSUEABCUJDUKEFKJIUKTUJTGHUGUH $.
$( A projective subspace sum is a superset of its second summand.
( ~ ssun2 analog.) (Contributed by NM, 3-Jan-2012.) $)
sspadd2 $p |- ( ( K e. B /\ X C_ A /\ Y C_ A ) -> X C_ ( Y .+ X ) ) $=
( vp vq vr wcel wss w3a cun cv cfv co wrex eqid cjn cple crab ssun2 ssun1
- wbr sstri wceq paddval 3com23 syl5sseqr ) DBLZEAMZFAMZNFEOZIPJPKPDUAQZRDU
+ wbr sstri wceq paddval 3com23 sseqtrrid ) DBLZEAMZFAMZNFEOZIPJPKPDUAQZRDU
BQZUFKESJFSIAUCZOZEFECRZEUOUSEFUDUOURUEUGULUNUMUTUSUHABCUPDUQFEKJIUQTUPTG
HUIUJUK $.
@@ -579662,7 +579739,7 @@ one element is a lattice line (expressed as the join ` P .\/ Q ` ).
$( A set of atoms is included in its projective subspace closure.
(Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) $)
pclssidN $p |- ( ( K e. V /\ X C_ A ) -> X C_ ( U ` X ) ) $=
- ( vy wcel wss wa cv cpsubsp cfv crab cint ssintub eqid pclvalN syl5sseqr
+ ( vy wcel wss wa cv cpsubsp cfv crab cint ssintub eqid pclvalN sseqtrrid
) CDIEAJKEHLJHCMNZOPEEBNHEUAQHAUABCDEFUARGST $.
$}
@@ -580271,7 +580348,7 @@ one element is a lattice line (expressed as the join ` P .\/ Q ` ).
( ._|_ ` ( ._|_ ` s ) ) = s ) } ) $=
( vk wcel cvv cv wss cfv wceq wa cab catm cpolN elex cpscN syl6eqr sseq2d
fveq2 fveq1d fveq12d eqeq1d abbidv df-psubclN cpw fvexi pwex selpw anbi1i
- anbi12d abbii ssab2 eqsstr3i ssexi fvmpt syl5eq syl ) DBKDLKZCFMZANZVEEOZ
+ anbi12d abbii ssab2 eqsstrri ssexi fvmpt syl5eq syl ) DBKDLKZCFMZANZVEEOZ
EOZVEPZQZFRZPDBUAVDCDUBOVKIJDVEJMZSOZNZVEVLTOZOZVOOZVEPZQZFRVKLUBVLDPZVSV
JFVTVNVFVRVIVTVMAVEVTVMDSOAVLDSUEGUCUDVTVQVHVEVTVPVGVOEVTVODTOEVLDTUEHUCZ
VTVEVOEWAUFUGUHUPUIJFUJVKAUKZAADSGULUMVKVEWBKZVIQZFRWBWDVJFWCVFVIFAUNUOUQ
@@ -603098,7 +603175,7 @@ are all translations (for a fiducial co-atom ` W ` ). (Contributed by
diassdvaN $p |- ( ( ( K e. Y /\ W e. H ) /\ ( X e. B /\ X .<_ W ) )
-> ( I ` X ) C_ V ) $=
( vf wcel wa cfv wbr cv ctrl crab eqid diaval ssrab2 wceq dvavbase adantr
- cltrn syl5sseqr eqsstrd ) EJRHCRSZIARIHFUASZSZIDTQUBHEUCTTZTIFUAZQHEUKTTZ
+ cltrn sseqtrrid eqsstrd ) EJRHCRSZIARIHFUASZSZIDTQUBHEUCTTZTIFUAZQHEUKTTZ
UDZGAUQUSQCDEFJHIKLMUSUEZUQUENUFUPUSUTGURQUSUGUNGUSUHUOUSBCEGHJMVAOPUIUJU
LUM $.
$}
@@ -603406,7 +603483,7 @@ are all translations (for a fiducial co-atom ` W ` ). (Contributed by
( cfv co cplusg ccom chlt wcel wceq eqid dvavadd syl12anc wne dia2dimlem4
wa simpld eqtr2d csubg clmod wss clvec dvalvec lveclmod lsssssubg syl cbs
3syl wbr atbase simprd dialss sseldd csn dia1dim2 dia2dimlem3 fveq2d eqss
- eqsstr3d dvavbase eleqtrrd lspsnel5 mpbird dia2dimlem2 lsmelvali syl22anc
+ eqsstrrd dvavbase eleqtrrd lspsnel5 mpbird dia2dimlem2 lsmelvali syl22anc
syl2anc sylib eqeltrd cabl lmodabl lsmcom syl3anc eleqtrd ) AKTNVIZJNVIZE
VJZYAXTEVJZAKCLUBVKVIZVJZYBAYECLVLZKAPVMVNUAMVNWAZCIVNZLIVNZYEYFVOUPVGVEY
DIUBCLMPVMUAUGUHUJYDVPZVQVRABCDFIKLMPQUAUCUFUGUHUPUSAKIVNDKVIDVSUTWBVEVFV
@@ -606069,7 +606146,7 @@ all translations (for a fiducial co-atom ` W ` ). (Contributed by NM,
/\ S .<_ ( Q .\/ X ) ) -> ( N ` { <. F , O >. } ) C_ ( I ` X ) ) $=
( chlt wcel wa wbr wn w3a cop csn cfv wceq simp1 simp21 simp22 ltrniotacl
co syl3anc dib1dim2 syl2anc wss cdlemn2 wb simp23 dibord syl121anc mpbird
- trlcl trlle eqsstr3d ) NULUMRKUMUNZCAUMCROUOUPUNZEAUMEROUOUPUNZSBUMSROUOU
+ trlcl trlle eqsstrrd ) NULUMRKUMUNZCAUMCROUOUPUNZEAUMEROUOUPUNZSBUMSROUOU
NZUQZECSMVFOUOZUQZJQURUSPUTZJDUTZLUTZSLUTZWFVTJFUMZWIWGVAVTWDWEVBZWFVTWAW
BWKWLVTWAWBWCWEVCVTWAWBWCWEVDACEFIJKNORUAUCUDUEUKVEVGZBDFGHJKLNPQRTUDUEUF
UGUIUHUJVHVIWFWIWJVJZWHSOUOZABCDEFIJKMNORSTUAUBUCUDUEUFUKVKWFVTWHBUMZWHRO
@@ -609581,7 +609658,7 @@ of phi(x) is independent of the atom q." (Contributed by NM,
wn wi syl clatglbcl syl2anc dihlss dihglblem5 adantr simpr lpssat ex ccnv
w3a crn simp1l dih1dimat adantlr 3adant3 dihcnvid2 wbr simp3l ssiin sylib
wral wb wf1o wf1 dihf11 f1f1orn 3syl f1ocnvdm sselda dihord sseq1d bitr3d
- syl3anc ralbidva mpbird simp1ll simp1rl clatleglb eqsstr3d pm2.21fal syld
+ syl3anc ralbidva mpbird simp1ll simp1rl clatleglb eqsstrrd pm2.21fal syld
simp3r rexlimdv3a mtoi dfpss3 notbii iman anclb 3bitr2i mpd eqss sylibr )
KUHUIZNIUIZUJZFCUKZFULUMZUJZUJZFHUNZJUNZAFAVBZJUNZUOZUKZYTYQUKZUJZYQYTUPY
OUUAUUCAUEUFCFUFVBUEVBNKUQUNZVCUPUEFURUFCUSZHIJNKUTUNUNZKLUUDNOPUUDVASTUU
@@ -609891,7 +609968,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $=
-> ( N ` X ) = ( I ` ( ._|_ ` ( `' I ` X ) ) ) ) $=
( vy vz wcel wa cfv wss wceq eqid wbr chlt crn cv cbs crab cglb ccnv cdvh
dihrnss dochval syldan cple clat ad2antrr hlclat ssrab2 clatglbcl sylancl
- hllat ccla dihcnvcl ssid dihcnvid2 syl5sseqr fveq2 sseq2d elrab clatglble
+ hllat ccla dihcnvcl ssid dihcnvid2 sseqtrrid fveq2 sseq2d elrab clatglble
a1i sylanbrc syl3anc wral adantr sseq1d simpll simpr dihord bitr3d biimpd
wb expimpd syl5bi ralrimiv clatleglb mpbird latasymd fveq2d eqtrd ) CUANZ
FANZOZGBUBNZOZGDPZGLUCZBPZQZLCUDPZUEZCUFPZPZEPZBPZGBUGPZEPZBPWKWLGFCUHPPZ
@@ -610262,7 +610339,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $=
(Contributed by NM, 7-Aug-2014.) $)
dochocsp $p |- ( ph -> ( ._|_ ` ( N ` X ) ) = ( ._|_ ` X ) ) $=
( cfv wcel wss syl2anc chlt wa dvhlmod lspssv lspssid dochss syl3anc cdih
- clmod crn wceq eqid dochcl dochoc dochssv dochspss eqsstr3d eqssd ) AIEQZ
+ clmod crn wceq eqid dochcl dochoc dochssv dochspss eqsstrrd eqssd ) AIEQZ
FQZIFQZADUARHCRUBZUSGSZIUSSZUTVASOABUIRZIGSZVCABCDHJKOUCZPIEGBMNUDTAVEVFV
DVGPIEGBMNUETBCDFGHIUSJKMLUFUGAVAVAFQZFQZUTAVBVAHDUHQQZUJRZVIVAUKOAVBVFVK
OPBCVJDFGHIJVJULZKMLUMTCVJDFHVAJVLLUNTAVBVHGSZUSVHSVIUTSOAVBVAGSZVMOAVBVF
@@ -610479,7 +610556,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $=
( wcel wss cfv a1i dochss syl3anc 3adant3 3adant2 chlt wa w3a simp1 simp2
cun cin simp3 unssd ssun1 ssun2 ssind cdih crn wceq eqid dochcl dihmeetcl
syl12anc dochoc syl2anc dochssv ssinss1 dochocss unss12 inss1 inss2 sstrd
- syl eqsstr3d eqssd ) CUAMFBMUBZGENZHENZUCZGHUFZDOZGDOZHDOZUGZVOVQVRVSVOVL
+ syl eqsstrrd eqssd ) CUAMFBMUBZGENZHENZUCZGHUFZDOZGDOZHDOZUGZVOVQVRVSVOVL
VPENZGVPNZVQVRNVLVMVNUDZVOGHEVLVMVNUEVLVMVNUHUIZWBVOGHUJPABCDEFGVPIJKLQRV
OVLWAHVPNZVQVSNWCWDWEVOHGUKPABCDEFHVPIJKLQRULVOVTVTDOZDOZVQVOVLVTFCUMOOZU
NZMZWGVTUOWCVOVLVRWIMZVSWIMZWJWCVLVMWKVNABWHCDEFGIWHUPZJKLUQSVLVNWLVMABWH
@@ -611824,7 +611901,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $=
( wcel cfv vv chlt adantr simpr dochsatshp c0g wne wrex csn cbs wceq cdih
wa cv crn wss eqid lssss syl dochcl syl2anc dochoc dvhlmod lshpne eqnetrd
clmod wb dochssv dochn0nv mpbird dochlss lssne0 mpbid w3a clspn lspsnel5a
- 3ad2ant1 simp2 lssel dihlsprn dochord eqsstr3d clvec dvhlvec simp1r simp3
+ 3ad2ant1 simp2 lssel dihlsprn dochord eqsstrrd clvec dvhlvec simp1r simp3
lsatlspsn2 syl3anc lshpcmp fveq2d eqeltrd rexlimdv3a mpd dochsat impbida
eqtrd ) ACBSZCHTZJSZAWQUMBCEFGHIJKMLOPAGUBSIFSUMZWQQUCAWQUDUEAWSUMZWRHTZB
SZWQXAUAUNZEUFTZUGZUAXBUHZXCXAXBXEUIUGZXGXAXHXBHTZEUJTZUGZXAXIWRXJAXIWRUK
@@ -613039,7 +613116,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
by NM, 16-Feb-2015.) $)
lcfl9a $p |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) $=
( cfv wceq c0g wa dochoc1 adantr wss dvhlmod lkrssv sneq fveq2d chlt wcel
- csn eqid doch0 syl sylan9eqr eqsstr3d eqssd 3eqtr4d simpr wne cdih dochcl
+ csn eqid doch0 syl sylan9eqr eqsstrrd eqssd 3eqtr4d simpr wne cdih dochcl
crn snssd syl2anc dochoc clsh clvec dvhlvec cdif simprl eldifsn dochsnshp
sylanbrc simprr wb lkrshp4 mpbid lshpcmp 3eqtr3d pm2.61da2ne ) ADGUBZHUBZ
HUBZWFUCKBUDUBZWFIAKWIUCZUEZIHUBZHUBZIWHWFAWMIUCZWJABEFHIJLNMORUFZUGWKWGW
@@ -613494,7 +613571,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
kernel of ` G ` , so the kernels of ` G ` and the sum are comparable.
(Contributed by NM, 18-Jan-2015.) $)
lclkrlem2r $p |- ( ph -> ( L ` G ) C_ ( L ` ( E .+ G ) ) ) $=
- ( cfv cin co wss wceq lclkrlem2p 3sstr4d sseqin2 sylib dvhlmod eqsstr3d
+ ( cfv cin co wss wceq lclkrlem2p 3sstr4d sseqin2 sylib dvhlmod eqsstrrd
csn lkrin ) ALPVKZJPVKZWDVLZJLDVMPVKAWDWEVNWFWDVOAUCWBSVKUBWBSVKWDWEABC
DEFGHIJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBVCVDVEVHVI
VJVPVGVFVQWDWEVRVSACDKJLPIULUTUMUNAIMOUAVAVCVEVTUQURWCWA $.
@@ -613506,7 +613583,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
( cfv clsh wcel co wceq csn chlt cdih crn wss snssd eqid dochcl syl2anc
wa dochoc ad2antrr lclkrlem2r clvec dvhlvec simplr simpr lshpcmp eqtr3d
mpbid fveq2d 3eqtr4d dochoc1 dvhlmod ldualvaddcl lkrshpor adantr lkrssv
- wo mpjaodan eqsstr3d eqssd ) ALPVKZIVLVKZVMZJLDVNZPVKZSVKZSVKZXLVOZXHTV
+ wo mpjaodan eqsstrrd eqssd ) ALPVKZIVLVKZVMZJLDVNZPVKZSVKZSVKZXLVOZXHTV
OZAXJWEZXLXIVMZXOXLTVOZXQXRWEZUCVPZSVKZSVKZSVKZYBXNXLAYDYBVOZXJXRAOVQVM
UAMVMWEZYBUAOVRVKVKZVSVMZYEVEAYFYATVTYHVEAUCTUPWAIMYGOSTUAYAVAYGWBZVCUE
VBWCWDMYGOSUAYBVAYIVBWFWDWGXTXMYCSXTXLYBSXTXHXLYBXTXHXLVTZXHXLVOAYJXJXR
@@ -613561,7 +613638,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
wceq dihprrn wss lssss syl dochoccl mpbid dochexmid dvhlvec cin csn cun
lclkrlem2n wa snssd dochdmj1 syl3anc df-pr fveq2i unssd dochocsp syl5eq
chlt ineq12d 3eqtr4d lkrin eqsstrd csubg clmod lsssssubg sseldd dochlss
- wb syl2anc lkrlss lsmlub mpbi2and eqsstr3d eqssd ) AIKCVIZOVJZSAJXTOSHU
+ wb syl2anc lkrlss lsmlub mpbi2and eqsstrrd eqssd ) AIKCVIZOVJZSAJXTOSHU
DUKUSAHLNTUTVBVDVKZABCJIKHUKULUMYBUPUQVLZVMASUAUBVNZQVJZYERVJZDVIZYAADH
VOVJZHLNRSTYEUTVAVBUDYHVPZVCVDAYHQSHUAUBUDYIURYBUNUOVQZAYETNVRVJVJZVSVT
YFRVJYEWAAHLYKNQSTUAUBUTVBUDURYKVPZVDUNUOWBAHLYKNRSTYEUTYLVBUDVAVDAYEYH
@@ -614058,7 +614135,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
lcfrlem6 $p |- ( ph -> ( X .+ Y ) e. E ) $=
( co cv cfv ciun wcel wrex syl6eleq eliun sylib clmod clss dvhlmod adantr
chlt cbs wss clfn eqid lssel sylan wceq ldualvbase eleqtrd lkrssv dochlss
- syl2anc simpr csn eqsstr3d lcfrlem4 lspsnel5 3imtr4d imp lssvacl syl22anc
+ syl2anc simpr csn eqsstrrd lcfrlem4 lspsnel5 3imtr4d imp lssvacl syl22anc
wa ex reximdva mpd sylibr syl6eleqr ) AOPCUKZFHFULZKUMZMUMZUNZGAWLWOUOZFH
UPZWLWPUOAOWOUOZFHUPZWRAOWPUOWTAOGWPUHUGUQFOHWOURUSAWSWQFHAWMHUOZWFZWSWQX
BWSWFEUTUOZWOEVAUMZUOZWSPWOUOZWQXBXCWSAXCXAAEIJNQSUEVBZVCZVCXBXEWSXBJVDUO
@@ -614448,7 +614525,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
inss2 dvhlvec lcfrlem22 lsatel clss dvhlmod clmod wcel clfn cv crab c0g
lcfrlem10 lkrlss syl2anc lsatssv sseldd ellkr2 mpbird lspsnel5a eqsstrd
csubg wa wb lsssssubg syl eldifad prssi dochlss lspprcl lcfrlem17 snssd
- chlt lssincl syl3anc syl5eqel mpbi2and eqsstr3d clsh dochsnshp wne cdif
+ chlt lssincl syl3anc syl5eqel mpbi2and eqsstrrd clsh dochsnshp wne cdif
lsmlub lcfrlem13 eldifsni lduallkr3 lshpcmp mpbid ) AUDUEHVKZVLZUAVMZUE
QVMZSVMZVNYMYOVOAYMUDUEVPZUAVMZFMVQVMZVKZYOAEFHYRMORTUAUBUCUDUEUFUGUHUI
UJUKULUMUNUOUPUQURUSYRVRZVSAYQYOVNZFYOVNZYSYOVNZAYQUDQVMSVMZYOVTYOABCDE
@@ -614626,7 +614703,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
cmulr dvhlvec cv crab c0g lcfrlem10 clss wcel dvhlmod lcfrlem22 lsatlssel
lssel syl2anc lcfrlem2 eqsstrd lcfrlem28 lsatel lcfrlem30 lkrlss lcfrlem3
clmod lspsnel5a csubg wa lsssssubg syl chlt eldifad dochlss sseldd lsmlub
- wb prssi syl3anc mpbi2and eqsstr3d clsh lcfrlem17 dochsnshp wne lcfrlem34
+ wb prssi syl3anc mpbi2and eqsstrrd clsh lcfrlem17 dochsnshp wne lcfrlem34
lduallkr3 mpbird lshpcmp mpbid ) AUGUHIVPZVQUDVRZGUAVRZVSYOYPVTAYOUGUHWAZ
UDVRZFNWBVRZVPZYPAEFIYSNQTUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBYSWCZWD
AYRYPVSZFYPVSZYTYPVSZAYRUGSVRZUAVRUHSVRZUAVRWEYPABCDEFIJKLMNOQRSTUAUCUDUE
@@ -614964,7 +615041,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
kernels. (Contributed by NM, 13-Mar-2015.) $)
lcdvbase $p |- ( ph -> V = B ) $=
( cld cfv cress co cbs chlt eqid lcdval2 fveq2d syl5eq wss wceq cv ssrab2
- crab eqsstri clmod dvhlmod ldualvbase syl5sseqr ressbas2 syl eqtr4d ) AKD
+ crab eqsstri clmod dvhlmod ldualvbase sseqtrrid ressbas2 syl eqtr4d ) AKD
UBUCZBUDUEZUFUCZBAKCUFUCVGPACVFUFABCVEDEFGHIJLUGMNOQRSVEUHZUATUIUJUKABVEU
FUCZULBVGUMAFBVIBEUNIUCZJUCJUCVJUMZEFUPFTVKEFUOUQAVEFVIDURRVHVIUHZADGHLMQ
UAUSUTVABVIVFVEVFUHVLVBVCVD $.
@@ -615659,7 +615736,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
mapdval2N $p |- ( ph
-> ( M ` T ) = { f e. C | E. v e. T ( O ` ( L ` f ) ) = ( N ` { v } ) } ) $=
( cfv cv wss crab csn wceq wrex chlt mapdvalc wcel wa clsa wi c0g dvhlmod
- clmod ad3antrrr simplr eqid islsati syl2anc simprr eqsstr3d adantr simprl
+ clmod ad3antrrr simplr eqid islsati syl2anc simprr eqsstrrd adantr simprl
cbs lspsnel5 mpbird reximssdv ex lss0cl simpr lspsn0 eqtr4d sneq rspceeqv
syl fveq2d adantlr a1d lcfl1lem simplbi adantl dochsat0 mpjaodan 3ad2ant1
w3a simp3 simp2 lspsnel5a eqsstrd rexlimdv3a impbid rabbidva eqtrd ) AEMU
@@ -616336,7 +616413,7 @@ Part of property (e) in [Baer] p. 40. (Contributed by NM,
mapdin $p |- ( ph -> ( M ` ( X i^i Y ) ) = ( ( M ` X ) i^i ( M ` Y ) ) ) $=
( cfv wss wcel mapdord cin inss1 clmod dvhlmod lssincl mpbiri inss2 ssind
syl3anc ccnv clcd eqid clss mapdcl2 eleqtrrd mapdincl mapdcnvid2 syl6eqss
- crn mapdrn2 lcdlmod mapdcnvcl mpbid mpbird eqsstr3d eqssd ) AHIUAZFQZHFQZ
+ crn mapdrn2 lcdlmod mapdcnvcl mpbid mpbird eqsstrrd eqssd ) AHIUAZFQZHFQZ
IFQZUAZAVHVIVJAVHVIRVGHRHIUBABCDEFGVGHJLMKNACUCSHBSIBSVGBSACDEGJLNUDOPBHI
CMUEUIZOTUFAVHVJRVGIRHIUGABCDEFGVGIJLMKNVLPTUFUHAVKVKFUJQZFQZVHADEFGVKJKN
AGEUKQQZCDEFGVIVJJKLVOULZNAVIVOUMQZFUSZAVOHBVQCDEFGJKLMVPVQULZNOUNZAVOVQD
@@ -617311,7 +617388,7 @@ be in (Fy)*..." (Contributed by NM, 18-Mar-2015.) $)
= ( ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { Z } ) )
i^i ( ( N ` { ( X .- Z ) } ) .(+) ( N ` { Y } ) ) ) ) $=
( vj va vb vd ve csn cfv cin clmod wcel cabl clvec lveclmod syl lmodabl
- co eldifad ablsubsub4 sneqd fveq2d lmodvsubcl syl3anc eqsstr3d ablsub32
+ co eldifad ablsubsub4 sneqd fveq2d lmodvsubcl syl3anc eqsstrrd ablsub32
wss lspsntrim eqtrd ssind cv wceq wrex wa lsmspsn anbi12d syl5bb wi w3a
elin simp11 cpr wn wne cdif simp12l simp12r simp2l simp2r baerlem5alem1
simp13 simp3 lmodvacl lspsneli eqeltrd 3exp rexlimdvv impd sylbid ssrdv
@@ -617339,7 +617416,7 @@ be in (Fy)*..." (Contributed by NM, 18-Mar-2015.) $)
i^i ( ( N ` { ( X .- ( Y .+ Z ) ) } ) .(+) ( N ` { X } ) ) ) ) $=
( vj va vb vd ve csn cfv cin clmod wcel wss clvec lveclmod syl lspsntri
co eldifad syl3anc lmodvacl lmodvsubcl lspsnsub lmodabl ablnncan fveq2d
- cabl sneqd eqtrd lspsntrim eqsstr3d ssind wceq wrex elin lsmspsn syl5bb
+ cabl sneqd eqtrd lspsntrim eqsstrrd ssind wceq wrex elin lsmspsn syl5bb
cv wa anbi12d wi w3a simp11 cpr wn simp12l simp12r simp2l simp2r simp13
wne cdif simp3 baerlem5blem1 cgrp crg lmodring ringgrp grpinvcl syl2anc
4syl lspsneli eqeltrd 3exp rexlimdvv impd sylbid ssrdv eqssd ) APRCVMZV
@@ -626455,7 +626532,7 @@ family of sets (implicit). (Contributed by Stefan O'Rear,
A. x e. NN0 ( F ` x ) C_ ( F ` ( x + 1 ) ) ) ->
E. y e. NN0 A. z e. ( ZZ>= ` y ) ( F ` z ) = ( F ` y ) ) $=
( vw va vb vc cfv wcel cn0 cv wss wral wceq wa wrex wb cnacs wf caddc w3a
- c1 co crn cuni fvssunirn simplrr syl5sseqr simpll3 simplrl simpr incssnn0
+ c1 co crn cuni fvssunirn simplrr sseqtrrid simpll3 simplrl simpr incssnn0
cuz syl3anc eqssd ralrimiva cipo cdrs cvv wne cun cpw frn 3ad2ant2 elpw2g
c0 3ad2ant1 mpbird elex syl cc0 wfn ffn 0nn0 fnfvelrn sylancl cr ad2antrl
ne0d nn0re ad2antll cle cz nn0z eluz syl2an biimpar adantll ssequn1 sylib
@@ -633614,7 +633691,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by
function. (Contributed by Stefan O'Rear, 18-Jan-2015.) $)
dnnumch2 $p |- ( ph -> A C_ ran F ) $=
( vx cv cres wf1o con0 wrex crn wss dnnumch1 wi wfo wceq f1ofo forn resss
- syl rnss mp1i eqsstr3d a1i rexlimdvw mpd ) AKLZDEUMMZNZKOPDEQZRZAKBCDEFGH
+ syl rnss mp1i eqsstrrd a1i rexlimdvw mpd ) AKLZDEUMMZNZKOPDEQZRZAKBCDEFGH
IJSAUOUQKOUOUQTAUODUNQZUPUOUMDUNUAURDUBUMDUNUCUMDUNUDUFUNERURUPRUOEUMUEUN
EUGUHUIUJUKUL $.
@@ -634964,7 +635041,7 @@ is the base set of some (relabeled) Abelian group. (Contributed by
S e. dom card ) $=
( vx va vc vd vb cfv cbs cgrp wcel cv wceq cvv wss wb wbr sseldd ad2antrr
wa co char cun cima wrex ccrd cdm wfn basfn ssv fvelimab mp2an cdom harcl
- cxp con0 onenon ax-mp xpnum cwdom ssun1 simpr syl5sseqr fvex ssex syl a1i
+ cxp con0 onenon ax-mp xpnum cwdom ssun1 simpr sseqtrrid fvex ssex syl a1i
cplusg w3a simp1l 3ad2ant1 simp2 ssun2 simp3 grpcl syl3anc simp1r eleqtrd
simplll simprl simprr simplr grplcan syl13anc grprcan wn harndom wdomnumr
eqid unxpwdom3 sylib numdom sylancr rexlimiva sylbi ) AAUAGZUBZHIUCJZBKZH
@@ -636053,7 +636130,7 @@ of ideals (the usual "pure ring theory" definition). (Contributed by
$( The ring-span of a set contains the set. (Contributed by Stefan O'Rear,
30-Nov-2014.) $)
rgspnssid $p |- ( ph -> A C_ U ) $=
- ( vt cv wss csubrg cfv crab cint ssintub rgspnval syl5sseqr ) ABLMNLDOPZQ
+ ( vt cv wss csubrg cfv crab cint ssintub rgspnval sseqtrrid ) ABLMNLDOPZQ
RBELBUBSALBCDEFGHIJKTUA $.
rgspnmin.sr $e |- ( ph -> S e. ( SubRing ` R ) ) $.
@@ -637451,7 +637528,7 @@ is in the span of P(i)(X), so there is an R-linear combination of
itgcl nnne0d cmul itgmulc2 cioo eqidd oveq1 oveq2d adantl ioossicc syldan
simpr a1i mulcld fvmptd itgeq2dv cdv crn ctg ccnfld ctopn reelprrecn 1nn0
cpr nn0addcld nn0cnd 1cnd addcld wf cdm fmpttd ssidd dvexp mpteq2dv eqtrd
- pncand feq1d mpbird fdmd syl5sseqr dvres3 syl22anc reseq1d resmpt 3eqtr3d
+ pncand feq1d mpbird fdmd sseqtrrid dvres3 syl22anc reseq1d resmpt 3eqtr3d
mp1i eqid tgioo2 cnt iccntr dvmptres2 ioossre sstri cncfmptc mulcncf cvol
eqeltrd ioombl iblss ftc2 wral fveq1d ralrimivw itgeq2 oveq1d cxr cle wbr
rexrd ubicc2 recnd lbicc2 oveq12d itgioo 3eqtr3rd mvllmuld ) AEUAUBKZBCDU
@@ -638575,6 +638652,47 @@ is in the span of P(i)(X), so there is an R-linear combination of
DFPACEGPHBCQR $.
$}
+$( May move to main section after ~ alephsuc . $)
+
+ $( ` ( aleph `` 1o ) ` is the least uncountable ordinal. (Contributed by RP,
+ 18-Nov-2023.) $)
+ aleph1min $p |- ( aleph ` 1o ) = |^| { x e. On | _om ~< x } $=
+ ( c1o cale cfv c0 csuc com cv csdm wbr con0 crab cint fveq2i char wcel wceq
+ df-1o ax-mp eqtri 0elon alephsuc aleph0 ccrd cdm omelon onenon harval2 ) BC
+ DEFZCDZGAHIJAKLMZBUICRNUJGODZUKUJECDZODZULEKPUJUNQUAEUBSUMGOUCNTGUDUEPZULUK
+ QGKPUOUFGUGSAGUHSTT $.
+
+$( May move to main section after ~ alephiso . $)
+
+ ${
+ $d x y z $.
+ $( ` aleph ` is a strictly order-preserving mapping of ` On ` onto the
+ class of all infinite cardinal numbers. (Contributed by RP,
+ 18-Nov-2023.) $)
+ alephiso2 $p |- aleph Isom _E , ~< ( On , { x e. ran card | _om C_ x } ) $=
+ ( vy vz con0 cv ccrd cfv wceq wa cab cep cale wiso csdm wf1o wb wral wcel
+ wbr df-isom com wss crn crab alephiso iscard4 anbi1ci abbii df-rab eqtr4i
+ f1oeq3 ax-mp wel alephon epelg alephord2 alephord 3bitr2d bibi2d ralbidva
+ mp1i ralbiia anbi12i 3bitr4i mpbi ) DUAAEZUBZVFFGVFHZIZAJZKKLMZDVGAFUCZUD
+ ZKNLMZAUEDVJLOZBEZCEZKSZVPLGZVQLGZKSZPZCDQZBDQZIDVMLOZVRVSVTNSZPZCDQZBDQZ
+ IVKVNVOWEWDWIVJVMHVOWEPVJVFVLRZVGIZAJVMVIWKAVHWJVGVFUFUGUHVGAVLUIUJVJVMDL
+ UKULWCWHBDVPDRZWBWGCDWLVQDRIZWAWFVRWMWAVSVTRZBCUMWFVTDRWAWNPWMVQUNVSVTDUO
+ VAVPVQUPVPVQUQURUSUTVBVCBCDVJKKLTBCDVMKNLTVDVE $.
+ $}
+
+ ${
+ $d x y $.
+ $( ` aleph ` is a strictly order-preserving mapping of ` On ` onto the
+ class of all infinite cardinal numbers. (Contributed by RP,
+ 18-Nov-2023.) $)
+ alephiso3 $p |- aleph Isom _E , ~< ( On , ( ran card \ _om ) ) $=
+ ( vx vy con0 com wss ccrd crn crab cep csdm cale wiso cdif alephiso2 wceq
+ cv wb wcel wn omelon cen wbr wral elrncard simplbi ontri1 sylancr rabbiia
+ dfdif2 eqtr4i isoeq5 ax-mp mpbi ) CDAPZEZAFGZHZIJKLZCUPDMZIJKLZANUQUSOURU
+ TQUQUNDRSZAUPHUSUOVAAUPUNUPRZDCRUNCRZUOVAQTVBVCBPUNUAUBSBUNUCBUNUDUEDUNUF
+ UGUHAUPDUIUJCUQUSIJKUKULUM $.
+ $}
+
$( May move to main section after ~ infdjuabs . $)
@@ -639504,7 +639622,7 @@ of all sets ( ~ inex1g ).
relation. (Contributed by RP, 30-Jul-2020.) $)
clrellem $p |- ( ph -> Rel |^| { x | ( X C_ x /\ ps ) } ) $=
( vy cv wss wa wrel ccnv cvv wcel 3syl wrex cint cnvexg wceq dfrel2 sylib
- wex cab cnvss eqsstr3d relcnv jca31 cleq2lem releq anbi12d rexab2 biimpri
+ wex cab cnvss eqsstrrd relcnv jca31 cleq2lem releq anbi12d rexab2 biimpri
a1i elabd relint ) AEDMZNBOZVAPZOZDUGZLMZPZLVBDUHZUAZVHUBPAVDEFQZQZNZCOZV
KPZODVKAFRSVJRSVKRSGFRUCVJRUCTAVLCVNAEEQZQZVKAEPVPEUDHEUEUFAEFNVOVJNVPVKN
JEFUIVOVJUITUJKVNAVJUKURULVAVKUDVBVMVCVNBCVAVKEIUMVAVKUNUOUSVIVEVBVGVCLDV
@@ -640017,7 +640135,7 @@ Transitive relations (not to be confused with transitive classes).
$( Concrete construction of a superclass of relation ` R ` which is a
transitive relation. (Contributed by RP, 20-Jul-2020.) $)
trrelsuperrel2dg $p |- ( ph -> ( R C_ S /\ ( S o. S ) C_ S ) ) $=
- ( wss ccom cdm crn cxp cun ssun1 syl5sseqr ccnv syl syl5eqssr sylib ax-mp
+ ( wss ccom cdm crn cxp cun ssun1 sseqtrrid ccnv syl syl5eqssr sylib ax-mp
wceq ssequn1 eqtri xptrrel ssun2 sstri a1i coeq12d coundir wrel cocnvcnv1
relcnv relssdmrn dmcnvcnv rncnvcnv xpeq12i syl6sseq coss1 cocnvcnv2 coss2
coundi syl6eq 3sstr4d jca ) ABCECCFZCEABBGZBHZIZJZBCBVEKDLAVEVEFZVFVBCVGV
@@ -640291,7 +640409,7 @@ natural numbers (including zero) is equivalent to the existence of at
$( A set is a subset of its image under the identity relation.
(Contributed by RP, 22-Jul-2020.) $)
fvilbd $p |- ( ph -> R C_ ( _I ` R ) ) $=
- ( cid cfv ssid cvv wcel wceq fvi syl syl5sseqr ) ABBBDEZBFABGHMBICBGJKL
+ ( cid cfv ssid cvv wcel wceq fvi syl sseqtrrid ) ABBBDEZBFABGHMBICBGJKL
$.
$}
@@ -640669,7 +640787,7 @@ over the natural numbers (including zero) is equivalent to the
( wcel cn wceq wa cv wss wi crelexp co c1 sseq1d imbi2d vx vy ccom ciun
cfv cvv simplr simpr oveq1d iuneq12d elex adantr nnex iunex a1i fvmptd2
ovex relexp1g anbi1d wral caddc oveq2 weq simprl w3a simp2l relexpaddnn
- simp1 1nn syl3anc simp2rr simp3 simp2rl trrelssd eqsstr3d 3exp ralrimiv
+ simp1 1nn syl3anc simp2rr simp3 simp2rl trrelssd eqsstrrd 3exp ralrimiv
a2d nnind com12 iunss sylibr ex sylbird sseq1 syl5ibr mpcom alrimiv ) B
EIZDJKZLZBFMZNZWLWLUCWLNZLZBAUEZWLNZOZFWPCJBCMZPQZUDZKZWKWRWKGBCDGMZWSP
QZUDXAUFAUFHWKXCBKZLZCDJXDWTWIWJXEUGXFXCBWSPWKXEUHUIUJWIBUFIWJBEUKULXAU
@@ -640773,7 +640891,7 @@ over the natural numbers (including zero) is equivalent to the
vx vy wa cun cres ccom w3a cfv ciun simplr simpr oveq1d iuneq12d adantr
cvv elex nn0ex ovex iunex a1i fvmptd2 cc0 relexp0g relexp1g 3anbi12d cn
wral wo elnn0 caddc weq simpr2 simp1 simp2l relexpaddnn syl3anc simp2r3
- 1nn simp3 simp2r2 trrelssd eqsstr3d 3exp a2d nnind simpr1 syl5ibr sylbi
+ 1nn simp3 simp2r2 trrelssd eqsstrrd 3exp a2d nnind simpr1 syl5ibr sylbi
jaoi com12 ralrimiv iunss sylibr ex sylbird sseq1 mpcom alrimiv ) BEIZD
JKZUFZUABUBBUCUGUHZFLZMZBXFMZXFXFUIXFMZUJZBAUKZXFMZNZFXKCJBCLZOPZULZKZX
DXMXDGBCDGLZXNOPZULXPURAURHXDXRBKZUFZCDJXSXOXBXCXTUMYAXRBXNOXDXTUNUOUPX
@@ -640838,7 +640956,7 @@ over the natural numbers (including zero) is equivalent to the
( wcel crelexp co cc0 wss wceq wi oveq2 oveq1d sseq1d imbi2d cid cun cres
c1 cvv ax-mp vx vy cn0 cn wo elnn0 cv weq relexp1g ssid syl6eqss w3a ccom
caddc simp2 simp1 wa relexpsucnnr syl2anc cdm crn ovex coexg relexp0g syl
- dmcoss rncoss unss12 mp2an ssres2 resundi ssun1 syl5sseqr adantr sseqtr4i
+ dmcoss rncoss unss12 mp2an ssres2 resundi ssun1 sseqtrrid adantr sseqtr4i
mpan ssun2 simpr unssd syl5eqss 3adant1 sstrd eqsstrd 3exp a2d relexp0idm
syl5ss nnind sylan9eq eqimss ex jaoi sylbi impcom ) BUCDZACDZABEFZGEFZAGE
FZHZWOBUDDZBGIZUEWPWTJZBUFXAXCXBWPAUAUGZEFZGEFZWSHZJWPAREFZGEFZWSHZJWPAUB
@@ -640896,7 +641014,7 @@ over the natural numbers (including zero) is equivalent to the
( wcel crelexp co ccom caddc wss cc0 wceq w3a syl cres ccnv oveq2d 3ad2ant3
c1 eqtrd cn0 cn wo wi elnn0 biimpi relexpaddnn eqimss 3exp cuz cfv elnn1uz2
c2 cid cdm crn wrel relco dfrel2 ax-mp cnvco cnvresid coeq2i coires1 3eqtri
- cun cnvss resss sstri eqsstr3i cnvcnvss a1i simp1 relexp0g relexp1g coeq12d
+ cun cnvss resss sstri eqsstrri cnvcnvss a1i simp1 relexp0g relexp1g coeq12d
simp2 oveq12d addid2d 3sstr4d cnvexg relexpuzrel syl2anc eluz2nn relexpnndm
1cnd cvv df-rn ssun2 syl6ss relssres syl5eq eluzge2nn0 relexpcnv 3eqtr4d wb
simp3 3adant1 cnveqb sylancr mpbird coeq1d oveq1d eluzelcn jaod syl5bi jaoi
@@ -640993,7 +641111,7 @@ over the natural numbers (including zero) is equivalent to the
dftrcl3 $p |- t+ = ( r e. _V |-> U_ n e. NN ( r ^r n ) ) $=
( vz va vk vt vs cvv cv wss ccom wa cmpt cn crelexp co ciun wcel c1 wceq
ctcl cab cint df-trcl cfv relexp1g 1nn oveq1 iuneq2d oveq2 cbviunv syl6eq
- nnex weq cbvmptv ov2ssiunov2 mp3an23 eqsstr3d cuz nnuz 1nn0 iunrelexpuztr
+ nnex weq cbvmptv ov2ssiunov2 mp3an23 eqsstrrd cuz nnuz 1nn0 iunrelexpuztr
cn0 wb wi wal fvex trcleq2lem a1i alrimiv elabgt sylancr mpbir2and intss1
syl wral vex elab eqid iunrelexpmin1 mpan2 19.21bi syl5bi ralrimiv sylibr
ssint eqssd ovex iunex fvmpt eqtrd mpteq2ia eqtri ) UABHBIZCIZJWOWOKWOJLZ
@@ -641116,7 +641234,7 @@ over the natural numbers (including zero) is equivalent to the
ctcl ccom cint csn cun cop df-br trclfv 3ad2ant3 elimasng 3adant3 3bitr4d
a1i breqd wrex intimasn 3ad2ant1 wal wex cxp simpl3 snex vex xpex sylancl
wi unexg trclfvlb unssad trclfvcotrg cin wne simpl1 inelcm syl2anc xpima2
- c0 snidg unssbd imass1 eqsstr3d imaundir simpr imassrn rnxpss sstri unssd
+ c0 snidg unssbd imass1 eqsstrrd imaundir simpr imassrn rnxpss sstri unssd
crn syl5eqss trclimalb2 eqssd sbcan csb sbcssg csbconstg csbvargi sseq12i
fvex bitri csbcog coeq12i anbi12i sbceq2g csbima12 imaeq1i imaeq2i 3eqtri
eqtri eqeq2i sylbbr syl21anc spesbcd ex eqeq1 imaeq1 eqeq2d rexab2 syl6bb
@@ -641217,7 +641335,7 @@ over the natural numbers (including zero) is equivalent to the
dfrtrcl3 $p |- t* = ( r e. _V |-> U_ n e. NN0 ( r ^r n ) ) $=
( vz va vk vt vs cvv cv wss ccom w3a cmpt cn0 crelexp ciun wcel cc0 sseq2
co crtcl cid cdm crn cun cres cab cint df-rtrcl relexp0g nn0ex 0nn0 oveq1
- cfv weq iuneq2d oveq2 cbviunv syl6eq cbvmptv ov2ssiunov2 mp3an23 eqsstr3d
+ cfv weq iuneq2d oveq2 cbviunv syl6eq cbvmptv ov2ssiunov2 mp3an23 eqsstrrd
c1 relexp1g 1nn0 cuz wceq nn0uz iunrelexpuztr wb wal fvex coeq12d sseq12d
wi id 3anbi123d a1i alrimiv elabgt sylancr mpbir3and intss1 syl wral elab
eqid iunrelexpmin2 mpan2 19.21bi syl5bi ralrimiv ssint sylibr eqssd iunex
@@ -641303,7 +641421,7 @@ over the natural numbers (including zero) is equivalent to the
( vi vj vk vd crelexp cc0 c1 cn cn0 cun wss wceq wcel ax-mp cv ciun oveq2
1nn co cvv va vb vc cpr crcl ctcl crtcl dfrcl4 dftrcl3 dfrtrcl3 prex nnex
df-n0 uncom df-pr uneq1i unass ssequn1 mpbi uneq2i 3eqtrri 3eqtri cbviunv
- csn snssi ss2iun relexp1g elv ssiun2s eqsstr3i ovex iunex 0nn0 1nn0 prssi
+ csn snssi ss2iun relexp1g elv ssiun2s eqsstrri ovex iunex 0nn0 1nn0 prssi
a1i mp2an sseli relexpss1d mprg eqsstri eqtr4i 1ex prid2 c0ex ssid unss12
prid1 iuneq1 iunxun iunxsn cin c0 wne vex nnssnn0 inelcm iunrelexp0 mp3an
eqtri uneq12i 3sstr4i comptiunov2i ) ABCEFGUDZHIUEUFUGUAUBUCDAUAUHBUBUICU
@@ -647915,7 +648033,7 @@ base set if and only if the neighborhoods (convergents) of every point
$( The image of a mapping from A is nonempty if A is nonempty.
(Contributed by Stanislas Polu, 9-Mar-2020.) $)
wnefimgd $p |- ( ph -> ( F " A ) =/= (/) ) $=
- ( cdm cin wss wceq ssid fdmd syl5sseqr sseqin2 sylib eqnetrd imadisjlnd
+ ( cdm cin wss wceq ssid fdmd sseqtrrid sseqin2 sylib eqnetrd imadisjlnd
c0 ) ADBADGZBHZBRABSITBJABBSBKABCDFLMBSNOEPQ $.
$}
@@ -649434,7 +649552,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing
( va cv wcel c0 cuni wa cpr wceq wss wrex csn eleq1 rexbidv 0ex prid1 a1i
anbi1d rspcdva adantr wn simprl simpr 0nep0 snnzg necomd nelprd elnelneqd
wne syl adantrr adantrl wo elpri eleq2s orcomd ord sylc unieqd snex unipr
- cun uncom un0 3eqtri syl6eq simprrr eqsstr3d snssg biimprd eleq2w anbi12d
+ cun uncom un0 3eqtri syl6eq simprrr eqsstrrd snssg biimprd eleq2w anbi12d
unieq sseq1d rexlimddvcbv ) ADBNZOZPMNZOZWIQZWGUAZRZPCNZOZWNQZWGUAZRZCMHA
GNZWNOZWQRZCHUBWRCHUBGPPUCZSZPWSPTZXAWRCHXDWTWOWQWSPWNUDUIUELPXCOAPXBUFUG
UHUJAWIHOZWMRZRZDFOZDUCZWGUAZWHAXHXFJUKXGXIWKWGXGWKPXISZQZXIXGWIXKXGXEWIX
@@ -649457,7 +649575,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing
( va cv wcel c0 csn wa cpr wceq cuni wrex eleq1 anbi1d rexbidv p0ex prid2
wss a1i rspcdva simpl simprl simpr wne 0nep0 necomi 0ex sneqr eqcomd nsyl
wn neqned nelprd adantr elnelneqd adantrr adantrl elpri eleq2s ord unieqd
- wo sylc cun snex unipr df-pr eqtr4i syl6eq simprrr eqsstr3d prssg sylancr
+ wo sylc cun snex unipr df-pr eqtr4i syl6eq simprrr eqsstrrd prssg sylancr
cvv wb biimprd simprd eleq2w unieq sseq1d anbi12d rexlimddvcbv ) AEBNZOZP
QZMNZOZWPUAZWMUHZRZWOCNZOZXAUAZWMUHZRZCMHAGNZXAOZXDRZCHUBXECHUBGPWOSZWOXF
WOTZXHXECHXJXGXBXDXFWOXAUCUDUELWOXIOAPWOUFUGUIUJAWPHOZWTRZRZPWMOZWNXMAPES
@@ -649702,7 +649820,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing
$( Lemma for ~ grumnud . (Contributed by Rohan Ridenour, 13-Aug-2023.) $)
grumnudlem $p |- ( ph -> G e. M ) $=
( vw vv va wcel cv cpw wss wel wa wrex cuni wi wral wal cgru gruss 3expia
- syl3an1 alrimiv pwss ccoll cun wceq w3a ssun1 simp3 syl5sseqr simp1l3 wex
+ syl3an1 alrimiv pwss ccoll cun wceq w3a ssun1 simp3 sseqtrrid simp1l3 wex
sylibr wbr simp1r simpr unieqd simpl eqtr4d adantll simpll3 simprd simpld
weq eqeltrd eleqtrrd 3jca simpl2 rr-spce simp1l1 syl simp2 gruuni syl2anc
rr-rspce simpl1 gruel syl3anc 3ad2ant1 sylan rexbidva mpbird rexex adantr
@@ -660848,7 +660966,7 @@ reference definition (axiom) in set.mm. Also, a theorem or
( vu vv vz copab cvv cv wcel wa vex cop wceq wex cab exbii eximi biimpi
wi cxp wss wrel pm3.2i a1i ssopab2i biantru abbii ax6ev equcom mpbi opeq2
idn1 e1a idn2 opeq1 eqeq1 biimprd e12 eqeq2 biimpd in2 in1 19.37iv 19.37v
- e2 ax-mp 19.9v ss2abi eqsstr3i sseqtri df-opab 3sstr4i df-xp eqcomi sstri
+ e2 ax-mp 19.9v ss2abi eqsstrri sseqtri df-opab 3sstr4i df-xp eqcomi sstri
syl df-rel biimpri e0a ) ABCGZHHUAZUBZWAUCZWABIZHJZCIZHJZKZBCGZWBAWIBCWIA
WFWHBLCLUDZUEUFWJDIZHJZEIZHJZKZDEGZWBFIZWEWGMZNZWIKZCOZBOZFPZWRWLWNMZNZWP
KZEOZDOZFPZWJWQXDXFEOZDOZFPZXJXDWTCOZBOZFPXMXOXCFXNXBBWTXACWIWTWKUGQQUHXO
@@ -671971,7 +672089,7 @@ closed under the multiplication ( ' X ' ) of a finite number of
mullimc $p |- ( ph -> ( X x. Y ) e. ( H limCC D ) ) $=
( cc crp vz vy vw vc va vd vb ve vf cmul climc wcel wne cmin cabs cfv clt
co cv wbr wa wi wral limccl sseldi mulcld simpr adantr mulcn2 syl3anc w3a
- wrex fmptd cdm dmmptd wf wss limcrcl simp2d eqsstr3d simp3d ellimc3 mpbid
+ wrex fmptd cdm dmmptd wf wss limcrcl simp2d eqsstrrd simp3d ellimc3 mpbid
syl simprd r19.21bi adantrr adantrl reeanv sylanbrc cle cif ifcl 3ad2ant2
nfv nfra1 nfan nf3an simp11l simp1rl 3ad2ant1 simp12 simp13l jca31 simp1r
simp2 simp3l simplll simp1lr simp3r simp1l sselda syl2anc cr rpre ltletrd
@@ -672089,7 +672207,7 @@ closed under the multiplication ( ' X ' ) of a finite number of
sseq1 anbi12d rspcev syl12anc ioossre jctil elioore snssd sylancr 3adant1
wceq isnei mpbird neeq1d rspccva syl2anc ralrimivv simplbda eleq2i biimpi
ineq1 3ad2ant2 simp1 simp3l simpr adantr snssg syl jca tg2 cxp cpw wf wfn
- 3exp ioof ovelrn mp2b simpll eleqtrd simplr eqsstr3d ex reximdv rexlimiva
+ 3exp ioof ovelrn mp2b simpll eleqtrd simplr eqsstrrd ex reximdv rexlimiva
ffn 3syl simpl3r sstr expcom anim2d reximdva rexlimdv adantlr nfra1 nfra2
nfan inss1 simp3r syl5ss inss2 ssind simpllr 3ad2ant1 simp2 rsp2 rexlimd
mpd syl3c ssn0 ralrimiva impbida bitrd ) ACBDUBLLMZJUHZBCUCZUDZUEZUIUFZJU
@@ -672230,7 +672348,7 @@ closed under the multiplication ( ' X ' ) of a finite number of
mullimcf $p |- ( ph -> ( B x. C ) e. ( H limCC D ) ) $=
( vz wcel cc clt wa crp vy vw vc va vd vb ve vf cmul co climc cv wne cmin
cabs cfv wbr wi wral limccl sseldi mulcld simpr adantr mulcn2 syl3anc w3a
- wrex cdm fdmd wss limcrcl syl simp2d eqsstr3d simp3d ellimc3 mpbid simprd
+ wrex cdm fdmd wss limcrcl syl simp2d eqsstrrd simp3d ellimc3 mpbid simprd
r19.21bi adantrr adantrl reeanv sylanbrc cle cif ifcl 3ad2ant2 nfra1 nfan
nfv nf3an simp11l simp1rl 3ad2ant1 jca simp12 simp13l jca31 simp1r simp3l
simp2 simplll simp1lr simp3r simp1l sselda syl2anc subcld abscld ad2antrl
@@ -672661,7 +672779,7 @@ closed under the multiplication ( ' X ' ) of a finite number of
$( The domain of a maps-to function with a limit. (Contributed by Glauco
Siliprandi, 11-Dec-2019.) $)
limcmptdm $p |- ( ph -> A C_ CC ) $=
- ( cdm cc dmmptd wf wss wcel climc co w3a limcrcl syl simp2d eqsstr3d ) AC
+ ( cdm cc dmmptd wf wss wcel climc co w3a limcrcl syl simp2d eqsstrrd ) AC
GKZLABGCDLHIMAUDLGNZUDLOZFLPZAEGFQRPUEUFUGSJFEGTUAUBUC $.
$}
@@ -678597,7 +678715,7 @@ distinct definitions for the same symbol (limit of a sequence).
mpdan leloe ltpnfd ssinss1 ioossre unima 3eqtrd 3adant2 elinel2 mpbir3and
xrnltled anim2i adantll simplbda 3adant3 3adant1r eqsstrd eqeq1d pm2.65da
simp1 ltled lenlt ltpnf ancom ancli vtoclg sylan9eqr neqne necomd leneltd
- sylc wne condan pm5.6 3adantll2 olci inss ax-mp sseli ralrimiva syl5sseqr
+ sylc wne condan pm5.6 3adantll2 olci inss ax-mp sseli ralrimiva sseqtrrid
funimass4 eqeltrrd nelneq necon3bi ad4antlr simp2d leltne simp3d ioossicc
necom inss2 sstri sslin rexlimdv dfss3 syld3an1 iscnp cncnp fnssres fvres
3exp eqfnfvd ) AFGHEUUAZUUBUGZUUCUGSZFCDUUDUGZUUEZEUHAWUDUIWUBFUUFZFUAUJZ
@@ -679038,7 +679156,7 @@ distinct definitions for the same symbol (limit of a sequence).
a1i cfv cif cxr rexrd iccid syl oveq2 sylan9req eqcomd eleqtrd mpteq12dva
elsni iftrued syl5eq recnd cncfdmsn eqeltrd sseldd oveq1d simpll wo eqcom
adantlr biimpi con3i adantl simplr pm4.56 lttrid mpbird ccnfld ctopn eqid
- ccn 0ss crest ctop cnfldtop rest0 ax-mp eqcomi cncfcn mp2an eqsstr3i icc0
+ ccn 0ss crest ctop cnfldtop rest0 ax-mp eqcomi cncfcn mp2an eqsstrri icc0
wb mpteq1d mpt0 3eqtrd 0cnf syl6eqel pm2.61dan ) ACDUAUBZGCDUCPZQUDPZRZAX
QUEZBCDEFGHYABUJJACUFRZXQKSADUFRZXQLSAXQUGAFCDUHPQUDPRXQMSAHFDUKPRXQNSAEF
CUKPZRXQOSUIAXQULZUEZCDUMZXTAYGXTYEAYGUEZGCUNZQUDPZXSYHYIEUNZUDPZYJGAYLYJ
@@ -679561,7 +679679,7 @@ distinct definitions for the same symbol (limit of a sequence).
( cres cdv co cfv cc wss wceq cnt wf syl22anc wfn ffn fnresdm 3syl oveq2d
dvres ctop wcel cuni crest cnfldtopon resttopon sylancr syl5eqel topontop
ctopon toponuni sseqtrd eqid ntridm syl2anc fveq2d 3eqtr3d reseq2d ntrss2
- syl eqsstr3d sstrd 3eqtr4rd ) ABCFNZOPZBCOPZFDUAQZQZNZVOBCGNOPZABRSZFRCUB
+ syl eqsstrrd sstrd 3eqtr4rd ) ABCFNZOPZBCOPZFDUAQZQZNZVOBCGNOPZABRSZFRCUB
ZFBSZWBVNVRTHJIIFFBDCELKUIUCAVMCBOAWACFUDVMCTJFRCUEFCUFUGUHAVOGVPQZNZVOGN
VSVRAWCGVOAVQVPQZVQWCGADUJUKZFDULZSZWEVQTADBUSQZUKZWFADEBUMPZWIKAERUSQUKV
TWKWIUKELUNHBERUOUPUQZBDURVIZAFBWGIAWJBWGTWLBDUTVIVAZFDWGWGVBZVCVDAVQGVPM
@@ -682142,7 +682260,7 @@ distinct definitions for the same symbol (limit of a sequence).
Glauco Siliprandi, 11-Dec-2019.) $)
iblsplit $p |- ( ph -> ( x e. U |-> C ) e. L^1 ) $=
( vk cmpt wcel cr cc0 cfv wa cc eqidd vy cibl cmbf cv ci cexp co cdiv cre
- cle wbr cif citg2 c3 cfz wral fmpttd cres cun syl5sseqr resmptd imdistani
+ cle wbr cif citg2 c3 cfz wral fmpttd cres cun sseqtrrid resmptd imdistani
ssun1 sseld isibl2 mpbid simpld eqeltrd ssun2 eqcomd mbfres2cn caddc cvol
syl cdm mbfdm2 adantr cin covol wceq cpnf cicc cxr adantlr ax-icn elfznn0
a1i expcld ad2antlr wne cz elfzelz expne0d divcld recld rexrd simpr pnfge
@@ -682919,7 +683037,7 @@ distinct definitions for the same symbol (limit of a sequence).
wral fdmd cncfperiod elrab simprr nfre1 nfan 3jca 3ad2ant1 mpbird rexlimd
nfv 3exp mpd sylan2b cmin resubcld pncand eqcomd lesub1dd eqbrtrd breqtrd
reseq2d oveq1d cibl a1i 1cnd ssid cvol cdv resmptd eqtrd cc0 3eqtrd eqrdv
- rspceeqv sylanbrc impbida eqsstr3d feqresmpt iccshift 3eltr4d ioosscn csn
+ rspceeqv sylanbrc impbida eqsstrrd feqresmpt iccshift 3eltr4d ioosscn csn
npcand constcncfg cxp fconstmpt ioovolcl iblconst syl5eqelr elind crn ctg
ioombl cnt oveq2d addcld fmpttd ccnfld ctopn tgioo2 dvres syl22anc iccntr
cpr reelprrecn dvmptid 0cnd dvmptc dvmptadd reseq1d ioossre 1p0e1 mpteq2i
@@ -689006,7 +689124,7 @@ approximated is nonnegative (this assumption is removed in a later
wss ax-resscn resmpt 3eqtrd ccncf constcncfg cncfmpt1f cncfmptssg addcncf
cdm fsumcncf difssd eldifsn mpbir2an divcncf eqeltrd cibl halfcld clt wbr
recn zcnd 0red elfzle1 ltletrd gt0ne0d fsumcl addcld dvmptid tgioo2 reopn
- 0lt1 2cnd simpr coscld mp1i eqcomd fmpttd wral ralrimiva dmmptg syl5sseqr
+ 0lt1 2cnd simpr coscld mp1i eqcomd fmpttd wral ralrimiva dmmptg sseqtrrid
wf dvsinax dmeqd sseqtr4d dvcnre reseq1d ax-mp syl6eq mpteq2dva dvmptfsum
divcan3d eqtrd dvmptadd iccssred iccntr dvmptres2 syl5eq fvmpt2d itgeq2dv
cnt 3eqtr4d ioosscn ssid coscn mulc1cncf csn cdif cncfmptc mp3an ioossicc
@@ -689169,7 +689287,7 @@ approximated is nonnegative (this assumption is removed in a later
ioossre cn nnred readdcld sselda remulcld resincld 2re remulcli rehalfcld
pire crn ctg iooretop cnt nncnd pm3.2i ccnfld ctopn tgioo2 dvres syl21anc
retop ccld cha rehaus elioored uniretop difopn isopn3i reseq2d reelprrecn
- sncld dvsinax syl5sseqr dvres3 syl22anc syl6eq eqimss oveq2i mulcli sseli
+ sncld dvsinax sseqtrrid dvres3 syl22anc syl6eq eqimss oveq2i mulcli sseli
cpr 2cn 1cnd div13d halfcl mulid1d dvasinbx mul32d recidd mulid2d resmptd
ccom wfn ralrimiva fnmpt eldifi eleq1w anbi2d neeq1d imbi12d elioore 3syl
oveq1 0re pipos gtneii sineq0 mtbid eqneltrd crp 2rp rpmulcl mp2an neqned
@@ -695699,7 +695817,7 @@ approximated is nonnegative (this assumption is removed in a later
crab rexlimdv ralrimiva sylibr simpll fourierdlem15 ioossicc fourierdlem1
3exp rabss sseli cncfperiod elrab simprbi nfre1 nfan elioo2 simp2d simp3d
nfv 3jca rexlimd mpd pncand ltsub1dd eqbrtrd npcand rspcev sylanbrc eqrdv
- breqtrd impbida reseq2d feqresmpt 3eltr3d syl5sseqr sseldi anbi2d eqeq12d
+ breqtrd impbida reseq2d feqresmpt 3eltr3d sseqtrrid sseldi anbi2d eqeq12d
eleq1 imbi12d limcperiod eleqtrd eliccre adantll ad2antlr iccgelb iccleub
chvarv necomd fvres 3eqtrrd mpteq2dva syl5eq syl6eq iftrued lbicc2 limccl
ltled gtned neneqd ubicc2 3eqtr4rd ad4ant14 stoic1a eliccd fssresd neqned
@@ -696650,7 +696768,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u
cioc simp3d breq12d rspccva ancli eleq1 anbi2d eqeq12d eleq1i rgenw rabbi
rexbii mpbi fourierdlem65 breqtrrd id sseq12d fourierdlem79 fourierdlem10
anabsi7 wne neqne adantl leneltd ltletrd eliood fvres ifeq2da cdm ioosscn
- fdm feq2d ioossre syl5sseqr syl5eqel iooshift eqsstr3d syl6breqr divcan1d
+ fdm feq2d ioossre sseqtrrid syl5eqel iooshift eqsstrrd syl6breqr divcan1d
c2 gt0ne0d negsubdi2d cmpt oveq2 redivcld remulcld readdcld fvmptd mulcld
flcld zred pncan2d divnegd 3eqtr4d divcan4d znegcld adantlr sylan2 ccnfld
simpr fperiodmul ctopn cico crest reseq2d eleq12d nfmpt1 nfcxfr nfcv nffv
@@ -696931,7 +697049,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u
simp3d breq12d rspccva eleq1 anbi2d eleq1i rgenw rabbi mpbi fourierdlem65
eqeq12d rexbii anabsi7 mpdan breqtrrd sseq12d fourierdlem79 fourierdlem10
c2 lelttrd wne neqne necomd adantl leneltd eliood fvres ifeq2da cdm feq2d
- fdm ioosscn ioossre syl5sseqr syl5eqel iooshift eqsstr3d gt0ne0d divcan1d
+ fdm ioosscn ioossre sseqtrrid syl5eqel iooshift eqsstrrd gt0ne0d divcan1d
syl6breqr negsubdi2d cmpt id oveq2 redivcld zred remulcld readdcld fvmptd
flcld zcnd mulcld pncan2d divnegd 3eqtr4d divcan4d znegcld adantlr sylan2
simpr fperiodmul ccnfld ctopn csn crest ancli reseq2d eleq12d sylc nfmpt1
@@ -697067,7 +697185,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u
vtoclg npcand ioossre feqresmpt iooshift mpteq1d biimpar elioore resubcld
eqtr3d eleq2d 3adant2 simp3 ioogtlb iooltub fourierdlem15 lelttrd ltletrd
eliood fvres ltled recn 3eqtr2rd mpteq2dva 3eqtrd ioosscn rexbidv cbvrexv
- eqeq2d syl6bb cbvrabv cncfshift eqeltrd cdm ffdm syl5sseqr eqcomi 3sstr4d
+ eqeq2d syl6bb cbvrabv cncfshift eqeltrd cdm ffdm sseqtrrid eqcomi 3sstr4d
syl5eqssr ioossicc sseli fourierdlem1 limcperiod syl5eq reseq2d pm2.61dan
fdm ) AUNIUOUPZBCIUQURZDIUQURZUSURZBUTZLVAZVBZBCDUSURZVUEVBZVCZAUYTVDZBCD
EFGUJUNOVEURZUJUTZFVAZIUQURZVFZIJKLUKJUTZFVAZVUPUUAUQURZFVAZUSURZUKUTZVUQ
@@ -699787,7 +699905,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u
jca32 vtoclg sylc dirkerper adantll resmpt ax-mp ccom rexrd breq2d npcand
reseq1i addcomd ioogtlb breq1d iooltub eliood fmptco constcncfg cncfmptid
ltadd2dd ssid addcncf ioosscn cncfmptssg cncfco feqmptd cncfss dirkercncf
- mp2an sseldi cncff mulcncf csn cdif crn wne syl5sseqr ssdmres sylib gtned
+ mp2an sseldi cncff mulcncf csn cdif crn wne sseqtrrid ssdmres sylib gtned
fdm eldifsn sylanbrc rnmptss eqidd leidd iccssred resmptd rescncf cnlimci
syl6ss eqcomi addcld limciccioolb limccog mullimcf adantllr ltned 3eltr3d
fcompt limcicciooub resabs1 fourierdlem110 subnegd itgeq1d 3eqtr3a ltleii
@@ -701593,7 +701711,7 @@ those for the more general case of a piecewise smooth function (see
( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) ) = ( F
` X ) ) $=
( co cr cc cioo crn ctg cfv cpi cneg cdm cdif cfn cdv cres dmeqi wss wceq
- c0 ioossre ccncf wcel wf cncff fdm syl5sseqr ssdmres sylib syl5eq difeq2d
+ c0 ioossre ccncf wcel wf cncff fdm sseqtrrid ssdmres sylib syl5eq difeq2d
3syl difid syl6eq 0fin syl6eqel rescncf mpsyl oveq1d 3eltr4d cv cico cpnf
a1i climc wne pire renegcli rexri icossre mp2an eldifi sseldi wa limcresi
cxr reseq1i resres eqtr2i oveq1i sseqtri adantr simpr cnlimci ne0d sylan2
@@ -713253,7 +713371,7 @@ its dimensional volume (the product of its length in each dimension,
( ( i ( H ` X ) ( B ` i ) ) \ ( i ( H ` X ) ( A ` i ) ) ) ) $=
( vk wcel wceq wa cr adantlr vf vh cv cfv cico co cixp cdif ciin wal wral
wb nfv nfcv nfixp1 nfel nfan cmnf cif wfn ixpfn ad2antlr wss fveq2 oveq2d
- cioo iftrue eqtr4d eqimss syl ioossre iffalse syl5sseqr pm2.61i cxr mnfxr
+ cioo iftrue eqtr4d eqimss syl ioossre iffalse sseqtrrid pm2.61i cxr mnfxr
wn a1i ffvelrnda rexrd icossre syl2anc simpl simpr oveq12d adantll sseldd
fvixp mnfltd clt wbr icoltub syl3anc eliood sseldi ralrimiva elixp sylibr
jca vex cfn cmpo cmpt equequ1 ifbid cbvixpv mpoeq3ia mpteq2i eqtri adantr
@@ -725765,7 +725883,7 @@ The modulo (remainder) operation - extension
elfzle1 1red elfzel1 zred letr syl3anc mpan2d syl5com syl3anbrc elfzel2
imp ad4antr nnred elfzle2 elfzolt2 lelttrd elfzo2 iccpartipre cmin cmap
cr cxr ad3antrrr fzoval wss elfzo0le 0le1 0red mpani sylbid mpd 0zd jca
- elfzoel2 ssfzo12bi mpbird eqsstr3d sselda iccpartimp simprd exp31 com23
+ elfzoel2 ssfzo12bi mpbird eqsstrrd sselda iccpartimp simprd exp31 com23
smonoord elfzuz iccpartiltu breq2 3imtr4d exp4c com13 com3r sylbi imp32
anbi2d ralrimivva ) ACUFZDUFZIJZUVEBKZUVFBKZIJZLZCDMEUANZUVLAUVEUVLOZUV
FUVLOZUVKAUVMUVEMUBZPEUANZUCZOZUVNUVKLZAUVLUVQUVEAUVLUVOMPUDNZEUANZUCZU
@@ -726266,6 +726384,26 @@ valid cases ( ` (/) ` is the last symbol) and invalid cases ( ` (/) `
$.
$}
+ ${
+ $d a x ph $. $d a y ph $.
+ $( Setvar variables are interchangeable in a wff they do not appear in.
+ (Contributed by SN, 23-Nov-2023.) $)
+ ichv $p |- [ x <> y ] ph $=
+ ( va wich wsb wb wal sbv sbbii bitri gen2 df-ich mpbir ) ABCEACDFZBCFZDBF
+ ZAGZCHBHRBCQADBFAPADBPABCFAOABCACDIJABCIKJADBIKLABCDMN $.
+ $}
+
+ ${
+ $d a x $. $d a y $. $d a ph $.
+ ichf.1 $e |- F/ x ph $.
+ ichf.2 $e |- F/ y ph $.
+ $( Setvar variables are interchangeable in a wff they are not free in.
+ (Contributed by SN, 23-Nov-2023.) $)
+ ichf $p |- [ x <> y ] ph $=
+ ( va wich wsb wb wal sbf sbbii bitri sbv gen2 df-ich mpbir ) ABCGACFHZBCH
+ ZFBHZAIZCJBJUABCTAFBHASAFBSABCHARABCACFEKLABCDKMLAFBNMOABCFPQ $.
+ $}
+
${
$d a ph $. $d a x $.
$( A setvar variable is always interchangeable with itself. (Contributed
@@ -726451,6 +726589,16 @@ valid cases ( ` (/) ` is the last symbol) and invalid cases ( ` (/) `
DETUAUJCDETUB $.
$}
+ $( Two setvar variables are always interchangeable when there are two
+ universal quantifiers. (Contributed by SN, 23-Nov-2023.) $)
+ ich2al $p |- [ x <> y ] A. x A. y ph $=
+ ( wal nfa1 nfa2 ichf ) ACDZBDBCHBEACBFG $.
+
+ $( Two setvar variables are always interchangeable when there are two
+ existential quantifiers. (Contributed by SN, 23-Nov-2023.) $)
+ ich2ex $p |- [ x <> y ] E. x E. y ph $=
+ ( wex nfe1 excom nfxfr ichf ) ACDZBDZBCIBEJABDZCDCABCFKCEGH $.
+
${
$d ph x y $. $d ps x y $. $d a b x y $.
$( If two setvar variables are interchangeable in two wffs, then they are
@@ -734110,7 +734258,7 @@ their group (addition) operations are equal for all pairs of elements of
resmgmhm2 $p |- ( ( F e. ( S MgmHom U ) /\ X e. ( SubMgm ` T ) )
-> F e. ( S MgmHom T ) ) $=
( vx vy cmgmhm co wcel cfv wa cmgm cbs wf cv cplusg wceq eqid wral simpld
- csubmgm mgmhmrcl submgmrcl anim12i wss mgmhmf submgmbas submgmss eqsstr3d
+ csubmgm mgmhmrcl submgmrcl anim12i wss mgmhmf submgmbas submgmss eqsstrrd
fss syl2an mgmhmlin 3expb adantlr ressplusg ad2antlr oveqd ralrimivva jca
eqtr4d ismgmhm sylanbrc ) DACIJKZEBUCLZKZMZANKZBNKZMAOLZBOLZDPZGQZHQZARLZ
JDLZVNDLZVODLZBRLZJZSZHVKUAGVKUAZMDABIJKVEVIVGVJVEVICNKACDUDUBEBUEUFVHVMW
@@ -738374,7 +738522,7 @@ and the morphisms (ring homomorphisms) to mappings of the corresponding
srhmsubc $p |- ( U e. V -> J e. ( Subcat ` ( RingCat ` U ) ) ) $=
( vx vy wcel cfv co wa crg crh wceq adantr vg vf vz cringc csubc cssc wbr
chomf cv ccid cop cco wral cin wss vtoclri ssriv sslin mp1i syl5eqss chom
- eleq1w ssid cbs simpl srhmsubclem2 adantrr adantrl ringchom syl5sseqr cvv
+ eleq1w ssid cbs simpl srhmsubclem2 adantrr adantrl ringchom sseqtrrid cvv
eqid cmpo a1i oveq12 adantl simprl simprr ovexd ovmpod homfval ralrimivva
3sstr4d cxp wfn ovex fnmpoi homffn id eqcomd sqxpeqd fneq2d mpbiri inex1g
ringcbas isssc mpbir2and cid cres elin2 sylbi idrhm ringcid simpr 3eltr4d
@@ -738621,7 +738769,7 @@ homomorphisms between unital rings (in the same universe). (Contributed
srhmsubcALTV $p |- ( U e. V -> J e. ( Subcat ` ( RingCatALTV ` U ) ) ) $=
( vx vy wcel cfv co wa crg crh wceq adantr vg vf vz cringcALTV csubc cssc
chomf wbr cv ccid cop cco wral cin wss eleq1w vtoclri ssriv mp1i syl5eqss
- sslin chom ssid cbs eqid simpl srhmsubcALTVlem1 adantrr adantrl syl5sseqr
+ sslin chom ssid cbs eqid simpl srhmsubcALTVlem1 adantrr adantrl sseqtrrid
ringchomALTV cvv oveq12 adantl simprl simprr ovexd ovmpod homfval 3sstr4d
cmpo a1i ralrimivva cxp ovex fnmpoi homffn id ringcbasALTV eqcomd sqxpeqd
wfn fneq2d mpbiri inex1g isssc mpbir2and cid cres elin2 sylbi ringcidALTV
@@ -739444,7 +739592,7 @@ Ordered group sum operation (extension)
$( Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.) $)
gsumsplit2f $p |- ( ph -> ( G gsum ( k e. A |-> X ) ) =
( ( G gsum ( k e. C |-> X ) ) .+ ( G gsum ( k e. D |-> X ) ) ) ) $=
- ( cmpt cgsu cres eqid fmptdf gsumsplit cun ssun1 syl5sseqr resmptd oveq2d
+ ( cmpt cgsu cres eqid fmptdf gsumsplit cun ssun1 sseqtrrid resmptd oveq2d
co ssun2 oveq12d eqtrd ) AHGBJUBZUCUMHUQDUDZUCUMZHUQEUDZUCUMZFUMHGDJUBZUC
UMZHGEJUBZUCUMZFUMABCDEFUQHIKMNOPQAGBJCUQLRUQUEUFSTUAUGAUSVCVAVEFAURVBHUC
AGBDJADEUHZDBDEUIUAUJUKULAUTVDHUCAGBEJAVFEBEDUNUAUJUKULUOUP $.
@@ -743261,7 +743409,7 @@ The natural logarithm on complex numbers (extension)
|-> ( ( F ` x ) / ( G ` x ) ) ) ) $=
( cc wf wcel co cc0 csupp cres cdiv cfv cvv wceq fex eqtrd syl2anc wfn cv
w3a cfdiv cof cmpt 3adant2 3adant1 wa fdivval offres wss ffn 3ad2ant1 cdm
- suppssdm fdm eqcomd 3ad2ant2 syl5sseqr fnssres ovexd inidm adantl offval
+ suppssdm fdm eqcomd 3ad2ant2 sseqtrrid fnssres ovexd inidm adantl offval
fvres ) BFCGZBFDGZBEHZUBZCDUCIZCDJKIZLZDVKLZMUDZIZAVKAUAZCNZVPDNZMIUEVICO
HZDOHZVJVOPVFVHVSVGBFECQUFVGVHVTVFBFEDQUGVSVTUHVJCDVNIVKLVOCDOOUIVKMCDOOU
JRSVIAVKVKVQVRMVKVLVMOOVICBTZVKBUKZVLVKTVFVGWAVHBFCULUMVIDUNZVKBDJUOVGVFB
@@ -743299,7 +743447,7 @@ The natural logarithm on complex numbers (extension)
fdivpm $p |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V )
-> ( F /_f G ) e. ( CC ^pm A ) ) $=
( cc wf wcel w3a cvv cc0 csupp cfdiv wss cpm cnex a1i simp3 fdivmptf cdm
- co suppssdm wceq fdm eqcomd 3ad2ant2 syl5sseqr elpm2r syl22anc ) AEBFZAEC
+ co suppssdm wceq fdm eqcomd 3ad2ant2 sseqtrrid elpm2r syl22anc ) AEBFZAEC
FZADGZHZEIGZUKCJKTZEBCLTZFUNAMUOEANTGUMULOPUIUJUKQABCDRULCSZUNACJUAUJUIAU
PUBUKUJUPAAECUCUDUEUFEAUNUOIDUGUH $.
@@ -743308,7 +743456,7 @@ The natural logarithm on complex numbers (extension)
refdivpm $p |- ( ( F : A --> RR /\ G : A --> RR /\ A e. V )
-> ( F /_f G ) e. ( RR ^pm A ) ) $=
( cr wf wcel w3a cvv cc0 csupp co cfdiv wss cpm reex a1i simp3 refdivmptf
- cdm suppssdm wceq fdm eqcomd 3ad2ant2 syl5sseqr elpm2r syl22anc ) AEBFZAE
+ cdm suppssdm wceq fdm eqcomd 3ad2ant2 sseqtrrid elpm2r syl22anc ) AEBFZAE
CFZADGZHZEIGZUKCJKLZEBCMLZFUNANUOEAOLGUMULPQUIUJUKRABCDSULCTZUNACJUAUJUIA
UPUBUKUJUPAAECUCUDUEUFEAUNUOIDUGUH $.
@@ -748182,7 +748330,7 @@ the relevant induction theorem (such as ~ tfi ) to the other class.
14-Mar-2022.) $)
setrecsres $p |- ( ph -> B = setrecs ( ( F |` ~P B ) ) ) $=
( vx cpw cres csetrecs cv wss cfv wi wa wceq id resss a1i setrecsss wcel
- syl6sseqr sylan9ssr selpw sylbir syl eqid cvv vex setrec1 adantl eqsstr3d
+ syl6sseqr sylan9ssr selpw sylbir syl eqid cvv vex setrec1 adantl eqsstrrd
fvres ex alrimiv setrec2v eqssd ) ABCBGZHZIZABUSCFDAFJZUSKZUTCLZUSKZMFAVA
VCAVANZVBUTURLZUSVDUTBKZVEVBOZVAAUTUSBVAPZAUSCIBAURCEURCKACUQQRSDUAZUBVFU
TUQTVGFBUCUTUQCULUDUEVAVEUSKAVAUTUSURUSUFUTUGTVAFUHRVHUIUJUKUMUNUOVIUP $.
@@ -748320,7 +748468,7 @@ the relevant induction theorem (such as ~ tfi ) to the other class.
elpglem2 $p |- ( ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg )
-> E. x ( x C_ Pg
/\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) $=
- ( c1st cfv cpg wss c2nd wa cv cpw wcel wceq wi fvex unex isseti syl5sseqr
+ ( c1st cfv cpg wss c2nd wa cv cpw wcel wceq wi fvex unex isseti sseqtrrid
cun elpw2 sylibr sseq1 unss syl6bbr biimprd ssun1 id vex ssun2 jca jctird
eximii 19.37iv ) BCDZEFBGDZEFHZAIZEFZUMUPJZKZUNURKZHZHZAUPUMUNRZLZUOVBMAA
VCUMUNBCNBGNOPVDUOUQVAVDUQUOVDUQVCEFUOUPVCEUAUMUNEUBUCUDVDUSUTVDUMUPFUSVD
@@ -750013,7 +750161,7 @@ to Davis and Putnam (1960). (Contributed by David A. Wheeler,
ccrg csubrg cdr resubdrg subrgring cghm cmhm cgim reloggim gimghm gsummhm
ghmmhm subrgsubg simprl mulcomd caofcom fsumrpcl eqeltrd eqeltrrd ringcmn
simprr remulcl cicc simpl iccssioo2 syl6sseq cico ioossico 0lt1 breqtrrid
- eqsstr3i logccv 3adant1 simp21 remulcld resubcld simp22 readdcld eliooord
+ eqsstrri logccv 3adant1 simp21 remulcld resubcld simp22 readdcld eliooord
elioore simpld elrpd simprd 1m0e1 syl6breqr ltsub13d rpaddcl ltnegd mpbid
cmin fvmptd oveq12d negdid eqtr4d 3brtr4d scvxcvx jensen div1d mulcl rpcn
submid ssriv fssd fmptco 3brtr3d lenegcon1d eqbrtrrd efle reeflogd ) ADCE
@@ -750189,7 +750337,6 @@ to Davis and Putnam (1960). (Contributed by David A. Wheeler,
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
-
$( (End of Larry Lesyna's mathbox.) $)
$(
@@ -750198,7 +750345,6 @@ to Davis and Putnam (1960). (Contributed by David A. Wheeler,
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
-
$( Replacement of a nested antecedent with an outer antecedent. Commuted
simplificated form of elimination of a nested antecedent. Also holds
intuitionistically. Polish prefix notation: CCCpqrCsCqr . (Contributed
@@ -750510,7 +750656,6 @@ with simplest antecedents (i.e., in the corresponding ordering of binary
( wi adh-minimp-ax1 adh-minimp-ax2 ax-mp ) AABCZCZAACZCZHGCZAGACCJAGDAGAEFH
IGCCJKCAABEHIGEFF $.
-
$( (End of Adhemar's mathbox.) $)
$( End $[ set-mbox.mm $] $)
";
althtmldef "card" as 'card';
latexdef "card" as "\mathrm{card}";
+htmldef "CHOICE" as "CHOICE";
+ althtmldef "CHOICE" as "CHOICE";
+ latexdef "CHOICE" as "\mathrm{CHOICE}";
htmldef "N." as "
";
althtmldef "N." as 'N';
latexdef "N." as "\mathcal{N}";
@@ -144589,35 +144897,6 @@ with definitions ( ` B ` is the definiendum that one wants to prove
ax-bj-d0cl $a |- DECID ph $.
$}
- $( Equivalence property for negation. TODO: minimize all theorems using
- ~ notbid and ~ notbii . (Contributed by BJ, 27-Jan-2020.)
- (Proof modification is discouraged.) $)
- bj-notbi $p |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) ) $=
- ( wb wn bi2 con3d bi1 impbid ) ABCZADBDIBAABEFIABABGFH $.
-
- ${
- bj-notbii.1 $e |- ( ph <-> ps ) $.
- $( Inference associated with ~ bj-notbi . (Contributed by BJ,
- 27-Jan-2020.) (Proof modification is discouraged.) $)
- bj-notbii $p |- ( -. ph <-> -. ps ) $=
- ( wb wn bj-notbi ax-mp ) ABDAEBEDCABFG $.
- $}
-
- ${
- bj-notbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
- $( Deduction form of ~ bj-notbi . (Contributed by BJ, 27-Jan-2020.)
- (Proof modification is discouraged.) $)
- bj-notbid $p |- ( ph -> ( -. ps <-> -. ch ) ) $=
- ( wb wn bj-notbi syl ) ABCEBFCFEDBCGH $.
- $}
-
- $( Equivalence property for ` DECID ` . TODO: solve conflict with ~ dcbi ;
- minimize ~ dcbii and ~ dcbid with it, as well as theorems using those.
- (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) $)
- bj-dcbi $p |- ( ( ph <-> ps ) -> ( DECID ph <-> DECID ps ) ) $=
- ( wb wn wo wdc id bj-notbi orbi12d df-dc 3bitr4g ) ABCZAADZEBBDZEAFBFLABMNL
- GABHIAJBJK $.
-
${
$d a x ph $.
$( ` Delta0 ` -classical logic and separation implies classical logic.
@@ -144626,9 +144905,9 @@ with definitions ( ` B ` is the definiendum that one wants to prove
bj-d0clsepcl $p |- DECID ph $=
( va vx wdc wex c0 cv wcel wb csn wel wal 0ex bj-snex zfauscl wceq anbi1d
wa eleq1 eximii bibi12d spcv snid biantrur bicomi bibi2i exbii ax-bj-d0cl
- mpbi bj-bd0el bj-dcbi mpbii bj-ex ax-mp ) ADZBEUOFBGZHZAIZUOBUQFFJZHZARZI
- ZBEURBECBKZCGZUSHZARZIZCLVBBACBUSFMNOVGVBCFMVDFPZVCUQVFVAVDFUPSVHVEUTAVDF
- USSQUAUBTVBURBVAAUQAVAUTAFMUCUDUEUFUGUIURUQDUOUQBUJUHUQAUKULTUOBUMUN $.
+ mpbi bj-bd0el dcbiit mpbii bj-ex ax-mp ) ADZBEUOFBGZHZAIZUOBUQFFJZHZARZIZ
+ BEURBECBKZCGZUSHZARZIZCLVBBACBUSFMNOVGVBCFMVDFPZVCUQVFVAVDFUPSVHVEUTAVDFU
+ SSQUAUBTVBURBVAAUQAVAUTAFMUCUDUEUFUGUIURUQDUOUQBUJUHUQAUKULTUOBUMUN $.
$}
@@ -145643,17 +145922,6 @@ truth value (as seen in theorems like ~ exmidexmid ), then this theorem
TCPQUSVASTUAUBUIUCUJBAUDUEUMUOUQUFUGUNUPUHUK $.
$}
- ${
- $d A x $.
- $( A slight strengthening of ~ pwtrufal . (Contributed by Mario Carneiro
- and Jim Kingdon, 12-Sep-2023.) $)
- pwntru $p |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) ) $=
- ( vx c0 csn wss wne wa cv wcel wex simpr neneqd simpll simpl sselda elsni
- wn wceq syl eqeltrrd snssd eqssd ex exlimdv mtod notm0 sylib ) ACDZEZAUHF
- ZGZBHZAIZBJZQACRUKUNAUHRZUKAUHUIUJKLUKUMUOBUKUMUOUKUMGZAUHUIUJUMMUPCAUPUL
- CAUPULUHIULCRUKAUHULUIUJNOULCPSUKUMKTUAUBUCUDUEBAUFUG $.
- $}
-
${
$d N x $.
pwle2.t $e |- T = U_ x e. N ( { x } X. 1o ) $.
@@ -145753,7 +146021,7 @@ a singleton (where the latter can also be thought of as representing
wn 1on ontrci peano2 syl df-2o syl6eleq trsucss mpsyl sseqtr4d word simpr
iftrue nnord ordtr unisucg mpbid wne neqned nnsucpred syl2anc suceq fveq2
3syl sseq12d fveq1 ralbidv df-nninf elrab2 simprbi cbvralv sylib ad2antrr
- rspcdva eqsstr3d eqsstrd peano3 neneqd ad2antlr iffalsed 3sstr4d mpjaodan
+ rspcdva eqsstrrd eqsstrd peano3 neneqd ad2antlr iffalsed 3sstr4d mpjaodan
weq wo exmiddc eqeq1 unieq ifbieq2d fvmptg ralrimiva sylanbrc fmpti ) CHH
BIBJZTKZLXTUAZCJZMZUBZUCZADYCHNZYFOIUDUEZNZEJZUFZYFMZYJYFMZPZEIUGZYFHNYGI
OYFUHZYIYGBIYEOYFYGXTINZUIZYALYDOLONZYRUOUJYRIOYBYCYGIOYCUHZYQYCUKZQYQYBI
@@ -146825,6 +147093,26 @@ Limited Principle of Omniscience (LPO) implies excluded middle, so we
$)
+$(
+=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
+ Testable propositions
+=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
+$)
+
+ $( A proposition is testable iff its negative or double-negative is true.
+ See Chapter 2 [Moschovakis] p. 2.
+
+ We do not formally define testability with a new token, but instead use
+ ` DECID -. ` before the formula in question. For example,
+ ` DECID -. x = y ` corresponds to " ` x = y ` is testable". (Contributed
+ by David A. Wheeler, 13-Aug-2018.) For statements about testable
+ propositions, search for the keyword "testable" in the comments of
+ statements, for instance using the Metamath command "MM> SEARCH *
+ "testable" / COMMENTS". (New usage is discouraged.) $)
+ dftest $p |- ( DECID -. ph <-> ( -. ph \/ -. -. ph ) ) $=
+ ( wn df-dc ) ABC $.
+
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Allsome quantifier
diff --git a/mmil.raw.html b/mmil.raw.html
index eef4545027..99a5b64da2 100644
--- a/mmil.raw.html
+++ b/mmil.raw.html
@@ -10876,15 +10876,7 @@
";
htmldef "b" as "
";
@@ -10194,7 +10199,6 @@ Kalmbach axioms (soundness proofs) that require WOML
htmldef "=" as ' = ';
*/
-
/* Definitions for Unicode version */
althtmldef "a" as 'a';
althtmldef "b" as 'b';
@@ -10290,7 +10294,6 @@ Kalmbach axioms (soundness proofs) that require WOML
*/
/* End of Unicode defintions */
-
latexdef "a" as "a";
latexdef "b" as "b";
latexdef "c" as "c";
@@ -10372,6 +10375,7 @@ Weakly distributive ortholattices (WDOL)
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
WDOL law
@@ -10409,7 +10413,6 @@ Weakly distributive ortholattices (WDOL)
wddi1 $p |- ( ( a ^ ( b v c ) ) == ( ( a ^ b ) v ( a ^ c ) ) ) = 1 $=
( wdcom wfh1 ) ABCABDACDE $.
-
$( The weak distributive law in WDOL. (Contributed by NM, 5-Mar-2006.) $)
wddi2 $p |- ( ( ( a v b ) ^ c ) == ( ( a ^ c ) v ( b ^ c ) ) ) = 1 $=
( wo wa wancom wddi1 w2or wr2 ) ABDZCECJEZACEZBCEZDZJCFKCAEZCBEZDNCABGOLPMC
@@ -10426,7 +10429,6 @@ Weakly distributive ortholattices (WDOL)
( wa wo wa2 wddi3 w2an wr2 ) ABDZCECJEZACEZBCEZDZJCFKCAEZCBEZDNCABGOLPMCAFC
BFHII $.
-
${
wdid0id5.1 $e |- ( a ==0 b ) = 1 $.
$( Show that quantum identity follows from classical identity in a WDOL.
@@ -10533,7 +10535,6 @@ Modular ortholattices (MOL)
mli $p |- ( ( a ^ b ) v c ) = ( a ^ ( b v c ) ) $=
( wa wo ancom ror orcom ml2i 3tr ran ) ABEZCFZCBFZAEZBCFZAEAQENBAEZCFCRFP
MRCABGHRCIABCDJKOQACBILQAGK $.
-
$}
${
@@ -10563,7 +10564,6 @@ Modular ortholattices (MOL)
ml3 $p |- ( a v ( b ^ ( c v a ) ) ) = ( a v ( c ^ ( b v a ) ) ) $=
( wo wa ml3le lebi ) ABCADEDACBADEDABCFACBFG $.
-
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 (Contributed by NM,
15-Mar-2010.) (Revised by NM, 31-Mar-2011.) $)
vneulem1 $p |- ( ( ( x v y ) v u ) ^ w )
@@ -11238,7 +11238,6 @@ Modular ortholattices (MOL)
UDBUBRST $.
$}
-
${
xdp41.c0 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
xdp41.c1 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
@@ -11335,7 +11334,6 @@ Modular ortholattices (MOL)
EBEUQUJVPWLUIVEWLBVPBEUHVLUGUOVFTVAUO $.
$}
-
${
xxdp.1 $e |- p2 =< ( a2 v b2 ) $.
xxdp.c0 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
@@ -11588,7 +11586,6 @@ Modular ortholattices (MOL)
IUOUIUJYFXIGDGVDUQXJYFUOVLYFDXJDGUNVHUMVBWPUPWDVB $.
$}
-
${
xxxdp.c0 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
xxxdp.c1 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
@@ -11726,7 +11723,6 @@ Modular ortholattices (MOL)
GDGVCUPXIYEUNVKYEDXIDGUMVGULVAWOUOWCVA $.
$}
-
${
3dp.c0 $e |- c0 = ( ( a1 v a1 ) ^ ( b1 v b1 ) ) $.
3dp.c1 $e |- c1 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
@@ -11808,7 +11804,6 @@ Modular ortholattices (MOL)
XFFDFVAUNXGYBULVIYBDXGDFUKVEUJUSWMUMWAUS $.
$}
-
${
oadp35lem.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
oadp35lem.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
@@ -11882,7 +11877,6 @@ Modular ortholattices (MOL)
$)
$}
-
${
oadp35.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
oadp35.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
diff --git a/set.mm b/set.mm
index cae1f55ff8..3814e09f3b 100644
--- a/set.mm
+++ b/set.mm
@@ -1854,6 +1854,15 @@ In classical logic (our logic) this is always true. In intuitionistic
( nsyl3 con2i ) CAABCDEFG $.
$}
+ ${
+ nsyl2.1 $e |- ( ph -> -. ps ) $.
+ nsyl2.2 $e |- ( -. ch -> ps ) $.
+ $( A negated syllogism inference. (Contributed by NM, 26-Jun-1994.)
+ (Proof shortened by Wolf Lammen, 14-Nov-2023.) $)
+ nsyl2 $p |- ( ph -> ch ) $=
+ ( wn nsyl3 con4i ) CAABCFDEGH $.
+ $}
+
$( Double negation introduction. Converse of ~ notnotr and one implication
of ~ notnotb . Theorem *2.12 of [WhiteheadRussell] p. 101. This was the
sixth axiom of Frege, specifically Proposition 41 of [Frege1879] p. 47.
@@ -1886,6 +1895,22 @@ In classical logic (our logic) this is always true. In intuitionistic
( wn notnot syl6 con4d ) ABCEZABECIEDCFGH $.
$}
+ $( Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. Its
+ associated inference is ~ con1i . (Contributed by NM, 29-Dec-1992.)
+ (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)
+ con1 $p |- ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) $=
+ ( wn wi id con1d ) ACBDZABGEF $.
+
+ ${
+ con1i.1 $e |- ( -. ph -> ps ) $.
+ $( A contraposition inference. Inference associated with ~ con1 . Its
+ associated inference is ~ mt3 . (Contributed by NM, 3-Jan-1993.)
+ (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (Proof shortened by
+ Wolf Lammen, 19-Jun-2013.) $)
+ con1i $p |- ( -. ps -> ph ) $=
+ ( wn id nsyl2 ) BDZBAGECF $.
+ $}
+
${
mt3d.1 $e |- ( ph -> -. ch ) $.
mt3d.2 $e |- ( ph -> ( -. ps -> ch ) ) $.
@@ -1904,13 +1929,8 @@ In classical logic (our logic) this is always true. In intuitionistic
$}
${
- nsyl2.1 $e |- ( ph -> -. ps ) $.
- nsyl2.2 $e |- ( -. ch -> ps ) $.
- $( A negated syllogism inference. (Contributed by NM, 26-Jun-1994.)
- (Proof shortened by Wolf Lammen, 14-Nov-2023.) $)
- nsyl2 $p |- ( ph -> ch ) $=
- ( wn nsyl3 con4i ) CAABCFDEGH $.
-
+ nsyl2OLD.1 $e |- ( ph -> -. ps ) $.
+ nsyl2OLD.2 $e |- ( -. ch -> ps ) $.
$( Obsolete version of ~ nsyl2 as of 14-Nov-2023. (Contributed by NM,
26-Jun-1994.) (Proof modification is discouraged.)
(New usage is discouraged.) $)
@@ -1918,22 +1938,6 @@ In classical logic (our logic) this is always true. In intuitionistic
( wn wi a1i mt3d ) ACBDCFBGAEHI $.
$}
- $( Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. Its
- associated inference is ~ con1i . (Contributed by NM, 29-Dec-1992.)
- (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)
- con1 $p |- ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) $=
- ( wn wi id con1d ) ACBDZABGEF $.
-
- ${
- con1i.1 $e |- ( -. ph -> ps ) $.
- $( A contraposition inference. Inference associated with ~ con1 . Its
- associated inference is ~ mt3 . (Contributed by NM, 3-Jan-1993.)
- (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (Proof shortened by
- Wolf Lammen, 19-Jun-2013.) $)
- con1i $p |- ( -. ps -> ph ) $=
- ( wn id nsyl2 ) BDZBAGECF $.
- $}
-
${
pm2.24i.1 $e |- ph $.
$( Inference associated with ~ pm2.24 . Its associated inference is
@@ -2137,32 +2141,13 @@ In classical logic (our logic) this is always true. In intuitionistic
( wi a1i pm2.61d ) ABCBCFAEGDH $.
$}
- ${
- ja.1 $e |- ( -. ph -> ch ) $.
- ja.2 $e |- ( ps -> ch ) $.
- $( Inference joining the antecedents of two premises. For partial
- converses, see ~ jarri and ~ jarli . (Contributed by NM, 24-Jan-1993.)
- (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) $)
- ja $p |- ( ( ph -> ps ) -> ch ) $=
- ( wi imim2i pm2.61d1 ) ABFACBCAEGDH $.
- $}
-
- ${
- jad.1 $e |- ( ph -> ( -. ps -> th ) ) $.
- jad.2 $e |- ( ph -> ( ch -> th ) ) $.
- $( Deduction form of ~ ja . (Contributed by Scott Fenton, 13-Dec-2010.)
- (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
- jad $p |- ( ph -> ( ( ps -> ch ) -> th ) ) $=
- ( wi wn com12 ja ) BCGADBCADGABHDEIACDFIJI $.
- $}
-
${
pm2.61i.1 $e |- ( ph -> ps ) $.
pm2.61i.2 $e |- ( -. ph -> ps ) $.
$( Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.)
- (Proof shortened by Wolf Lammen, 12-Sep-2013.) $)
+ (Proof shortened by Wolf Lammen, 19-Nov-2023.) $)
pm2.61i $p |- ps $=
- ( wi id ja ax-mp ) AAEBAFAABDCGH $.
+ ( nsyl4 pm2.18i ) BABBCDEF $.
$}
${
@@ -2197,6 +2182,35 @@ In classical logic (our logic) this is always true. In intuitionistic
( wn wi a1d pm2.61ii pm2.61i ) CDHABCIZDJEADNFKBDNGKLM $.
$}
+ ${
+ ja.1 $e |- ( -. ph -> ch ) $.
+ ja.2 $e |- ( ps -> ch ) $.
+ $( Inference joining the antecedents of two premises. For partial
+ converses, see ~ jarri and ~ jarli . (Contributed by NM, 24-Jan-1993.)
+ (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) $)
+ ja $p |- ( ( ph -> ps ) -> ch ) $=
+ ( wi imim2i pm2.61d1 ) ABFACBCAEGDH $.
+ $}
+
+ ${
+ jad.1 $e |- ( ph -> ( -. ps -> th ) ) $.
+ jad.2 $e |- ( ph -> ( ch -> th ) ) $.
+ $( Deduction form of ~ ja . (Contributed by Scott Fenton, 13-Dec-2010.)
+ (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
+ jad $p |- ( ph -> ( ( ps -> ch ) -> th ) ) $=
+ ( wi wn com12 ja ) BCGADBCADGABHDEIACDFIJI $.
+ $}
+
+ ${
+ pm2.61iOLD.1 $e |- ( ph -> ps ) $.
+ pm2.61iOLD.2 $e |- ( -. ph -> ps ) $.
+ $( Obsolete version of ~ pm2.61i as of 19-Nov-2023. (Contributed by NM,
+ 5-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.)
+ (Proof modification is discouraged.) (New usage is discouraged.) $)
+ pm2.61iOLD $p |- ps $=
+ ( wi id ja ax-mp ) AAEBAFAABDCGH $.
+ $}
+
$( Weak Clavius law. If a formula implies its negation, then it is false. A
form of "reductio ad absurdum", which can be used in proofs by
contradiction. Theorem *2.01 of [WhiteheadRussell] p. 100. Provable in
@@ -2495,8 +2509,8 @@ statements not containing the new symbol (or new combination) should
~ dfbi1ALT for an unusual version proved directly from axioms.
(Contributed by NM, 29-Dec-1992.) $)
dfbi1 $p |- ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) $=
- ( wb wi wn df-bi pm2.21 mt3 impbi impi impbii ) ABCZABDZBADZEDEZLODZPOLDEZD
- ABFPQGHMNLABIJK $.
+ ( wb wi wn df-bi impbi con3rr3 mt3 ) ABCZABDBADEDEZCZJKDZKJDZEDABFMNLJKGHI
+ $.
$( Alternate proof of ~ dfbi1 . This proof, discovered by Gregory Bush on
8-Mar-2004, has several curious properties. First, it has only 17 steps
@@ -3486,7 +3500,7 @@ intermediate steps that are essentially incomprehensible to humans (other
$( A mixed syllogism inference from an implication and a biconditional.
(Contributed by Wolf Lammen, 16-Dec-2013.) $)
sylnib $p |- ( ph -> -. ch ) $=
- ( wb a1i mtbid ) ABCDBCFAEGH $.
+ ( biimpri nsyl ) ABCDBCEFG $.
$}
${
@@ -4304,7 +4318,7 @@ use disjunction (although this is not required since definitions are
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
imp5g $p |- ( ( ph /\ ps ) -> ( ( ( ch /\ th ) /\ ta ) -> et ) ) $=
- ( wa wi imp imp4c ) ABHCDEFABCDEFIIIGJK $.
+ ( wa wi imp4b impd ) ABHCDHEFABCDEFIGJK $.
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
imp55 $p |- ( ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) /\ ta ) -> et ) $=
@@ -4468,7 +4482,7 @@ use disjunction (although this is not required since definitions are
$( Deduction commuting conjunction in antecedent. (Contributed by NM,
12-Dec-2004.) $)
ancomsd $p |- ( ph -> ( ( ch /\ ps ) -> th ) ) $=
- ( wa ancom syl5bi ) CBFBCFADCBGEH $.
+ ( expcomd impd ) ACBDABCDEFG $.
$}
${
@@ -4957,7 +4971,7 @@ use disjunction (although this is not required since definitions are
$( A wff is equivalent to its conjunction with truth. (Contributed by NM,
3-Aug-1994.) $)
biantrur $p |- ( ps <-> ( ph /\ ps ) ) $=
- ( wa wb ibar ax-mp ) ABABDECABFG $.
+ ( biantru biancomi ) BABABCDE $.
$}
${
@@ -5005,7 +5019,7 @@ use disjunction (although this is not required since definitions are
Carneiro, 11-Sep-2015.) (Proof shortened by Wolf Lammen,
19-Jan-2020.) $)
rbaibr $p |- ( ch -> ( ps <-> ph ) ) $=
- ( wa iba syl6bbr ) CBBCEACBFDG $.
+ ( biancomi baibr ) ACBACBDEF $.
$( Move conjunction outside of biconditional. (Contributed by Mario
Carneiro, 11-Sep-2015.) (Proof shortened by Wolf Lammen,
@@ -5024,7 +5038,7 @@ use disjunction (although this is not required since definitions are
$( Move conjunction outside of biconditional. (Contributed by Mario
Carneiro, 11-Sep-2015.) $)
rbaibd $p |- ( ( ph /\ th ) -> ( ps <-> ch ) ) $=
- ( wa iba bicomd sylan9bb ) ABCDFZDCEDCJDCGHI $.
+ ( biancomd baibd ) ABDCABDCEFG $.
$}
${
@@ -5167,7 +5181,7 @@ use disjunction (although this is not required since definitions are
Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct
reversed). (Contributed by NM, 1-Dec-2003.) $)
pm4.71ri $p |- ( ph <-> ( ps /\ ph ) ) $=
- ( wi wa wb pm4.71r mpbi ) ABDABAEFCABGH $.
+ ( pm4.71i biancomi ) ABAABCDE $.
$}
${
@@ -5182,7 +5196,7 @@ use disjunction (although this is not required since definitions are
Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed
by NM, 10-Feb-2005.) $)
pm4.71rd $p |- ( ph -> ( ps <-> ( ch /\ ps ) ) ) $=
- ( wi wa wb pm4.71r sylib ) ABCEBCBFGDBCHI $.
+ ( pm4.71d biancomd ) ABCBABCDEF $.
$}
$( Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM,
@@ -5250,7 +5264,7 @@ use disjunction (although this is not required since definitions are
3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) $)
pm5.3 $p |- ( ( ( ph /\ ps ) -> ch ) <->
( ( ph /\ ps ) -> ( ph /\ ch ) ) ) $=
- ( wa wi impexp imdistan bitri ) ABDZCEABCEEIACDEABCFABCGH $.
+ ( wa simpl biantrurd pm5.74i ) ABDZCACDHACABEFG $.
$( Distribution of implication over biconditional. Theorem *5.32 of
[WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) $)
@@ -5393,7 +5407,7 @@ use disjunction (although this is not required since definitions are
$( Syllogism inference combined with a biconditional. (Contributed by BJ,
25-Apr-2019.) $)
sylanblrc $p |- ( ph -> th ) $=
- ( wa biimpri sylancl ) ABCDEFDBCHGIJ $.
+ ( a1i sylanbrc ) ABCDECAFHGI $.
$}
${
@@ -5629,7 +5643,7 @@ use disjunction (although this is not required since definitions are
$( Deduction joining nested implications to form implication of
conjunctions. (Contributed by NM, 29-Feb-1996.) $)
im2anan9 $p |- ( ( ph /\ th ) -> ( ( ps /\ ta ) -> ( ch /\ et ) ) ) $=
- ( wa wi adantr adantl anim12d ) ADIBCEFABCJDGKDEFJAHLM $.
+ ( wa adantrd adantld anim12ii ) ABEICDFABCEGJDEFBHKL $.
$( Deduction joining nested implications to form implication of
conjunctions. (Contributed by NM, 29-Feb-1996.) $)
@@ -5723,7 +5737,7 @@ use disjunction (although this is not required since definitions are
conjunct on the right-hand side of an equivalence. Deduction form.
(Contributed by Peter Mazsa, 22-May-2021.) $)
anbi1cd $p |- ( ph -> ( ( th /\ ps ) <-> ( ch /\ th ) ) ) $=
- ( wb wa anbi2 biancomd syl ) ABCFZDBGZCDGFEKLCDBCDHIJ $.
+ ( wa anbi2d biancomd ) ADBFCDABCDEGH $.
$}
$( Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM,
@@ -5766,14 +5780,14 @@ use disjunction (although this is not required since definitions are
$( Swap two conjuncts. (Contributed by Peter Mazsa, 18-Sep-2022.) $)
an21 $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
- ( wa ancom anbi1i anass bitri ) ABDZCDBADZCDBACDDIJCABEFBACGH $.
+ ( wa biid bianassc bicomi ) BACDZDABDCDHACBHEFG $.
$( Swap two conjuncts. Note that the first digit (1) in the label refers to
the outer conjunct position, and the next digit (2) to the inner conjunct
position. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Peter
Mazsa, 18-Sep-2022.) $)
an12 $p |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
- ( wa anass an21 bitr3i ) ABCDDABDCDBACDDABCEABCFG $.
+ ( wa ancom bianass biancomi ) ABCDZDBACDHCBABCEFG $.
$( A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof
shortened by Wolf Lammen, 25-Dec-2012.) $)
@@ -5835,8 +5849,7 @@ use disjunction (although this is not required since definitions are
$( Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.) $)
an4 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
( ( ph /\ ch ) /\ ( ps /\ th ) ) ) $=
- ( wa an12 anbi2i anass 3bitr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHAC
- LHI $.
+ ( wa anass an12 bianass bitri ) ABECDEZEABJEZEACEBDEZEABJFKCLABCDGHI $.
$( Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.) $)
an42 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
@@ -6014,7 +6027,7 @@ use disjunction (although this is not required since definitions are
sylanl2.2 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
$( A syllogism inference. (Contributed by NM, 1-Jan-2005.) $)
sylanl2 $p |- ( ( ( ps /\ ph ) /\ th ) -> ta ) $=
- ( wa anim2i sylan ) BAHBCHDEACBFIGJ $.
+ ( adantl syldanl ) BACDEACBFHGI $.
$}
${
@@ -6040,7 +6053,7 @@ use disjunction (although this is not required since definitions are
$( A syllogism deduction combined with conjoining antecedents.
(Contributed by Alan Sare, 28-Oct-2011.) $)
syl6an $p |- ( ph -> ( ch -> ta ) ) $=
- ( wa jctild syl6 ) ACBDIEACDBGFJHK $.
+ ( ex sylsyld ) ABCDEFGBDEHIJ $.
$}
${
@@ -6061,7 +6074,7 @@ use disjunction (although this is not required since definitions are
$( ~ syl2an with antecedents in standard conjunction form. (Contributed by
Alan Sare, 27-Aug-2016.) $)
syl2an2 $p |- ( ( ch /\ ph ) -> ta ) $=
- ( wa syl2an anabss7 ) CAEABDECAIFGHJK $.
+ ( wa adantl syl2anc ) CAIBDEABCFJGHK $.
$}
${
@@ -6267,7 +6280,7 @@ use disjunction (although this is not required since definitions are
$( Detach truth from conjunction in biconditional. Deduction form.
(Contributed by Peter Mazsa, 24-Sep-2022.) $)
mpbiran2d $p |- ( ph -> ( ps <-> ch ) ) $=
- ( wa biantrud bitr4d ) ABCDGCFADCEHI $.
+ ( biancomd mpbirand ) ABDCEABDCFGH $.
$}
${
@@ -6285,7 +6298,7 @@ use disjunction (although this is not required since definitions are
$( Detach truth from conjunction in biconditional. (Contributed by NM,
22-Feb-1996.) $)
mpbiran2 $p |- ( ph <-> ps ) $=
- ( wa biantru bitr4i ) ABCFBECBDGH $.
+ ( biancomi mpbiran ) ACBDACBEFG $.
$}
${
@@ -6402,7 +6415,7 @@ use disjunction (although this is not required since definitions are
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
19-Oct-1999.) $)
ad2antrl $p |- ( ( ch /\ ( ph /\ th ) ) -> ps ) $=
- ( wa adantr adantl ) ADFBCABDEGH $.
+ ( adantl adantrr ) CABDABCEFG $.
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
19-Oct-1999.) $)
@@ -6530,7 +6543,7 @@ use disjunction (although this is not required since definitions are
$( Deduction adding 1 conjunct to antecedent. (Contributed by Alan Sare,
17-Oct-2017.) $)
adantl3r $p |- ( ( ( ( ( ph /\ et ) /\ ps ) /\ ch ) /\ th ) -> ta ) $=
- ( wa wi ex adantllr imp ) AFHBHCHDEABCDEIFABHCHDEGJKL $.
+ ( wa id adantlr sylanl1 ) AFHBHABHZCDEABLFLIJGK $.
$}
${
@@ -6538,7 +6551,7 @@ use disjunction (although this is not required since definitions are
$( Deduction adding conjuncts to antecedent. (Contributed by Alan Sare,
17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
ad4ant13 $p |- ( ( ( ( ph /\ th ) /\ ps ) /\ ta ) -> ch ) $=
- ( wa adantlr adantr ) ADGBGCEABCDFHI $.
+ ( wa adantr adantllr ) ABECDABGCEFHI $.
$( Deduction adding conjuncts to antecedent. (Contributed by Alan Sare,
17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
@@ -6548,12 +6561,12 @@ use disjunction (although this is not required since definitions are
$( Deduction adding conjuncts to antecedent. (Contributed by Alan Sare,
17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
ad4ant23 $p |- ( ( ( ( th /\ ph ) /\ ps ) /\ ta ) -> ch ) $=
- ( wa adantll adantr ) DAGBGCEABCDFHI $.
+ ( wa adantr adantlll ) ABECDABGCEFHI $.
$( Deduction adding conjuncts to antecedent. (Contributed by Alan Sare,
17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
ad4ant24 $p |- ( ( ( ( th /\ ph ) /\ ta ) /\ ps ) -> ch ) $=
- ( wa adantll adantlr ) DAGBCEABCDFHI $.
+ ( adantlr adantlll ) AEBCDABCEFGH $.
$}
${
@@ -7102,7 +7115,7 @@ _praeclarum theorema_ (splendid theorem). (Contributed by NM,
$( Syllogism combined with contraction. (Contributed by Jeff Hankins,
1-Aug-2009.) $)
syl12anc $p |- ( ph -> ta ) $=
- ( wa jca32 syl ) ABCDJJEABCDFGHKIL $.
+ ( wa jca syl2anc ) ABCDJEFACDGHKIL $.
$}
${
@@ -7110,7 +7123,7 @@ _praeclarum theorema_ (splendid theorem). (Contributed by NM,
$( Syllogism combined with contraction. (Contributed by Jeff Hankins,
1-Aug-2009.) $)
syl21anc $p |- ( ph -> ta ) $=
- ( wa jca31 syl ) ABCJDJEABCDFGHKIL $.
+ ( wa jca syl2anc ) ABCJDEABCFGKHIL $.
$}
${
@@ -7134,7 +7147,7 @@ _praeclarum theorema_ (splendid theorem). (Contributed by NM,
unification theorem uses left-nested conjunction. (Contributed by Alan
Sare, 17-Oct-2017.) $)
syl1111anc $p |- ( ph -> et ) $=
- ( wi exp41 syl3c mpd ) AEFJABCDEFLGHIBCDEFKMNO $.
+ ( wa jca syl21anc ) ABCLDEFABCGHMIJKN $.
$}
${
@@ -7145,7 +7158,7 @@ _praeclarum theorema_ (splendid theorem). (Contributed by NM,
mpsyl4anc.5 $e |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ta ) -> et ) $.
$( An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) $)
mpsyl4anc $p |- ( th -> et ) $=
- ( wi exp41 mp2 mpsyl ) CDEFIJABCEFLLGHABCEFKMNO $.
+ ( a1i syl1111anc ) DABCEFADGLBDHLCDILJKM $.
$}
$( Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM,
@@ -7170,8 +7183,8 @@ _praeclarum theorema_ (splendid theorem). (Contributed by NM,
19-Jan-2020.) $)
a2and $p |- ( ph -> ( ( ( ps /\ ch ) -> ta )
-> ( ( ps /\ rh ) -> th ) ) ) $=
- ( wa wi expd imdistand imp embantd ex com23 ) ABFIZBCIZEJZDAQSDJAQIREDAQR
- ABFCABFCHKLMAQEDJGMNOP $.
+ ( wa wi expd imdistand imim1 com3l syl6c com23 ) ABFIZBCIZEJZDAQEDJZRSDJG
+ ABFCABFCHKLSTRDREDMNOP $.
$}
${
@@ -7836,7 +7849,7 @@ an illustration of the conservativity of definitions (definitions do not
$( Theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by NM,
3-Jan-2005.) $)
pm5.14 $p |- ( ( ph -> ps ) \/ ( ps -> ch ) ) $=
- ( wi wn conax1 pm2.21d orri ) ABDZBCDIEBCABFGH $.
+ ( wi pm2.521g orri ) ABDBCDABCEF $.
$( Theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by NM,
3-Jan-2005.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) $)
@@ -16379,7 +16392,6 @@ modal logic (the other standard formulation being ~ extru ). Note: This
( spvv mpg ) ABCABCDEGFH $.
$}
-
${
$d x y $.
$( Version of ~ equs4 with a disjoint variable condition, which requires
@@ -18388,8 +18400,15 @@ Converse of the inference rule of (universal) generalization ~ ax-gen .
$( Consequence of the definition of not-free. (Contributed by Mario
Carneiro, 26-Sep-2016.) ~ df-nf changed. (Revised by Wolf Lammen,
- 11-Sep-2021.) $)
+ 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.) $)
nf5r $p |- ( F/ x ph -> ( ph -> A. x ph ) ) $=
+ ( wex wnf wal 19.8a id nfrd syl5 ) AABCABDZABEABFJABJGHI $.
+
+ $( Obsolete version of ~ nfrd as of 23-Nov-2023. (Contributed by Mario
+ Carneiro, 26-Sep-2016.) ~ df-nf changed. (Revised by Wolf Lammen,
+ 11-Sep-2021.) (Proof modification is discouraged.)
+ (New usage is discouraged.) $)
+ nf5rOLD $p |- ( F/ x ph -> ( ph -> A. x ph ) ) $=
( wex wnf wal 19.8a wi df-nf biimpi syl5 ) AABCZABDZABEZABFLKMGABHIJ $.
${
@@ -32993,13 +33012,6 @@ Such interpretation is rarely needed (see also ~ df-ral ). (Contributed
elabg $p |- ( A e. V -> ( A e. { x | ph } <-> ps ) ) $=
( cv cab wcel wb nfab1 nfel2 nfv nfbi wceq eleq1 bibi12d abid vtoclg1f )
CGZACHZIZAJDUAIZBJCDEUCBCCDUAACKLBCMNTDOUBUCABTDUAPFQACRS $.
-
- $( Obsolete version of ~ elabg as of 23-Nov-2022. Membership in a class
- abstraction, using implicit substitution. Compare Theorem 6.13 of
- [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
- (Proof modification is discouraged.) (New usage is discouraged.) $)
- elabgOLD $p |- ( A e. V -> ( A e. { x | ph } <-> ps ) ) $=
- ( nfcv nfv elabgf ) ABCDECDGBCHFI $.
$}
${
@@ -33154,13 +33166,6 @@ Such interpretation is rarely needed (see also ~ df-ral ). (Contributed
cv ) DACEGZHDIHZDEHZBJZDTKUBUABDEKLCSZEHZAJUCCDTIUDDMUEUBABUDDENFOACEPQR
$.
- $( Obsolete version of ~ elrab as of 23-Nov-2022. Membership in a
- restricted class abstraction, using implicit substitution. (Contributed
- by NM, 21-May-1999.) (Proof modification is discouraged.)
- (New usage is discouraged.) $)
- elrabOLD $p |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) $=
- ( nfcv nfv elrabf ) ABCDECDGCEGBCHFI $.
-
$( Membership in a restricted class abstraction, using implicit
substitution. (Contributed by NM, 5-Oct-2006.) $)
elrab3 $p |- ( A e. B -> ( A e. { x e. B | ph } <-> ps ) ) $=
@@ -36403,7 +36408,7 @@ technically classes despite morally (and provably) being sets, like ` 1 `
eqsstr3.2 $e |- B C_ C $.
$( Substitution of equality into a subclass relationship. (Contributed by
NM, 19-Oct-1999.) $)
- eqsstr3i $p |- A C_ C $=
+ eqsstrri $p |- A C_ C $=
( eqcomi eqsstri ) ABCBADFEG $.
$}
@@ -36435,11 +36440,11 @@ technically classes despite morally (and provably) being sets, like ` 1 `
$}
${
- eqsstr3d.1 $e |- ( ph -> B = A ) $.
- eqsstr3d.2 $e |- ( ph -> B C_ C ) $.
+ eqsstrrd.1 $e |- ( ph -> B = A ) $.
+ eqsstrrd.2 $e |- ( ph -> B C_ C ) $.
$( Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.) $)
- eqsstr3d $p |- ( ph -> A C_ C ) $=
+ eqsstrrd $p |- ( ph -> A C_ C ) $=
( eqcomd eqsstrd ) ABCDACBEGFH $.
$}
@@ -36571,11 +36576,11 @@ technically classes despite morally (and provably) being sets, like ` 1 `
$}
${
- syl5sseqr.1 $e |- B C_ A $.
- syl5sseqr.2 $e |- ( ph -> C = A ) $.
+ sseqtrrid.1 $e |- B C_ A $.
+ sseqtrrid.2 $e |- ( ph -> C = A ) $.
$( Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.) $)
- syl5sseqr $p |- ( ph -> B C_ C ) $=
+ sseqtrrid $p |- ( ph -> B C_ C ) $=
( wss a1i sseqtr4d ) ACBDCBGAEHFI $.
$}
@@ -37945,7 +37950,7 @@ technically classes despite morally (and provably) being sets, like ` 1 `
Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM,
27-Apr-1994.) $)
inss2 $p |- ( A i^i B ) C_ B $=
- ( cin incom inss1 eqsstr3i ) ABCBACBBADBAEF $.
+ ( cin incom inss1 eqsstrri ) ABCBACBBADBAEF $.
${
$d x A $. $d x B $. $d x C $.
@@ -43942,7 +43947,7 @@ either the empty set or a singleton ( ~ uniintsn ). (Contributed by NM,
30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
unissint $p |- ( U. A C_ |^| A <-> ( A = (/) \/ U. A = |^| A ) ) $=
( cuni cint wss c0 wo wn wa simpl wne df-ne intssuni sylbir adantl eqssd ex
- wceq orrd cvv ssv int0 sseqtr4i inteq syl5sseqr eqimss jaoi impbii ) ABZACZ
+ wceq orrd cvv ssv int0 sseqtr4i inteq sseqtrrid eqimss jaoi impbii ) ABZACZ
DZAEQZUHUIQZFUJUKULUJUKGZULUJUMHUHUIUJUMIUMUIUHDZUJUMAEJUNAEKALMNOPRUKUJULU
KECZUHUIUHSUOUHTUAUBAEUCUDUHUIUEUFUG $.
@@ -47505,8 +47510,18 @@ holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc]
$( In the Separation Scheme ~ zfauscl , we require that ` y ` not occur in
` ph ` (which can be generalized to "not be free in"). Here we show
special cases of ` A ` and ` ph ` that result in a contradiction if that
- requirement is not met. (Contributed by NM, 8-Feb-2006.) $)
+ requirement is not met. (Contributed by NM, 8-Feb-2006.) (Proof
+ shortened by BJ, 18-Nov-2023.) $)
notzfaus $p |- -. E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) $=
+ ( cv wcel wa wb wal wn wex c0 wne csn 0ex snnz eqnetri mpbi ibar syl5rbbr
+ n0 pm5.19 bibi2d mtbiri eximii exnal nex ) BGZCGHZUJDHZAIZJZBKZCUNLZBMUOL
+ ULUPBDNOULBMDNPNENQRSBDUCTULUNUKUKLZJUKUDULUMUQUKUQAULUMFULAUAUBUEUFUGUNB
+ UHTUI $.
+
+ $( Obsolete proof of ~ notzfaus as of 18-Nov-2023. (Contributed by NM,
+ 8-Feb-2006.) (Proof modification is discouraged.)
+ (New usage is discouraged.) $)
+ notzfausOLD $p |- -. E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) $=
( cv wcel wa wb wal wn wex c0 wne csn 0ex snnz eqnetri mpbi n0 biimt iman
wi anbi2i xchbinxr syl6bb xor3 sylibr eximii exnal nex ) BGZCGHZUMDHZAIZJ
ZBKZCUQLZBMURLUOUSBDNOUOBMDNPNENQRSBDUATUOUNUPLZJUSUOUNUOUNUDZUTUOUNUBVAU
@@ -52956,7 +52971,7 @@ Contrast with domain (defined in ~ df-dm ). For alternate definitions,
$( The domain of the universe is the universe. (Contributed by NM,
8-Aug-2003.) $)
dmv $p |- dom _V = _V $=
- ( cvv cdm ssv cid dmi wss dmss ax-mp eqsstr3i eqssi ) ABZAKCADBZKEDAFLKFD
+ ( cvv cdm ssv cid dmi wss dmss ax-mp eqsstrri eqssi ) ABZAKCADBZKEDAFLKFD
CDAGHIJ $.
$}
@@ -53873,7 +53888,7 @@ the restriction (of the relation) to the singleton containing this
$( A diagonal set as a subset of a Cartesian product. (Contributed by
Thierry Arnoux, 29-Dec-2019.) (Proof shortened by BJ, 9-Sep-2022.) $)
idssxp $p |- ( _I |` A ) C_ ( A X. A ) $=
- ( cid cres cxp cin idinxpres inss2 eqsstr3i ) BACBAADZEIAFBIGH $.
+ ( cid cres cxp cin idinxpres inss2 eqsstrri ) BACBAADZEIAFBIGH $.
${
$d A x y $.
@@ -54866,7 +54881,7 @@ the restriction (of the relation) to the singleton containing this
relationship for it components. (Contributed by NM, 17-Dec-2008.) $)
ssxpb $p |- ( ( A X. B ) =/= (/) -> ( ( A X. B ) C_ ( C X. D ) <->
( A C_ C /\ B C_ D ) ) ) $=
- ( cxp c0 wne wss wa cdm wceq xpnz dmxp adantl sylbir adantr eqsstr3d syl6ss
+ ( cxp c0 wne wss wa cdm wceq xpnz dmxp adantl sylbir adantr eqsstrrd syl6ss
dmss crn dmxpss rnxp rnss rnxpss jca ex xpss12 impbid1 ) ABEZFGZUICDEZHZACH
ZBDHZIZUJULUOUJULIZUMUNUPAUKJZCUPAUIJZUQUJURAKZULUJAFGZBFGZIZUSABLZVAUSUTAB
MNOPULURUQHUJUIUKSNQCDUARUPBUKTZDUPBUITZVDUJVEBKZULUJVBVFVCUTVFVAABUBPOPULV
@@ -59996,7 +60011,7 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ).
( F |` B ) : B -onto-> D ) ->
( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) ) $=
( ccnv wfun cres wfo cdif cima wf1o wceq wb ax-mp crn df-ima wa forn syl5eq
- w3a cdm wss fofun difss fof fdmd syl5sseqr fores syl2anc cin resres reseq2i
+ w3a cdm wss fofun difss fof fdmd sseqtrrid fores syl2anc cin resres reseq2i
indif eqtri foeq1 rneqi 3eqtr4i foeq3 bitri funres11 dff1o3 biimpri syl2anr
sylib 3adant3 anim12i imadif difeq12 sylan9eq sylan2 3impb f1oeq3d mpbid )
EFGZACEAHZIZBDEBHZIZUAZABJZEWAKZEWAHZLZWACDJZWCLVOVQWDVSVQWAWBWCIZWCFGZWDVO
@@ -62051,7 +62066,7 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ).
(Contributed by FL, 25-Jan-2007.) $)
fimacnv $p |- ( F : A --> B -> ( `' F " B ) = A ) $=
( wf ccnv cima crn imassrn cdm dfdm4 fdm ssid syl6eqss syl5eqssr syl5ss wss
- fimass wfun wb ffun syl5sseqr funimass3 syl2anc mpbid eqssd ) ABCDZCEZBFZAU
+ fimass wfun wb ffun sseqtrrid funimass3 syl2anc mpbid eqssd ) ABCDZCEZBFZAU
FUHUGGZAUGBHUFUICIZACJUFUJAAABCKZALZMNOUFCAFBPZAUHPZABCAQUFCRAUJPUMUNSABCTU
FAAUJULUKUAABCUBUCUDUE $.
@@ -63129,7 +63144,7 @@ from the cartesian product of two singletons onto a singleton (case where
Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro,
22-Dec-2016.) $)
fnsnr $p |- ( F Fn { A } -> ( B e. F -> B = <. A , ( F ` A ) >. ) ) $=
- ( csn wfn wcel cfv cop wceq cres fnresdm wfun fnfun funressn eqsstr3d sseld
+ ( csn wfn wcel cfv cop wceq cres fnresdm wfun fnfun funressn eqsstrrd sseld
wss syl elsni syl6 ) CADZEZBCFBAACGHZDZFBUCIUBCUDBUBCCUAJZUDUACKUBCLUEUDQUA
CMACNROPBUCST $.
@@ -69099,7 +69114,7 @@ associative structure (such as a group). (Contributed by NM,
ofrfval $p |- ( ph -> ( F oR R G <-> A. x e. S C R D ) ) $=
( wcel vf vg cofr wbr cv cfv cdm cin wral cvv wb fnex syl2anc wceq dmeq
wfn wa ineqan12d fveq1 breqan12d raleqbidv df-ofr brabga ineq12d syl6eq
- fndmd raleqdv inss1 eqsstr3i sseli sylan2 inss2 breq12d ralbidva 3bitrd
+ fndmd raleqdv inss1 eqsstrri sseli sylan2 inss2 breq12d ralbidva 3bitrd
) AIJGUCZUDZBUEZIUFZVRJUFZGUDZBIUGZJUGZUHZUIZWABHUIEFGUDZBHUIAIUJTZJUJT
ZVQWEUKAICUPCKTWGMOCKIULUMAJDUPDLTWHNPDLJULUMVRUAUEZUFZVRUBUEZUFZGUDZBW
IUGZWKUGZUHZUIWEUAUBIJVPUJUJWIIUNZWKJUNZUQWMWABWPWDWQWRWNWBWOWCWIIUOWKJ
@@ -69115,7 +69130,7 @@ associative structure (such as a group). (Contributed by NM,
Carneiro, 20-Jul-2014.) $)
ofval $p |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) ) $=
( cfv vx wcel wa cof co cv cmpt wceq eqidd offval fveq1d adantr oveq12d
- fveq2 eqid ovex fvmpt adantl inss1 eqsstr3i sseli sylan2 inss2 3eqtrd
+ fveq2 eqid ovex fvmpt adantl inss1 eqsstrri sseli sylan2 inss2 3eqtrd
cin ) ALGUBZUCZLHIFUDUEZTZLUAGUAUFZHTZVJITZFUEZUGZTZLHTZLITZFUEZDEFUEAV
IVOUHVFALVHVNAUABCVKVLFGHIJKMNOPQAVJBUBUCVKUIAVJCUBUCVLUIUJUKULVFVOVRUH
AUALVMVRGVNVJLUHVKVPVLVQFVJLHUNVJLIUNUMVNUOVPVQFUPUQURVGVPDVQEFVFALBUBV
@@ -69125,7 +69140,7 @@ associative structure (such as a group). (Contributed by NM,
Carneiro, 28-Jul-2014.) $)
ofrval $p |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D ) $=
( wcel vx cofr wbr w3a cfv wa cv wral eqidd ofrfval biimpa wceq breq12d
- wi fveq2 rspccv syl 3impia simp1 cin inss1 eqsstr3i simp3 syl2anc inss2
+ wi fveq2 rspccv syl 3impia simp1 cin inss1 eqsstrri simp3 syl2anc inss2
sseldi 3brtr3d ) AHIFUBUCZLGTZUDZLHUEZLIUEZDEFAVHVIVKVLFUCZAVHUFUAUGZHU
EZVNIUEZFUCZUAGUHZVIVMUNAVHVRAUABCVOVPFGHIJKMNOPQAVNBTUFVOUIAVNCTUFVPUI
UJUKVQVMUALGVNLULVOVKVPVLFVNLHUOVNLIUOUMUPUQURVJALBTVKDULAVHVIUSZVJGBLG
@@ -69194,7 +69209,7 @@ associative structure (such as a group). (Contributed by NM,
$( The function operation produces a function. (Contributed by Mario
Carneiro, 20-Jul-2014.) $)
off $p |- ( ph -> ( F oF R G ) : C --> U ) $=
- ( vz cv cfv co cof ffnd wcel wa eqidd offval wral wf inss1 eqsstr3i sseli
+ ( vz cv cfv co cof ffnd wcel wa eqidd offval wral wf inss1 eqsstrri sseli
cin ffvelrn syl2an inss2 ralrimivva adantr ovrspc2v syl21anc fmpt3d ) AUA
FUAUBZKUCZVELUCZGUDZJKLGUEUDAUADEVFVGGFKLMNADHKPUFAEILQUFRSTAVEDUGZUHVFUI
AVEEUGZUHVGUIUJAVEFUGZUHVFHUGZVGIUGZBUBCUBGUDJUGZCIUKBHUKZVHJUGADHKULVIVL
@@ -69213,7 +69228,7 @@ associative structure (such as a group). (Contributed by NM,
of the operation itself. (Contributed by Mario Carneiro,
15-Sep-2014.) $)
ofres $p |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) ) $=
- ( vx co cfv wcel cvv wfn cof cv cmpt cres eqidd offval wss inss1 eqsstr3i
+ ( vx co cfv wcel cvv wfn cof cv cmpt cres eqidd offval wss inss1 eqsstrri
wa cin fnssres sylancl inss2 ssexg sylancr inidm wceq fvres adantl eqtr4d
) AFGEUAZPODOUBZFQZVCGQZEPUCFDUDZGDUDZVBPAOBCVDVEEDFGHIJKLMNAVCBRUJVDUEAV
CCRUJVEUEUFAODDVDVEEDVFVGSSAFBTDBUGZVFDTJDBCUKZBNBCUHUIZBDFULUMAGCTDCUGVG
@@ -69288,7 +69303,7 @@ associative structure (such as a group). (Contributed by NM,
ofco $p |- ( ph ->
( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) ) $=
( cfv vx vy cof co ccom cv cmpt ffvelrnda feqmptd wcel eqidd offval fveq2
- wa oveq12d fmptco wfn wf wss cin inss1 eqsstr3i fss sylancl fnfco syl2anc
+ wa oveq12d fmptco wfn wf wss cin inss1 eqsstrri fss sylancl fnfco syl2anc
wceq inss2 inidm ffnd fvco2 sylan eqtr4d ) AGHFUCZUDZIUEUAEUAUFZITZGTZVQH
TZFUDZUGGIUEZHIUEZVNUDAUAUBEDVQUBUFZGTZWCHTZFUDVTIVOAEDVPIOUHAUAEDIOUIAUB
BCWDWEFDGHJKMNPQSAWCBUJUNWDUKAWCCUJUNWEUKULWCVQVGWDVRWEVSFWCVQGUMWCVQHUMU
@@ -74888,7 +74903,7 @@ any sets (which usually are functions) and any element (even not
by AV, 28-May-2019.) $)
suppssof1 $p |- ( ph -> ( ( A oF O B ) supp Z ) C_ L ) $=
( vx cof co csupp cv cfv cmpt ffnd inidm wcel eqidd offval oveq1d feqmptd
- wa cvv eqsstr3d fvexd ffvelrnda suppssov1 eqsstrd ) ACDIUAUBZMUCUBTETUDZC
+ wa cvv eqsstrrd fvexd ffvelrnda suppssov1 eqsstrd ) ACDIUAUBZMUCUBTETUDZC
UEZVBDUEZIUBUFZMUCUBHAVAVEMUCATEEVCVDIECDKKAEJCPUGAEFDQUGRREUHAVBEUIUNZVC
UJVFVDUJUKULATBVCVDEFHIUOGLMATEVCUFZLUCUBCLUCUBHACVGLUCATEJCPUMULNUPOVFVB
CUQAEFVBDQURSUSUT $.
@@ -75527,7 +75542,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
( F o. ( x e. ( ( _V X. _V ) u. { (/) } ) |-> U. `' { x } ) ) $=
( vy vz vw cvv csn cun cv ccnv cuni wss wceq cop wcel wbr wa wex vex wb
ctpos cxp cmpt ccom cdm df-tpos cres wrel relcnv df-rel mpbi unss1 resmpt
- c0 mp2b resss eqsstr3i coss2 ax-mp eqsstri relco opelco eleq1 sneq cnveqd
+ c0 mp2b resss eqsstrri coss2 ax-mp eqsstri relco opelco eleq1 sneq cnveqd
unieqd eqeq2d anbi12d anbi2d df-mpt brab wi simplr breldm adantl eqeltrrd
eqeq1 opswap eleq1i opelcnv bitr4i eleq1d bibi12d mpbiri exlimivv biimpcd
elvv sylbi elun1 syl6 syl elun2 a1i wo simpll sylib mpjaod simpr eqbrtrrd
@@ -76689,7 +76704,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) $)
smores2 $p |- ( ( Smo F /\ Ord A ) -> Smo ( F |` A ) ) $=
( vy vx wsmo word wa cdm con0 wf cv cfv wcel wral crn wss dfsmo2 syl wceq
- adantr cres wfun simp1bi ffund funres funfnd cima df-ima imassrn eqsstr3i
+ adantr cres wfun simp1bi ffund funres funfnd cima df-ima imassrn eqsstrri
wfn frnd syl5ss df-f sylanbrc smodm cin ordin wb dmres ordeq ax-mp sylibr
ancoms sylan resss simp3bi ssralv mpsyl wel wi ordtr1 inss1 eqsstri sseli
dmss syl6 expcomd imp31 fvresd ad2antlr eleq12d ralbidva mpbird syl3anbrc
@@ -77043,7 +77058,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
( ( x g u /\ x h v ) -> u = v ) ) $=
( vz va vw cv wcel cfv wceq wral wa con0 wfn cres wrex wi tfrlem3a reeanv
wbr vex w3a cin fveq2 eqeq12d onin 3ad2ant1 wfun cdm wss simp2ll syl fndm
- fnfun inss1 syl5sseqr simp2rl inss2 simp2lr ssralv simp2rr tfrlem1 simp3l
+ fnfun inss1 sseqtrrid simp2rl inss2 simp2lr ssralv simp2rr tfrlem1 simp3l
jca mpsyl fnbr syl2anc simp3r rspcdva funbrfv sylc 3eqtr3d 3exp rexlimivv
elind sylbir syl2anb ) GNZEOWEKNZUAZLNZWEPZWEWHUBIPQZLWFRZSZKTUCZHNZMNZUA
ZWHWNPZWNWHUBIPQZLWORZSZMTUCZANZDNZWEUGZXBCNZWNUGZSZXCXEQZUDZWNEOABKLEFIW
@@ -78841,7 +78856,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
(Contributed by NM, 6-Dec-2004.) $)
oaword1 $p |- ( ( A e. On /\ B e. On ) -> A C_ ( A +o B ) ) $=
( con0 wcel wa c0 coa co wceq oa0 adantr wss 0ss 0elon oaword 3com13 mp3an3
- wb mpbii eqsstr3d ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBAJKUCFBLZUDUELZBMUAUBFCDZU
+ wb mpbii eqsstrrd ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBAJKUCFBLZUDUELZBMUAUBFCDZU
FUGRZNUHUBUAUIFBAOPQST $.
$( An ordinal is less than or equal to its sum with another. Theorem 21 of
@@ -79215,7 +79230,7 @@ A C_ ( A .o B ) ) $=
omword2 $p |- ( ( ( A e. On /\ B e. On ) /\ (/) e. B ) ->
A C_ ( B .o A ) ) $=
( con0 wcel wa c0 c1o comu co wceq om1r ad2antrr wss eloni ordgt0ge1 biimpa
- word sylan adantll wi 1on omwordri mp3an1 ancoms adantr mpd eqsstr3d ) ACDZ
+ word sylan adantll wi 1on omwordri mp3an1 ancoms adantr mpd eqsstrrd ) ACDZ
BCDZEZFBDZEZAGAHIZBAHIZUHUMAJUIUKAKLULGBMZUMUNMZUIUKUOUHUIBQZUKUOBNUQUKUOBO
PRSUJUOUPTZUKUIUHURGCDUIUHURUAGBAUBUCUDUEUFUG $.
@@ -79904,7 +79919,7 @@ A C_ ( B .o A ) ) $=
( con0 wcel co va vd ve c2o cdif c1o wa coe comu coa wceq w3a wb wss csuc
eldifi adantr ad2antrr simprl oecl syl2anc om1 syl c0 csn df1o2 wne dif1o
simprbi ad2antll onelon on0eln0 mpbird snssd syl5eqss 1on omwordi syl3anc
- wi a1i eqsstr3d omcl simplrl oaword1 simplrr sseqtrd sstrd oeeulem simp3d
+ wi a1i eqsstrrd omcl simplrl oaword1 simplrr sseqtrd sstrd oeeulem simp3d
mpd simp1d suceloni ontr2 mp2and simplll oeord simp2d word eloni ordsucss
onsssuc sylc dif20el syl21anc omword syl31anc mpbid oaord eqeltrrd oa1suc
oen0 odi oveq2d 3eqtr3d eleqtrrd sseldd oesuc ad2antrl jca eqeltrd simprr
@@ -80301,7 +80316,7 @@ general ordinal versions of these theorems (in this case ~ oa0r ) so
(Revised by Mario Carneiro, 15-Nov-2014.) $)
nnaword1 $p |- ( ( A e. _om /\ B e. _om ) -> A C_ ( A +o B ) ) $=
( com wcel wa c0 coa co wceq adantr wss 0ss wb peano1 nnaword 3com13 mp3an3
- nna0 mpbii eqsstr3d ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBARJUCFBKZUDUEKZBLUAUBFCD
+ nna0 mpbii eqsstrrd ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBARJUCFBKZUDUEKZBLUAUBFCD
ZUFUGMZNUHUBUAUIFBAOPQST $.
$( Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) $)
@@ -85208,7 +85223,7 @@ equivalent to the law of the excluded middle (LEM), and in ILE the LEM
cid cuni cpw csn cxp c1st f1f1orn 3ad2ant1 simp2 rnexg syl pwexg 1stconst
uniexg cen wbr wa difexg 3ad2ant3 disjen syl2anc simpld disjdif a1i f1oun
3syl syl22anc undif2 wf frnd ssequn1 sylib syl5eq f1oeq3d mpbid f1ocnv wb
- f1f f1oeq1 sylibr wfo f1ofo forn mpbird ssun1 syl5sseqr cores dmres f1odm
+ f1f f1oeq1 sylibr wfo f1ofo forn mpbird ssun1 sseqtrrid cores dmres f1odm
ssid ineq2d syl6eq wrel relres reldm0 uneq2d cnvun eqtri reseq1i resundir
cdm df-rn reseq2i relcnv resdm uneq1i 3eqtrri un0 3eqtr3g coeq1d f1cocnv1
eqtrd syl5eqr 3jca ) ABCUAZAEHZBFHZUBZBDIZDJZAYBKDCLZUDAMZNYAYCBABCIZUCZA
@@ -88456,7 +88471,7 @@ intersections of elements of the argument (see ~ elfi2 ). (Contributed
ssexi cmpt mpoeq12 anidms rneqd frsucmpt sylancl sylib ex ralrimiva sseq2
a1i imbi12d sseq1 rspc2v syl3c sseld simprr onsseleq rzal biantrud bitr3d
syl2an2 ssfii csn id inidm abeq2i ssriv simpl inss1 cbvmpov cbvmptv eqtri
- rdgeq1 reseq1i syl5sseqr sstr2 syl5com ralimdv ralsn jctird df-suc raleqi
+ rdgeq1 reseq1i sseqtrrid sstr2 syl5com ralimdv ralsn jctird df-suc raleqi
ralunb bitri syl6ibr fiin ss2ralv mpi jctild expimpd finds2 impcom simprd
frnd r19.21bi eqimss jaod sylbid ssun1 ssun2 peano2 ssiun2s 3syl 2rexbidv
eqeq1 elab syl6eleqr eleqtrrd sseldd syl6eleq syl2and rexlimdvva sylan2br
@@ -92074,7 +92089,7 @@ of all inductive sets (which is the smallest inductive set, since
ad2antrl pm2.27 cantnfvalf ffvelrni ad2antlr ad3antrrr suppssdm fssdm
coa sucidg sseldd oif onelon oecl oaword2 cantnfsuc ad4ant13 sseqtr4d
simpr omcl expcom adantrr syld expr fveq2d f1ocnvfv2 ad2antrr eqtr3d
- oveq2d oveq12d oaword1 eqsstr3d a1dd jaod syl5bi expimpd com23 finds2
+ oveq2d oveq12d oaword1 eqsstrrd a1dd jaod syl5bi expimpd com23 finds2
vtoclga mpcom mpd cantnfval om0 0ss syl6eqss pm2.61ne ) ACEUCUDZEHUEZ
UFUDZHCDUGUDUEZRUUSSUFUDZUVBRUUTSUUTSUHUVAUVCUVBUUTSUUSUFUIUJAUUTSUKZ
ULZUVAIUMZJUEZUVBUVEEIUNZUEZUVFTZUVAUVGRZUVEHSUOUDZUVFEUVHUVEUVFUVLIU
@@ -92351,7 +92366,7 @@ of all inductive sets (which is the smallest inductive set, since
ffvelrnd oicl ordelon ontri1 sucidg eleqtrrd isorel 3bitr3g f1ocnvfv2
nnon syl12anc mtbid onss sstrd imnan onelon elsni notbid eldifn velsn
elun2 sylbir nsyl ssun1 sscon sseli sylan2 elun mpjaod ioran sylanbrc
- ordn2lp ordtri3 mpbird oveq2d nnord sssucid syl5sseqr wfo f1ofo foima
+ ordn2lp ordtri3 mpbird oveq2d nnord sssucid sseqtrrid wfo f1ofo foima
cres ffn fnsnfv sneqd difeq12d cin disjsn disj3 difun2 syl6eqr df-suc
syl6eq difeq1d eqtr4d imaeq2d dff1o3 simprbi imadif ssneldd c1o dif1o
wfun mpbiran bicomd anbi2d sylbird mpand syl5bir necon1bd mpd fveqeq2
@@ -92606,7 +92621,7 @@ of all inductive sets (which is the smallest inductive set, since
mpbir2and cdif wn eldifn adantl iffalsed suppss2 ifeq1da syl3anc sseldd
fveq2 eleq1w ifbieq1d eqid fvex fvmpt ifeq2d syl6reqr mpteq2ia cantnfp1
ifex eqtr3d ifor omsuc word eloni simp2d ordsucss sylc suceloni omwordi
- mpd eqsstr3d cantnflt2 oaord eqeltrd ) AFOUGUHZOLUIZUJUHZONUKZUIZULMUIZ
+ mpd eqsstrrd cantnflt2 oaord eqeltrd ) AFOUGUHZOLUIZUJUHZONUKZUIZULMUIZ
BGBUMZOUNZUVLKUIZUOUPZUQZFGURUHZUIZAUVHUVHUVJMUIZVCUHZUVKAUVHUSUTZUVSUS
UTZUVHUVTUNAUVFUSUTZUVGUSUTZUWAAFUSUTZOUSUTZUWCRAGUSUTZOGUTZUWFSAUWHOKU
IZUVGUTZOEUMZUTUWKKUIUWKLUIVAVDEGVBZABCDEFGHIKLOPQRSTUAUBUCUDVEZVFZGOVG
@@ -93430,7 +93445,7 @@ a Cantor normal form (and injectivity, together with coherence
( vv vu wss cv com cfv wrex c0 wcel wceq fveq2 wtr wa wal ciun peano1 cvv
wi cuni cun cmpt crdg cres fveq1i fr0g ax-mp eqtr2i eqimssi sseq2d rspcev
mp2an ssiun sseqtr4i dftr2 eliun anbi2i r19.42v bitr4i elunii ssun2 uniex
- csuc fvex unex unieq uneq12d frsucmpt2 mpan2 syl5sseqr sseld syl5 reximia
+ csuc fvex unex unieq uneq12d frsucmpt2 mpan2 sseqtrrid sseld syl5 reximia
weq id sylbi peano2 eleq2d syl rexlimiv cbvrexv sylibr ax-gen mpgbir treq
ex mpbir wral sseq1d eqtri sseq1i biimpri adantr uniss df-tr sstr2 syl5bi
wb anc2li unss syl6ib biimprd syl9r com23 adantld finds2 ralrimiv cbviunv
@@ -93587,7 +93602,7 @@ a Cantor normal form (and injectivity, together with coherence
$( Defining property of the transitive closure function: it contains its
argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.) $)
tcid $p |- ( A e. V -> A C_ ( TC ` A ) ) $=
- ( vx wcel cv wss wtr wa cab cint ctc cfv ssmin tcvalg syl5sseqr ) ABDACEZ
+ ( vx wcel cv wss wtr wa cab cint ctc cfv ssmin tcvalg sseqtrrid ) ABDACEZ
FPGZHCIJAAKLQCAMCABNO $.
$( Defining property of the transitive closure function: it is transitive.
@@ -93874,7 +93889,7 @@ of the previous layer (and the union of previous layers when the
( vy cr1 wcel cfv wa con0 wss word ax-mp syl wi wceq eleq1 eleq2d imbi12d
fveq2 wb syl2anc vx cdm csuc simpl wlim wfun r1funlim simpri limord sseli
ordsson onelon sylan suceloni ordsucss imp cv cpw fvex pwid limsuc r1sucg
- eloni sylbir syl5eleqr a1i wtr r1tr dftr4 mpbi syl5sseqr a2i syl5bir wral
+ eloni sylbir syl5eleqr a1i wtr r1tr dftr4 mpbi sseqtrrid a2i syl5bir wral
sseld ciun wrex simprl simplr sucelon sylibr ad2antrr mpbird mpbid simprr
ordelsuc ordtr1 rspcev eliun simpll r1limg eleqtrrd expr a1d tfindsg impr
syl22anc ex ) BDUBZEZABEZADFZBDFZEZWTXAGZBHEZAUCZHEZXGBIZWTXDXEWTXFWTXAUD
@@ -93928,7 +93943,7 @@ of the previous layer (and the union of previous layers when the
value at the successor. JFM CLASSES1 Th. 39. (Contributed by FL,
20-Apr-2011.) $)
r1sssuc $p |- ( A e. On -> ( R1 ` A ) C_ ( R1 ` suc A ) ) $=
- ( con0 wcel cr1 cfv cpw csuc wtr wss r1tr dftr4 mpbi r1suc syl5sseqr ) ABCA
+ ( con0 wcel cr1 cfv cpw csuc wtr wss r1tr dftr4 mpbi r1suc sseqtrrid ) ABCA
DEZFZOAGDEOHOPIAJOKLAMN $.
${
@@ -93974,7 +93989,7 @@ of the previous layer (and the union of previous layers when the
nfv biimpac wfun r1funlim simpri limsuc sylibr r1sucg eqtrd vex syl5eleqr
sucid ssiun2 ex a1d rexlimd imp r1limg wral wtr r1tr dftr4 mpbi ralrimivw
ss2iun adantrl w3o word limord ordsson sseli onzsl sylib mpjao3dan ordtr1
- ancoms ordelord mpan adantr ordelsuc syl2anc mpbid simpl r1ord3g eqsstr3d
+ ancoms ordelord mpan adantr ordelsuc syl2anc mpbid simpl r1ord3g eqsstrrd
mpd ralrimiva iunss eqssd ) BCUAZDZBCEZABAUBZCEZUCZFZXDBGHZXEXIIZBXFUDZHZ
AJUEZBUFDZBRZKZXDXJKZXEGXIXRXEGCEGXRBGCXDXJLMUGUHGXIIXRXIUINOXDXNXKXDXMXK
AJXDAUNAXEXIAXEUJABXHUKULXDXMXKPXFJDXDXMXKXDXMKZXEXHXIXSXEXLCEZXHXSBXLCXD
@@ -94497,7 +94512,7 @@ of the previous layer (and the union of previous layers when the
22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) $)
onwf $p |- On C_ U. ( R1 " On ) $=
( vx con0 cr1 cdm cima cuni wceq r1fnon fndm ax-mp cv wcel crnk rankonidlem
- wfn cfv simpld ssriv eqsstr3i ) BCDZCBEFZCBOTBGHBCIJATUAAKZTLUBUALUBMPUBGUB
+ wfn cfv simpld ssriv eqsstrri ) BCDZCBEFZCBOTBGHBCIJATUAAKZTLUBUALUBMPUBGUB
NQRS $.
${
@@ -94838,7 +94853,7 @@ suc suc ( ( rank ` A ) u. ( rank ` B ) ) $=
rankr1id $p |- ( A e. dom R1 <-> ( rank ` ( R1 ` A ) ) = A ) $=
( cr1 cdm wcel cfv crnk wceq wss ssid con0 cima cuni csuc cpw fvex r1sucg
wb pwid syl5eleqr r1elwf syl rankr1bg mpancom mpbii biimpi onssr1 rankssb
- rankonid sylc eqsstr3d eqssd id rankdmr1 syl6eqelr impbii ) ABCZDZABEZFEZ
+ rankonid sylc eqsstrrd eqssd id rankdmr1 syl6eqelr impbii ) ABCZDZABEZFEZ
AGZUQUSAUQURURHZUSAHZURIURBJKLDZUQVAVBQUQURAMZBEZDVCUQURURNVEURABORAPSURV
DTUAZURAUBUCUDUQAAFEZUSUQVGAGAUHUEUQVCAURHVGUSHVFAUFAURUGUIUJUKUTAUSUPUTU
LURUMUNUO $.
@@ -95050,7 +95065,7 @@ C_ suc suc suc ( rank ` ( A u. B ) ) $=
wlim limuni2 cun cv 0ellim n0i unieq uni0 syl6eq con3i 3syl rankon onsuci
elexi sucid ontri1 mp2an con2bii mpbi rankxpu sstr mpan2 reeanv wi simprl
mto simpr rankuni unieqi eqtri wne df-ne xpex rankeq0 notbii bitr2i sylib
- unixp syl fveq2d syl5reqr eqimss eqsstr3d adantrr limuni sseqtr4d cvv vex
+ unixp syl fveq2d syl5reqr eqimss eqsstrrd adantrr limuni sseqtr4d cvv vex
word onordi orduni ax-mp ordelsuc sylibr limsuc mpbid eqeltrd ordsucelsuc
onsucuni2 mpan ad2antll eleqtrd onsucssi ex a1d rexlimdvv syl5bir mtoi wo
ianor un00 animorl sylbir xpeq0 unex w3o ordzsl 3ori sylan orim12d syl5bi
@@ -96702,7 +96717,7 @@ several of their earlier lemmas available (which would otherwise be
dmex rnex cres imadmres ccnv cin crab inss2 cpr ssun1 cop elinel2 1st2nd2
wral syl elinel1 eqeltrrd opeldm sseldi ssun2 opelrn prssd ordunpr sseldd
rgen ssrab mpbir2an dmres fdmi ineq2i eqtri mptpreima 3sstr4i wfun funmpt
- wb resss dmss ax-mp funimass3 mp2an mpbir eqsstr3i ssexi fnwe epse vuniex
+ wb resss dmss ax-mp funimass3 mp2an mpbir eqsstrri ssexi fnwe epse vuniex
cuni cpw pwex xpex cab df-rab adantr elssuni adantl elpw jca unssad elxp6
unssbd sylanbrc abssi eqsstri fnse mptru ) JJUAZEUBZUUIEUCZKLUUJUUKLCDIUU
IJMFEAUUIAUDZNOZUULPOZUFZUEZECUDZUUIQZDUDZUUIQZKZUUQNOZUUQPOZUFZUUSNOZUUS
@@ -99498,7 +99513,7 @@ our Axiom of Choice (in the form of ~ ac2 ). The proof does not depend
dju1dif $p |- ( ( A e. V /\ B e. ( A |_| 1o ) )
-> ( ( A |_| 1o ) \ { B } ) ~~ A ) $=
( wcel c1o cdju wa csn cdif cop cen wbr cvv simpl 1oex djuex cxp 0ex wceq
- c0 sylancl simpr cun df1o2 xpeq2i xpsn eqtri ssun2 eqsstr3i opex snss mpbir
+ c0 sylancl simpr cun df1o2 xpeq2i xpsn eqtri ssun2 eqsstrri opex snss mpbir
wss df-dju eleqtrri a1i difsnen syl3anc difeq1i xp01disjl disj3 mpbi difun2
cin difeq2i 3eqtr2i eqtr4i xpsnen2g sylancr eqbrtrid entr syl2anc ) ACDZBAE
FZDZGZVNBHIZVNETJZHZIZKLZVTAKLVQAKLVPVNMDZVOVRVNDZWAVPVMEMDWBVMVONZOAECMPUA
@@ -99852,7 +99867,7 @@ Because we use a disjoint union for cardinal addition (as explained in the
unctb $p |- ( ( A ~<_ _om /\ B ~<_ _om ) -> ( A u. B ) ~<_ _om ) $=
( com cdom wbr wa cun cdju cvv wcel ctex undjudom sylancl domtr syl2anc cxp
omex wss c2o mp2an syl2an djudom1 sylan2 simpr djudom2 cen xpex xp2dju word
- ordom 2onn ordelss xpss1 ax-mp eqsstr3i ssdomg mp2 xpomen domentr ) ACDEZBC
+ ordom 2onn ordelss xpss1 ax-mp eqsstrri ssdomg mp2 xpomen domentr ) ACDEZBC
DEZFZABGZABHZDEZVDCDEZVCCDEUTAIJBIJZVEVAAKBKZABIILUAVBVDCCHZDEZVICDEZVFVBVD
CBHZDEZVLVIDEZVJVAUTVGVMVHACBIUBUCVBVACIJVNUTVAUDQBCCIUEMVDVLVINOVICCPZDEZV
OCUFEVKVOIJVIVORVPCCQQUGVISCPZVOCUHSCRZVQVORCUISCJVRUJUKCSULTSCCUMUNUOVIVOI
@@ -100025,7 +100040,7 @@ Because we use a disjoint union for cardinal addition (as explained in the
wf com co ccrd cdm w3a cv oveq2 breq2 anbi2d abbidv breq12d wne brrelex1i
cxp simpl2 brrelex2i xpcomeng simpl3 simpr mapdom3 numdom simpl1 infxpabs
syl3anc syl22anc entr ssenen relen abid2 elmapi fssxp wfun ffun vex ensym
- fundmen 3syl fdm breqtrd jca ss2abi eqsstr3i ssdomg domentr crn cmpt ovex
+ fundmen 3syl fdm breqtrd jca ss2abi eqsstrri ssdomg domentr crn cmpt ovex
mpisyl mptex rnex wrex wex wf1o ad2antll bren f1of adantl simplrl fssd wb
sylib elmapd ad2antrr mpbird wfo f1ofo eqcomd ex eximdv mpd df-rex sylibr
forn ss2abdv eqid rnmpt syl6sseqr mpsyl wfn wral rgenw mp1i fodomnum sylc
@@ -100220,7 +100235,7 @@ Because we use a disjoint union for cardinal addition (as explained in the
cv wn w3a simpr1 ackbij1lem3 simpr3 ackbij1lem1 syl6eqss csn ackbij1lem10
inss2 con0 ffvelrni nnon onpsssuc 4syl ackbij1lem14 psseq2d mpbird simpll
inss1 ackbij1lem11 sylancl cun ssun1 simpr2 ackbij1lem2 psssstrd sspsstrd
- syl5sseqr pssned necomd neneqd ) CHIUAJZKZDVOKZLZFUBZHKZVSCKZVSDKUCZUDZLZ
+ sseqtrrid pssned necomd neneqd ) CHIUAJZKZDVOKZLZFUBZHKZVSCKZVSDKUCZUDZLZ
CVSMZJZENZDWEJZENZWDWIWGWDWIWGWDWIVSENZWGWDVSVOKZWHVSOWIWJOWDVTWKVRVTWAWB
UEZVSUFPZWDWHDVSJZVSWDWBWHWNQVRVTWAWBUGVSDUHPDVSULUIABWHVSEGRSWDWJVSUJZEN
ZWGWDWJWPTWJWJMZTZWDWKWJHKWJUMKWRWMVOHVSEABEGUKUNWJUOWJUPUQWDWPWQWJWDVTWP
@@ -100439,7 +100454,7 @@ Because we use a disjoint union for cardinal addition (as explained in the
bitr4i df-rex funfn mpbi rdgdmlim limomss ax-mp fvelimab mp2an f1ofo forn
wfn 3syl eqsstrd rneq sseq1d syl5ibcom rexlimiv sylbi sselda exlimiv csuc
wfo peano2 fnfvima mp3an12i cvv vex cardnn wbr simpri sseli onssr1 ssdomg
- cdom mpsyl con0 nnon onenon finnum carddom2 mpbird eqsstr3d 4syl eleqtrrd
+ cdom mpsyl con0 nnon onenon finnum carddom2 mpbird eqsstrrd 4syl eleqtrrd
sucssel eleq1 eleq2d anbi12d spcev impbii 3bitri dff1o5 mpbir2an f1oeq1
eqriv eqtri ) KLUBUCZLEMZUULLDUDUEZLUBZUCZMZUUQUULLUUPNZUUPUGZLOUURILIUFZ
KPZUHZLILUUTUUNPZUHZNZUVALUVCNZUVCJUFZUUNPZQZUVHUVCQZUIZJLUJZRUVEILIJLUVC
@@ -101060,7 +101075,7 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality
eliun 3adant2 mpbird 3expa ralrimiva expl rexbiia syl6bb anbi12d syl21anc
cfflb ontri1 mpbid pm2.21dd expcomd 3impib 3adant1l ordunel eqeltrd 3impa
jaod syldc tfis2 mpcom onssi fneq1i mpbiran onordi sucid eliuni syl6eleqr
- a1i mpan2 elun2 eleqtrrd 3eltr4d rgen issmo2 mp3an23 sylc ssun1 syl5sseqr
+ a1i mpan2 elun2 eleqtrrd 3eltr4d rgen issmo2 mp3an23 sylc ssun1 sseqtrrid
sseq12d vtoclga sstr reximia ralimi ad2antlr fnex smoeq 3anbi123d exlimdv
) DMNZDUAOZDFUBZUCZAUBZBUBZUYEOZPZBUYDUDZADQZRZFUEUYDDEUBZUFZUYNUGZUYGUYH
UYNOZPZBUYDUDZADQZUHZEUEZABDFUIUYCUYMVUBFUYMUYCVUBUYMUYCRUYDDHUFZHUGZUYGU
@@ -103102,7 +103117,7 @@ version of the pigeonhole principle (for aleph-null pigeons and 2 holes)
ccnv dmfex fssdm sselpwd cres ccom wfun wf1o fin1a2lem4 f1cnv f1ofo fofun
wf1 ax-mp resex cofunexg mp2an cdm wss fores sylancl foimacnv mpan2 foeq3
f1f frn syl mpbid foco fowdom cnvex imaex con2bii sylib fin1a2lem6 f1ocnv
- isfin3-2 difss syl5sseqr syl2anc funcnvcnv imadif imaeq2d fimacnv difeq1d
+ isfin3-2 difss sseqtrrid syl2anc funcnvcnv imadif imaeq2d fimacnv difeq1d
fdmd 3eqtr3rd difexg con2bid wa eleq1 difeq2 eleq1d orbi12d notbid syl6bb
3syl ioran rspcev syl12anc rexnal exlimiv sylbi con2i syl5ibr imp ) CFJZB
UAZKJZCYGLZKJZUBZBCUEZUCZCKJZYMYNYFMCUDUFZNYOYMYOCMIUAZOZIUGZYMNZUHMJMUHU
@@ -103394,7 +103409,7 @@ version of the pigeonhole principle (for aleph-null pigeons and 2 holes)
~ trcl . (Contributed by Stefan O'Rear, 11-Feb-2015.) $)
itunitc $p |- ( TC ` A ) = U. ran ( U ` A ) $=
( va vb vc cvv wcel ctc cfv crn cuni wceq cv fveq2 wss c0 com eqeq12d wtr
- rneqd unieqd ituni0 elv fvssunirn eqsstr3i dftr3 wrex wfn itunifn fnunirn
+ rneqd unieqd ituni0 elv fvssunirn eqsstrri dftr3 wrex wfn itunifn fnunirn
wb vex mp2b elssuni csuc itunisuc syl6ss rexlimivw sylbi mprgbir wa tcmin
wi mp2an unissb fvelrnb itunitc1 sseq1 syl5ibcom rexlimiv eqssi vtoclg wn
a1i rn0 unieqi uni0 eqtr2i fvprc 3eqtr4a pm2.61i ) CIJZCKLZCDLZMZNZOZFPZK
@@ -104484,7 +104499,7 @@ a sequence whose values can only be shown to exist (but cannot be
dcomex expcom mtod vex difss eqsstri ssdomg mp2 jctil sylibr entr mpancom
bren2 ensym wf1o bren cuni wf csuc f1of peano1 ffvelrn eldifn eleq2s fvex
co elsn notbii neq0 bitr2i syl w3a cxp cpw elunii sylan2 ffvelrnda difabs
- difeq1i 3eqtr4i pwuni ssdif eqsstr3i sseli ralrimivw ralrimiva fmpo sylib
+ difeq1i 3eqtr4i pwuni ssdif eqsstrri sseli ralrimivw ralrimiva fmpo sylib
adantl difexi eqeltri uniex axdc4 syl2anc suceq fveq2d fveq2 3ad2ant3 imp
exlimiv 3adant1 3adant2 3ad2ant1 3simpb eximi ex mpcom velsn eleq2i eldif
necon3bbii sylbbr sylan2br simpl wrex wfo f1ofo foelrn sylan ccnv oveq12d
@@ -105498,7 +105513,7 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario
ttukeylem7 $p |- ( ph -> E. x e. A ( B C_ x /\ A. y e. A -. x C. y ) ) $=
( cfv wcel wss wn wa c0 con0 wceq cuni cdif ccrd wpss wral wrex csuc fvex
va cv sucid ttukeylem6 mpan2 ttukeylem4 w3a cardon 0ss 3pm3.2i ttukeylem5
- 0elon eqsstr3d wi simprr cun ssun1 undif1 sseqtr4i ccnv simpl wf1o f1ocnv
+ 0elon eqsstrrd wi simprr cun ssun1 undif1 sseqtr4i ccnv simpl wf1o f1ocnv
wf f1of 3syl adantr eldifi ad2antll simprll elunii eldifn eldifd ffvelrnd
syl2anc onelon sylancr suceloni syl a1i word onordi ordsucss syl13anc csn
mpsyl ssun2 eloni ordunisuc fveq2d f1ocnvfv2 eqtr2d velsn ordelss eqsstrd
@@ -109460,7 +109475,7 @@ prove that every set is contained in a weak universe in ZF (see
weq pweq preq12d preq2 cbvmptv preq1 mpteq2dv syl5eq rneqd cbviunv mpteq1
id uneq2d iuneq12d frsucmpt2 sylancl sseqtr4d fvssunirn rexlimdvaa syl5bi
sseqtr4i ralrimiv dftr3 sylibr con0 1on unexg mpan2 fveq1i fr0g syl6eqssr
- syl unssbd 1n0 ssn0 syl5sseqr sstrd unssad vpwex vuniex prss simprd fveq2
+ syl unssbd 1n0 ssn0 sseqtrrid sstrd unssad vpwex vuniex prss simprd fveq2
ssiun2s sseq2d vtoclga findsg sseldd wi imbi12d eqeltri simpld word ordom
simplrl ordunel mp3an2i ssidd suceq fveq2d sseq12d ad2antrr sstr2 syl5com
syl2anc simplrr biantrud bicomd eleq1w anbi2d sseq1 anbi12d sseq1d chvarv
@@ -109557,7 +109572,7 @@ prove that every set is contained in a weak universe in ZF (see
$( The weak universe closure of a set contains the set. (Contributed by
Mario Carneiro, 2-Jan-2017.) $)
wuncid $p |- ( A e. V -> A C_ ( wUniCl ` A ) ) $=
- ( vu wcel cv wss cwun crab cint cwunm cfv ssintub wuncval syl5sseqr ) ABD
+ ( vu wcel cv wss cwun crab cint cwunm cfv ssintub wuncval sseqtrrid ) ABD
ACEFCGHIAAJKCAGLCABMN $.
$( The weak universe closure of a set is a weak universe. (Contributed by
@@ -110353,7 +110368,7 @@ values in the universe (see ~ gruiun for a more intuitive version).
( cgru wcel wfun cima wss w3a wa cdm cin cres wrel wceq simpl2 wf syl3anc
crn 3syl funrel resres resdm reseq1d syl5eqr rneqd df-ima syl6reqr simpl1
simpr inss2 a1i gruss wfn wfo funforn fof sylbi fssres sylancl ffn simpl3
- inss1 eqsstr3d df-f sylanbrc grurn eqeltrd ex ) BDEZCFZCAGZBHZIZABEZVLBEV
+ inss1 eqsstrrd df-f sylanbrc grurn eqeltrd ex ) BDEZCFZCAGZBHZIZABEZVLBEV
NVOJZVLCCKZALZMZSZBVPVKCNZVLVTOVJVKVMVOPZCUAWAVTCAMZSVLWAVSWCWAVSCVQMZAMW
CCVQAUBWAWDCACUCUDUEUFCAUGUHTZVPVJVRBEZVRBVSQZVTBEVJVKVMVOUIZVPVJVOVRAHZW
FWHVNVOUJWIVPVQAUKULAVRBUMRVPVSVRUNZVTBHWGVPVKVRCSZVSQZWJWBVKVQWKCQZVRVQH
@@ -124305,7 +124320,7 @@ subset of complex numbers ( ~ nnsscn ), in contrast to the more elementary
dfnn3 $p |- NN =
|^| { x | ( x C_ RR /\ 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $=
( vz cv cr wss c1 wcel caddc wral wa cab cint cn eleq2 raleqbi1dv anbi12d
- wceq dfnn2 pm3.2i co w3a wb eqeq2i sylbir nnssre eqsstr3i peano2nn intabs
+ wceq dfnn2 pm3.2i co w3a wb eqeq2i sylbir nnssre eqsstrri peano2nn intabs
1nn rgen 3anass abbii inteqi 3eqtr4ri ) ADZEFZGUPHZBDZGIUAZUPHZBUPJZKZKZA
LZMVCALMUQURVBUBZALZMNVCGCDZHZUTVHHZBVHJZKZGNHZUTNHZBNJZKZACEUPVHRURVIVBV
KUPVHGOVAVJBUPVHUPVHUTOPQUPVLCLMZRUPNRZVCVPUCNVQUPCBSZUDVRURVMVBVOUPNGOVA
@@ -147802,7 +147817,7 @@ Proper unordered pairs and triples (sets of size 2 and 3)
by AV, 12-Nov-2021.) $)
hashdmpropge2 $p |- ( ph -> 2 <_ ( # ` dom F ) ) $=
( va wcel vb cdm cvv cv wne wrex chash cfv cle wbr dmexd cpr wss cop wceq
- c2 dmpropg syl2anc dmss eqsstr3d wa wb prssg wi neeq1 neeq2 rspc2ev 3expa
+ c2 dmpropg syl2anc dmss eqsstrrd wa wb prssg wi neeq1 neeq2 rspc2ev 3expa
syl expcom sylbird mpd hashge2el2difr ) AFUBZUCTSUDZUAUDZUEZUAVNUFSVNUFZU
PVNUGUHUIUJAFKPUKABCULZVNUMZVRAVSBDUNCEUNULZUBZVNADITEJTWBVSUONOBDCEIJUQU
RAWAFUMWBVNUMRWAFUSVIUTAVTBVNTZCVNTZVAZVRABGTCHTWEVTVBLMBCVNGHVCURABCUEZW
@@ -151427,7 +151442,6 @@ computer programs (as last() or lastChar()), the terminology used for
YNIZYMUVTYNYMYSYLYMWFQUYHUWBYSUVTYNUWBYSUWCQYSYNXJUWDWGYEWSQYCUWERWHWHTXM
UVEUVFUXSXNXOFUUCUUEUWRYIRXRYFYJWS $.
-
$( The subword of a concatenation is either a subword of the first
concatenated word or a subword of the second concatenated word or a
concatenation of a suffix of the first word with a prefix of the second
@@ -154809,7 +154823,7 @@ the symbol at any position is repeated at multiples of L (modulo the
cotr2 $p |- ( ( R o. R ) C_ R
<-> A. x e. A A. y e. B A. z e. C
( ( x R y /\ y R z ) -> x R z ) ) $=
- ( crn cdm cin incom eqsstr3i cotr2g ) ABCGGGDEFHGKZGLZMRQMERQNIOJP $.
+ ( crn cdm cin incom eqsstrri cotr2g ) ABCGGGDEFHGKZGLZMRQMERQNIOJP $.
$}
${
@@ -155129,7 +155143,7 @@ the symbol at any position is repeated at multiples of L (modulo the
$( The transitive closure of a relation has a lower bound. (Contributed by
RP, 28-Apr-2020.) $)
trclfvlb $p |- ( R e. V -> R C_ ( t+ ` R ) ) $=
- ( vr wcel cv wss ccom wa cab cint ctcl cfv ssmin trclfv syl5sseqr ) ABDAC
+ ( vr wcel cv wss ccom wa cab cint ctcl cfv ssmin trclfv sseqtrrid ) ABDAC
EZFPPGPFZHCIJAAKLQCAMCABNO $.
$}
@@ -155758,7 +155772,7 @@ the symbol at any position is repeated at multiples of L (modulo the
by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) $)
rtrclreclem1 $p |- ( ph -> ( _I |` U. U. R ) C_ ( t*rec ` R ) ) $=
( vr vn cuni crtrcl cfv wss wi cvv cn0 cv crelexp co ciun cc0 wcel wceq
- cres cmpt wrex 0nn0 ssid relexp0d syl5sseqr oveq2 sseq2d rspcev sylancr
+ cres cmpt wrex 0nn0 ssid relexp0d sseqtrrid oveq2 sseq2d rspcev sylancr
cid ssiun syl nn0ex ovex iunex oveq1 iuneq2d fvmptg sylancl sseqtr4d wb
eqid df-rtrclrec fveq1 imbi2d ax-mp mpbir ) AULBGGUAZBHIZJZKZAVJBELFMEN
ZFNZOPZQZUBZIZJZKZAVJFMBVOOPZQZVSAVJWBJZFMUCZVJWCJARMSVJBROPZJZWEUDAVJV
@@ -160519,7 +160533,7 @@ Superior limit (lim sup)
( y <_ z -> ( abs ` ( B - C ) ) < R ) ) $=
( vx cv wbr clt wral cr cc wf wcel cle cmin co cabs cfv wi wrex crp breq2
wceq imbi2d rexralbidv cmpt crli cdm rlimf syl eqid fmpt sylib fdmd feq2d
- mpbid sylibr wss rlimss eqsstr3d rlimcl rlim2 rspcdva ) ABMCMUANZEFUBUCUD
+ mpbid sylibr wss rlimss eqsstrrd rlimcl rlim2 rspcdva ) ABMCMUANZEFUBUCUD
UEZLMZONZUFZCDPBQUGZVKVLGONZUFZCDPBQUGLUHGVMGUJZVOVRBCQDVSVNVQVKVMGVLOUIU
KULACDEUMZFUNNZVPLUHPKALBCDEFADRVTSZERTCDPAVTUOZRVTSZWBAWAWDKFVTUPUQAWCDR
VTADHVTAEHTCDPDHVTSICDHEVTVTURZUSUTVAZVBVCCDREVTWEUSVDADWCQWFAWAWCQVEKFVT
@@ -160532,7 +160546,7 @@ Superior limit (lim sup)
( y <_ z -> ( abs ` ( B - C ) ) < R ) ) $=
( cv wbr co wral wrex cr wss cle cmin cabs cfv wi cpnf cico rlimi wcel wb
clt cmpt cdm wfn wceq eqid fnmpt fndm 3syl crli rlimss syl rexico syl2anc
- eqsstr3d mpbird ) ABNCNUAOEFUBPUCUDHUKOZUECDQZBGUFUGPRZVHBSRZABCDEFHIJKLU
+ eqsstrrd mpbird ) ABNCNUAOEFUBPUCUDHUKOZUECDQZBGUFUGPRZVHBSRZABCDEFHIJKLU
HADSTGSUIVIVJUJADCDEULZUMZSAEIUICDQVKDUNVLDUOJCDEVKIVKUPUQDVKURUSAVKFUTOV
LSTLFVKVAVBVEMVGDGBCVCVDVF $.
$}
@@ -160591,7 +160605,7 @@ Superior limit (lim sup)
lo1bdd $p |- ( ( F e. <_O(1) /\ F : A --> RR ) -> E. x e. RR
E. m e. RR A. y e. A ( x <_ y -> ( F ` y ) <_ m ) ) $=
( clo1 wcel cr wf wa cv cle wbr cfv wi wral wrex simpl wss wb wceq adantl
- simpr cdm fdm lo1dm adantr eqsstr3d ello12 syl2anc mpbid ) EFGZCHEIZJZULA
+ simpr cdm fdm lo1dm adantr eqsstrrd ello12 syl2anc mpbid ) EFGZCHEIZJZULA
KBKZLMUOENDKLMOBCPDHQAHQZULUMRUNUMCHSULUPTULUMUCUNCEUDZHUMUQCUAULCHEUEUBU
LUQHSUMEUFUGUHABCDEUIUJUK $.
$}
@@ -160736,7 +160750,7 @@ Superior limit (lim sup)
o1bdd $p |- ( ( F e. O(1) /\ F : A --> CC ) -> E. x e. RR
E. m e. RR A. y e. A ( x <_ y -> ( abs ` ( F ` y ) ) <_ m ) ) $=
( co1 wcel cc wf wa cv cle wbr cfv cabs wi wral cr wrex wss simpl wb wceq
- simpr cdm fdm adantl o1dm adantr eqsstr3d elo12 syl2anc mpbid ) EFGZCHEIZ
+ simpr cdm fdm adantl o1dm adantr eqsstrrd elo12 syl2anc mpbid ) EFGZCHEIZ
JZUNAKBKZLMUQENONDKLMPBCQDRSARSZUNUOUAUPUOCRTUNURUBUNUOUDUPCEUEZRUOUSCUCU
NCHEUFUGUNUSRTUOEUHUIUJABCDEUKULUM $.
@@ -161062,7 +161076,7 @@ Superior limit (lim sup)
( vj vk wbr clt wa cr wrex wcel cc syl adantr cv cle cfv cmin cabs cdiv
co c2 wi wral wceq crli rlimcl ad2antrr subcld abscld ffvelrnda adantlr
ltnrd abssubd breq1d anbi1d abs3lem syl22anc sylbid imim2d impcomd mtod
- wn nrexdv r19.29r nsyl cxr csup cpnf wss wb fdmd rlimss eqsstr3d ressxr
+ wn nrexdv r19.29r nsyl cxr csup cpnf wss wb fdmd rlimss eqsstrrd ressxr
cdm syl6ss supxrunb1 mpbird r19.29 ex wne wf ffvelrn ralrimiva absrpcld
simpr subne0d rphalfcld feqmptd eqbrtrrd rexanre mpbir2and necon1bd mpd
cmpt rlimi ) AJUAZKUAZUBLZXEEUCZCUDUGUEUCZCDUDUGZUEUCZUHUFUGZMLZXGDUDUG
@@ -161517,7 +161531,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
nfel1 csbeq1a eleq1d rspc sylc rlimi ad2antrr rpred wss ad4ant14 sseldd
ad3antrrr subcld abscld nfcv nfral oveq2 fveq2d breq12d ralbidv fvoveq1
nfbr breq2d rspcv lensymd imp nsyl nrexdv cxr csup cpnf cdm eqid dmmptd
- id wb rlimss eqsstr3d ressxr syl6ss supxrunb1 mpbird r19.29 expcom mtod
+ id wb rlimss eqsstrrd ressxr syl6ss supxrunb1 mpbird r19.29 expcom mtod
r19.21bi condan ) AGHQZPUAZBUAZRUBZFGUCUDZUEUFZCGIUGZUHUBZUIZBESZPTUJAX
MULZUMZPBEFGXSHAFHQZBESYCAYEBEOUKUNYDGUOHUPZQZIUQQZCYFSZXSUQQZYDGUOHYDB
EFURZGUSUBZGUOQZAYLYCKUNZGYKUTVAZAYCVBVCZAYIYCAYHCYFMUKUNYHYJCGYFCXSUQC
@@ -161658,7 +161672,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
bounded function. (Contributed by Mario Carneiro, 12-May-2016.) $)
o1co $p |- ( ph -> ( F o. G ) e. O(1) ) $=
( vn wcel cle cfv cr wrex wa vz ccom co1 cv wbr cabs wi wral cc wf wss wb
- cdm fdmd o1dm syl eqsstr3d elo12 syl2anc mpbid reeanv ad3antrrr ffvelrnda
+ cdm fdmd o1dm syl eqsstrrd elo12 syl2anc mpbid reeanv ad3antrrr ffvelrnda
wceq breq2 2fveq3 breq1d imbi12d rspcva sylan an32s adantr fveq2d sylibrd
fvco3 imim2d ralimdva expimpd ancomsd reximdva mpand rexlimdva mpd mpbird
syl5bir fco ) AGHUBZUCOZBUDZCUDZPUEZWJWGQZUFQZNUDZPUEZUGZCEUHZNRSZBRSZAFU
@@ -161711,7 +161725,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
fveq2 co cabs clt wi wral cr wrex crp wcel fvexd ralrimiva simpr eqbrtrrd
wa cvv ad2antrr rlimi wceq fvoveq1 imbrov2fvoveq simplrr ad4ant14 rspcdva
breq1d imim2d ralimdva reximdv expr mpid rexlimdva mpd syldan fdmd rlimss
- cdm wss syl eqsstr3d ffvelrnd rlim2 mpbird eqbrtrd ) AGHUBOEOQZHRZGRZUCZF
+ cdm wss syl eqsstrrd ffvelrnd rlim2 mpbird eqbrtrd ) AGHUBOEOQZHRZGRZUCZF
GRZUDAOUAEIWNUAQZGRWOHGAEIWMHJUEZAOEIHJUFZAUAIUGGMUFWRWNGUJUHAWPWQUDSPQWM
UISZWOWQTUKULRBQZUMSZUNZOEUOZPUPUQZBURUOAXFBURAXBURUSZVDZDQZFTUKULRZCQZUM
SZXIGRWQTUKULRXBUMSUNZDIUOZCURUQXFNXHXNXFCURXHXKURUSZVDZXNXAWNFTUKULRZXKU
@@ -161759,7 +161773,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
rlimcn2 $p |- ( ph -> ( z e. A |-> ( B F C ) ) ~~>r ( R F S ) ) $=
( vc va vb co cmpt crli wbr cv cle cmin cabs cfv clt wi wral cr wrex wcel
crp wa ralrimiva adantr simprl rlimi simprr reeanv r19.26 prth wb simplrl
- cif simplrr wss cdm eqid dmmptd rlimss syl eqsstr3d ad2antrr sselda maxle
+ cif simplrr wss cdm eqid dmmptd rlimss syl eqsstrrd ad2antrr sselda maxle
syl3anc imbi1d syl5ibr ralimdva ifcl ancoms ad2antlr adantlr wceq fvoveq1
jca breq1d anbi1d oveq1 fvoveq1d imbi12d anbi2d oveq2 rspc2va sylan an32s
imim2d breq1 rspceaimv syl6an ex com23 syld syl5bir rexlimdvva mp2and imp
@@ -162356,7 +162370,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
(Contributed by Mario Carneiro, 26-May-2016.) $)
o1add2 $p |- ( ph -> ( x e. A |-> ( B + C ) ) e. O(1) ) $=
( cmpt caddc co co1 cvv cr wss wcel syl eqidd cof wral ralrimiva dmmptg
- cdm wceq o1dm eqsstr3d reex ssex offval2 o1add syl2anc eqeltrrd ) ABCDK
+ cdm wceq o1dm eqsstrrd reex ssex offval2 o1add syl2anc eqeltrrd ) ABCDK
ZBCEKZLUAMZBCDELMKNABCDELUOUPOFFACPQCORACUOUEZPADFRZBCUBURCUFAUSBCGUCBC
DFUDSAUONRZURPQIUOUGSUHCPUIUJSGHAUOTAUPTUKAUTUPNRUQNRIJUOUPULUMUN $.
@@ -162364,7 +162378,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
(Contributed by Mario Carneiro, 26-May-2016.) $)
o1mul2 $p |- ( ph -> ( x e. A |-> ( B x. C ) ) e. O(1) ) $=
( cmpt cmul co co1 cvv cr wss wcel syl eqidd wral wceq ralrimiva dmmptg
- cof cdm o1dm eqsstr3d reex ssex offval2 o1mul syl2anc eqeltrrd ) ABCDKZ
+ cof cdm o1dm eqsstrrd reex ssex offval2 o1mul syl2anc eqeltrrd ) ABCDKZ
BCEKZLUEMZBCDELMKNABCDELUOUPOFFACPQCORACUOUFZPADFRZBCUAURCUBAUSBCGUCBCD
FUDSAUONRZURPQIUOUGSUHCPUIUJSGHAUOTAUPTUKAUTUPNRUQNRIJUOUPULUMUN $.
@@ -162372,7 +162386,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
(Contributed by Mario Carneiro, 15-Sep-2014.) $)
o1sub2 $p |- ( ph -> ( x e. A |-> ( B - C ) ) e. O(1) ) $=
( cmpt cmin co co1 cvv cr wss wcel syl eqidd wral wceq ralrimiva dmmptg
- cof cdm o1dm eqsstr3d reex ssex offval2 o1sub syl2anc eqeltrrd ) ABCDKZ
+ cof cdm o1dm eqsstrrd reex ssex offval2 o1sub syl2anc eqeltrrd ) ABCDKZ
BCEKZLUEMZBCDELMKNABCDELUOUPOFFACPQCORACUOUFZPADFRZBCUAURCUBAUSBCGUCBCD
FUDSAUONRZURPQIUOUGSUHCPUIUJSGHAUOTAUPTUKAUTUPNRUQNRIJUOUPULUMUN $.
$}
@@ -162384,7 +162398,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
bounded. (Contributed by Mario Carneiro, 26-May-2016.) $)
lo1add $p |- ( ph -> ( x e. A |-> ( B + C ) ) e. <_O(1) ) $=
( vc vm vn wcel cle wi wral cr wrex wa vp cmpt clo1 caddc co wbr reeanv
- cv wss wb cdm wceq ralrimiva dmmptg syl eqsstr3d adantr rexanre readdcl
+ cv wss wb cdm wceq ralrimiva dmmptg syl eqsstrrd adantr rexanre readdcl
lo1dm adantl lo1mptrcl adantlr simplrl simplrr le2add syl22anc ralimdva
imim2d imbi2d ralbidv rspcev syl6an reximdv sylbird rexlimdvva ello1mpt
breq2 syl5bir rexcom syl6bb anbi12d readdcld 3imtr4d mp2and ) ABCDUBZUC
@@ -162404,7 +162418,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
(Contributed by Mario Carneiro, 26-May-2016.) $)
lo1mul $p |- ( ph -> ( x e. A |-> ( B x. C ) ) e. <_O(1) ) $=
( vc vm vn wcel cle wbr cr wrex wa vp cmpt clo1 cmul co wral reeanv wss
- cv wi wb cdm ralrimiva dmmptg syl lo1dm eqsstr3d adantr rexanre cc0 cif
+ cv wi wb cdm ralrimiva dmmptg syl lo1dm eqsstrrd adantr rexanre cc0 cif
wceq simprl simprr ifcl sylancl remulcld simplrr max2 sylancr lo1mptrcl
0re adantlr syl3anc mpan2d jca simplrl lemul12b syl22anc sylan2d imim2d
letr max1 breq2 imbi2d ralbidv rspcev syl6an reximdv sylbird rexlimdvva
@@ -162444,7 +162458,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
o1dif $p |- ( ph ->
( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> C ) e. O(1) ) ) $=
( cmpt co1 wcel cmin co cof syl cvv cc cr eqidd caddc wi o1sub expcom wss
- wral wceq cv wa subcld ralrimiva dmmptg o1dm eqsstr3d reex offval2 nncand
+ wral wceq cv wa subcld ralrimiva dmmptg o1dm eqsstrrd reex offval2 nncand
cdm ssex mpteq2dva eqtrd eleq1d sylibd o1add ex npcand impbid ) ABCDIZJKZ
BCEIZJKZAVHVGBCDELMZIZLNMZJKZVJAVLJKZVHVNUAHVHVOVNVGVLUBUCOAVMVIJAVMBCDVK
LMZIVIABCDVKLVGVLPQQACRUDCPKACVLUQZRAVKQKZBCUEVQCUFAVRBCABUGCKUHZDEFGUIZU
@@ -162734,7 +162748,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
18-May-2016.) $)
rlimneg $p |- ( ph -> ( k e. A |-> -u B ) ~~>r -u C ) $=
( cc0 cmin co cmpt cneg crli cc wcel cr wss wbr syl cv wa 0cnd rlimmptrcl
- cdm wral wceq ralrimiva dmmptg rlimss eqsstr3d 0cn sylancl rlimsub df-neg
+ cdm wral wceq ralrimiva dmmptg rlimss eqsstrrd 0cn sylancl rlimsub df-neg
rlimconst mpteq2i 3brtr4g ) AEBICJKZLIDJKEBCMZLDMNAEBICIDOAEUABPUBUCABCDE
FGHUDABQRIOPEBILINSABEBCLZUEZQACFPZEBUFVBBUGAVCEBGUHEBCFUITAVADNSVBQRHDVA
UJTUKULEBIUPUMHUNEBUTUSCUOUQDUOUR $.
@@ -162771,7 +162785,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
rlimsqzlem $p |- ( ph -> ( x e. A |-> C ) ~~>r E ) $=
( vz vy wbr wcel wa cr cmpt crli cv cle cmin co cabs cfv clt wi wral cpnf
cico wrex crp ad3antrrr wb ad2antrr elicopnf syl simprbda adantrr wss cdm
- eqid dmmptd rlimss eqsstr3d adantr sselda simplbda letrd anassrs adantllr
+ eqid dmmptd rlimss eqsstrrd adantr sselda simplbda letrd anassrs adantllr
cc simprr syldan subcld abscld ad4ant13 rlimcl rpre ad3antlr lelttr mpand
syl3anc expr an32s a2d ralimdva reximdva ralrimiva rlim3 3imtr4d mpd ) AB
CDUAZFUBQZBCEUAGUBQZKAOUCZBUCZUDQZDFUEUFZUGUHZPUCZUIQZUJZBCUKZOHULUMUFZUN
@@ -162831,7 +162845,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
function. (Contributed by Mario Carneiro, 26-May-2016.) $)
lo1le $p |- ( ph -> ( x e. A |-> C ) e. <_O(1) ) $=
( vm wcel cle wbr wi cr wrex wa vy vz cmpt clo1 cv cif simpr adantr ifcld
- wral wb ad2antrr simplr wss cdm wceq ralrimiva dmmptg syl eqsstr3d simprr
+ wral wb ad2antrr simplr wss cdm wceq ralrimiva dmmptg syl eqsstrrd simprr
lo1dm sseldd maxle syl3anc syl6bi imim1d adantlr adantrll simpl lo1mptrcl
syl2an simprll letr mpand expr adantrd a2d syld anassrs ralimdva reximdva
sylbid breq1 imbi1d rexralbidv rspcev syl6an rexlimdva ello1mpt 3imtr4d
@@ -162876,7 +162890,7 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $=
( vc vy wcel cle wbr cr wrex wa wfal c1 cc0 adantr cmpt co1 cabs cfv wral
cv wi fal cdiv co cmin cif clt cc reccld ralrimiva simpr 1re ifcl sylancl
crp 1rp a1i max1 sylancr rpgecld rpreccld crli wss wb cdm wceq dmmptg syl
- rlimi rlimss eqsstr3d cxr csup cpnf ressxr syl6ss supxrunb1 mpbird r19.29
+ rlimi rlimss eqsstrrd cxr csup cpnf ressxr syl6ss supxrunb1 mpbird r19.29
rexanre r19.29r adantlr wne subid1d fveq2d 1cnd 0le1 absidd oveq1d 3eqtrd
absdivd ad2antrr rprecred absrpcld max2 lediv2ad lensymd eqnbrtrd pm2.21d
rpred letrd expimpd ancomsd imim2d impcomd rexlimdva syl5 rexlimdvw mpand
@@ -164944,7 +164958,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/
fsumsplit $p |- ( ph -> sum_ k e. U C =
( sum_ k e. A C + sum_ k e. B C ) ) $=
( csu caddc co wcel cc0 wss cc wceq syldan wa cv cif wral cuz cfv cfn cun
- wo ssun1 syl5sseqr sselda ralrimiva olcd sumss2 syl21anc oveq12d 0cn ifcl
+ wo ssun1 sseqtrrid sselda ralrimiva olcd sumss2 syl21anc oveq12d 0cn ifcl
ssun2 sylancl fsumadd eleq2d elun syl6bb biimpa iftrue adantl wn noel cin
wi c0 elin syl5rbbr mtbii imnan sylibr imp iffalsed addid1d eqtrd addid2d
con2d jaodan sumeq2dv 3eqtr2rd ) ABDFKZCDFKZLMEFUAZBNZDOUBZFKZEWICNZDOUBZ
@@ -166413,7 +166427,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/
wb abscld adantrr rprege0 ad2antrl absid oveq2d eqtrd cmul adantlr adantr
remulcld fveq2i fsumabs eqbrtrid cun ssun2 nnred flle simprr letrd fznnfl
flge1nn mpbir2and syldan recnd absge0d chash cfn cn0 3syl reflcl breqtrd
- clt fzsplit syl5sseqr sselda mulid2d fsumge0 jca simprd syl31anc eqbrtrrd
+ clt fzsplit sseqtrrid sselda mulid2d fsumge0 jca simprd syl31anc eqbrtrrd
lemul1a hashcl nn0re elfzuz peano2nnd eluznn sylan peano2re fllep1 eluzle
cuz simpllr nfv nffv nfbr breq2 fveq2d breq1d imbi12d syl3c sylan2 fsumle
nfim fsumconst syl2anc biidd 0red mpan9 nnzd uzid rspcdva ssdomg hashdomi
@@ -170552,7 +170566,7 @@ seq n ( x. ,
fprodsplit $p |- ( ph -> prod_ k e. U C =
( prod_ k e. A C x. prod_ k e. B C ) ) $=
( cprod cmul co wcel c1 wa cc wceq adantl iffalsed cv cif iftrue prodeq2i
- ssun1 syl5sseqr sselda syldan eqeltrd cdif eldifn fprodss syl5eqr oveq12d
+ ssun1 sseqtrrid sselda syldan eqeltrd cdif eldifn fprodss syl5eqr oveq12d
cun ssun2 ax-1cn ifcl sylancl fprodmul wo eleq2d elun syl6bb biimpa c0 wn
cin disjel sylan mulid1d eqtrd ex con2d mulid2d jaodan prodeq2dv 3eqtr2rd
imp ) ABDFKZCDFKZLMEFUAZBNZDOUBZFKZEWBCNZDOUBZFKZLMEWDWGLMZFKEDFKAVTWEWAW
@@ -183689,7 +183703,7 @@ reduced fraction representation (no common factors, denominator
cn nnne0 simpr necon3ai 3syl gcdn0cl syl21anc simp2d dvdsmultr1d nnmulcld
wn simprd wi dvdslegcd syl31anc mp2and simprbi wb nnle1eq1 mpbid sylanbrc
breqtrd eqtrd dvdsmul2 dvdstr syl3anc opelxpd eqeltrd ralrimiva wfun wf1o
- cdm wfn crth f1ofn fnfun ssrab3 fndm syl5sseqr syl2an cmpt eqtri eqeltrrd
+ cdm wfn crth f1ofn fnfun ssrab3 fndm sseqtrrid syl2an cmpt eqtri eqeltrrd
eqtr3d sylib dfphi2 fveq2i syl6eqr funimass4 mpbird xpss12 sseqtr4i sseli
ccnv f1ocnvfv2 wf f1ocnv f1of ffvelrn cbvmptv opelxp funfvima2 imp syldan
rpmul eqelssd wf1 f1of1 elexi f1imaen sylancl eqbrtrrd xpfi hashen sylibr
@@ -188742,7 +188756,7 @@ with complex numbers (gaussian integers) instead, so that we only have
ord nn0nnaddcl fmptd nnex elmap sylibr addcld nnm1nn0 elfznn0 nn0mulcld
adantl add4d 1cnd add32d mul12d 3eqtr4d rspceeqv biimpar sylan2 vdwlem4
mpan2 vdwlem3 eqeltrd mptex eleq1 fveqeq2 rexlimdva 3imtr4d ssrdv ssun1
- r19.21bi cun fzsuc syl5sseqr sselda eqeq1 ifbieq2d nnred ltp1d peano2re
+ r19.21bi cun fzsuc sseqtrrid sselda eqeq1 ifbieq2d nnred ltp1d peano2re
clt ltnled mpbid breq1 notbid con2d elfzle2 impel iffalse add12d oveq1i
cr subcld ax-1cn sylancl npcand 3eqtr3d syl5eq addcomd 3eqtr2d cnvimass
subcl 3eqtrd fssdm vdwapid1 3sstr4d ex eqtr4d c0 wne eluzfz1 elfzuz3 cz
@@ -189873,7 +189887,7 @@ with complex numbers (gaussian integers) instead, so that we only have
cmin adantl eqbrtrrd cn ffvelrnd nnred 1red cr cn0 ssfid hashcl nn0re
3syl lesubaddd mpbid fveq2 snidg syl sseld eldifn syl6 mt2d hashunsng
wn wi mp2and breqan12rd mpbird crab sylancl hashbcval syl2anc simpl1l
- eqsstr3d simpr simpl3 rabid sylanbrc sseldd elpw sylibr fveq2d eqtr3d
+ eqsstrrd simpr simpl3 rabid sylanbrc sseldd elpw sylibr fveq2d eqtr3d
sylib cc breq2d oveq1 sseq1d anbi12d rspcev syl12anc snfi unfi nnnn0d
w3a simpl2 elpwid vex eleqtrrd nnm1nn0 fveqeq2 uncom syl6sseq ssundif
difexi diffi nn0cn ax-1cn pncan undif1 eldifd elpwunsn ssequn2 oveq1d
@@ -193504,7 +193518,7 @@ C_ dom ( S sSet <. I , E >. ) ) $=
(Revised by Mario Carneiro, 30-Apr-2015.) $)
ressbasss $p |- ( Base ` R ) C_ B $=
( cvv wcel cbs cfv wss cin ressbas inss2 syl6eqssr wn c0 cress reldmress
- co ovprc2 syl5eq fveq2d base0 0ss eqsstr3i syl6eqss pm2.61i ) AGHZCIJZBKU
+ co ovprc2 syl5eq fveq2d base0 0ss eqsstrri syl6eqss pm2.61i ) AGHZCIJZBKU
IUJABLBABCGDEFMABNOUIPZUJQIJZBUKCQIUKCDARTQEDARSUAUBUCULQBUDBUEUFUGUH $.
$}
@@ -193545,7 +193559,7 @@ C_ dom ( S sSet <. I , E >. ) ) $=
the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.) $)
ressinbas $p |- ( A e. X -> ( W |`s A ) = ( W |`s ( A i^i B ) ) ) $=
( wcel cvv cress co cin wceq elex wss w3a eqid ressid2 syl3an eqtr4d csts
- wn wa ssid incom df-ss biimpi syl5eq syl5sseqr inex1g 3expb cnx cbs inass
+ wn wa ssid incom df-ss biimpi syl5eq sseqtrrid inex1g 3expb cnx cbs inass
cfv cop inidm ineq2i eqtr2i opeq2i oveq2i ressval2 inss1 sstr mpan2 con3i
3eqtr4a pm2.61ian c0 reldmress ovprc1 adantr syl ) ADFAGFZCAHIZCABJZHIZKZ
ADLCGFZVLVPBAMZVQVLUAVPVRVQVLVPVRVQVLNVMCVOABVMCGGVMOZEPVRBVNMZVQVQVLVNGF
@@ -195238,7 +195252,7 @@ C_ dom ( S sSet <. I , E >. ) ) $=
( cfv cop va vc vd ve cds cmpt crn cc0 csn cun cxr clt csup cmpo cplusg
cv co cxp c2nd chom cixp c1st cco cpr wss cple wbr wa copab cvsca cmulr
wral cip cgsu ctopn ccom cpt cbs eqid prdsplusg prdsmulr prdsvsca eqidd
- prdsbas prdsval pleid cvv wcel fvexi xpex prss anbi1i opabssxp eqsstr3i
+ prdsbas prdsval pleid cvv wcel fvexi xpex prss anbi1i opabssxp eqsstrri
vex opabbii ssexi a1i cnx cts csca snsstp2 ssun1 sstri ssun2 prdsvallem
ctp ) AJCGHCCBIBUPZGUPZSZXHHUPZSZXHESZUESUQUFUGUHUIUJUKULUMUNZDUOSZFUAU
BCCURZCUCUDUBUPZUAUPZUSSZGHCCBIXJXLXMUTSUQVAUNZUQXRXTSBIXHUCUPSXHUDUPSX
@@ -195255,7 +195269,7 @@ C_ dom ( S sSet <. I , E >. ) ) $=
Mario Carneiro, 16-Aug-2015.) $)
prdsless $p |- ( ph -> .<_ C_ ( B X. B ) ) $=
( vf vg vx cv cfv cpr wss cple wbr wral wa copab cxp prdsle wcel anbi1i
- vex prss opabbii opabssxp eqsstr3i syl6eqss ) AGPSZQSZUABUBZRSZURTVAUST
+ vex prss opabbii opabssxp eqsstrri syl6eqss ) AGPSZQSZUABUBZRSZURTVAUST
VADTUCTUDRFUEZUFZPQUGZBBUHZARBCDEPQFGHIJKLMNOUIVDURBUJUSBUJUFZVBUFZPQUG
VEVGVCPQVFUTVBURUSBPULQULUMUKUNVBPQBBUOUPUQ $.
$}
@@ -196336,7 +196350,7 @@ topology is based on the order and not the extended metric (which would
eliun opth syl5bi eqeq2 biimprd syl6 3expa rexlimdvva sylbid ralrimivva
impd alrimiv mo2icl fofn opeq2 breq1d mobidv ralrn ralbidv mpbird opeq1
breq1 ralxp sylibr ssralv sylc dffun7 sylanbrc eqimss2 iunss sylib snss
- opeldm sylbir ralimi sylbi eleq1d eleq1 dfss3 eqsstr3d eqssd df-fn ) AC
+ opeldm sylbir ralimi sylbi eleq1d eleq1 dfss3 eqsstrrd eqssd df-fn ) AC
UBZCUCZBBUDZQCUUFUEACULZNUFZOUFZCUGZOUHZNUUERZUUDAUUGHFGFHUFZEUIZGUFZEU
IZUJZUUMUUODUKZEUIZUJZUMZUNZUNZULZUVDUVBULZHFRUVEHFUVEUVAULZGFRUVFGFUUQ
UUSUUNUUPUOZUUREVEZUPUQGFUVAURUSUQHFUVBURUSACUVCMUTVAAUUEEVBZUVIUDZVCUU
@@ -196470,7 +196484,7 @@ topology is based on the order and not the extended metric (which would
weq mobidv mpbird ralxp sylanbrc eqtr3d adantl elsni simpl2im ex sylan2br
impel eqtr4d anassrs rexlimdva sylbid alrimiv mo2icl breq1d ssralv dffun7
fofn breq1 sylc eqimss2 r19.21bi adantrl syl5eqssr simprl opelxpi sylancl
- snid sseldd eleq1d eleq1 dfss3 eqsstr3d eqssd df-fn ) ADUHZDUIZIBUJZUKDUV
+ snid sseldd eleq1d eleq1 dfss3 eqsstrrd eqssd df-fn ) ADUHZDUIZIBUJZUKDUV
QULADUMZUDUNZUEUNZDUOZUEUPZUDUVPUQZUVOAUVRLJMUDILUNZGURZVBZMUNZUWDEUSZGUR
ZUTZVAZUMZUWLUWJUMZLJUQUWMLJUWJIUWFUJZULZUWMMUDIUWFUWIUWJUWJVCZUWHGVNZVDZ
UWNUWJVEVFVGLJUWJVHVIADUWKAUDBCDEFGHIJKLMOPQRSTUAUBVJZVKVLAUVPIGVMZUJZVOU
@@ -197572,7 +197586,7 @@ the closure of the set with the element removed (Section 0.6 in
$( A set is closed iff it is equal to its closure. (Contributed by Stefan
O'Rear, 31-Jan-2015.) $)
mrcidb $p |- ( C e. ( Moore ` X ) -> ( U e. C <-> ( F ` U ) = U ) ) $=
- ( cmre cfv wcel wceq mrcid wa simpr mrcssv adantr eqsstr3d mrccl eqeltrrd
+ ( cmre cfv wcel wceq mrcid wa simpr mrcssv adantr eqsstrrd mrccl eqeltrrd
wss syldan impbida ) ADFGHZBAHBCGZBIZABCDEJUAUCKZUBBAUAUCLZUAUCBDRUBAHUDB
UBDUEUAUBDRUCABCDEMNOABCDEPSQT $.
@@ -197589,7 +197603,7 @@ the closure of the set with the element removed (Section 0.6 in
$( The closure of a set is a superset. (Contributed by Stefan O'Rear,
31-Jan-2015.) $)
mrcssid $p |- ( ( C e. ( Moore ` X ) /\ U C_ X ) -> U C_ ( F ` U ) ) $=
- ( vs cmre cfv wcel wss wa cv crab cint ssintub mrcval syl5sseqr ) ADGHIBD
+ ( vs cmre cfv wcel wss wa cv crab cint ssintub mrcval sseqtrrid ) ADGHIBD
JKBFLJFAMNBBCHFBAOABCDFEPQ $.
$( A set is closed iff it contains its closure. (Contributed by Stefan
@@ -197853,7 +197867,7 @@ the closure of the set with the element removed (Section 0.6 in
mrieqv2d $p |- ( ph -> ( S e. I <->
A. s ( s C. S -> ( N ` s ) C. ( N ` S ) ) ) ) $=
( vx wcel wpss cfv wa w3a 3ad2ant1 adantr cvv cv wi wal wn wex pssnel csn
- 3ad2ant3 cdif cmre wceq simprr difsnb sylib simpl3 pssssd ssdifd eqsstr3d
+ 3ad2ant3 cdif cmre wceq simprr difsnb sylib simpl3 pssssd ssdifd eqsstrrd
simpl2 mrissd ssdifssd mrcssd difssd mrcssidd ismri2dad sspsstrd exlimddv
simprl sseldd ssnelpssd 3expia alrimiv ex wral wne elfvexd wss difexg syl
ssexd simp1r difsnpss simp2 psseq1d mpbird simp3 mpd fveq2d mpbid spcimdv
@@ -198038,7 +198052,7 @@ u C_ ( N ` ( v u. t ) ) /\ ( u u. t ) e. I )
( ( F \ { Y } ) u. ( H u. { g } ) ) e. I ) ) $=
( cv wcel csn cdif wn cun wa wex wrex cfv wss cmre simpr ssun2 difundir
adantr wceq cin c0 incom ssdifin0 syl syl5eqr minel difsnb sylib uneq2d
- syl2anc syl5eq syl5sseqr mrissd ssdifssd mrcssidd sstrd mrcssvd mrcidmd
+ syl2anc syl5eq sseqtrrid mrissd ssdifssd mrcssidd sstrd mrcssvd mrcidmd
unssd mrcssd sseqtrd sseldd sseldi ismri2dad pm2.65da nss simprl simprr
ssun1 ssneldd unass wral cpw difss unss1 mp1i mrissmrid fveq2d neleqtrd
difss2d mreexmrid syl5eqelr jca32 ex eximdv mpd df-rex sylibr ) AEUCZGU
@@ -198096,7 +198110,7 @@ f C_ ( N ` ( g u. h ) ) /\ ( f u. h ) e. I ) ->
csn wral wss animorrl mreexexlem3d wne n0 biimpi adantl wn mreexexlem2d
wex simpr w3a 3anass cvv ad2antrr elfvexd simpr2 difsnb ssdifssd ssdifd
sylib difun1 syl6sseqr simpr1 uncom uneq2i unass difsnid uneq1d syl5eqr
- eqsstr3d syl5eq syl fveq2d sseqtr4d simpr3 csuc wo com wi simplr dif1en
+ eqsstrrd syl5eq syl fveq2d sseqtr4d simpr3 csuc wo com wi simplr dif1en
3anan12 sylbir expcom syl2anc orim12d mpd mreexexlemd ad3antrrr difss2d
wal ssexd simprl simplr1 snssd unssd sselpwd ad3antlr cin simprrl en2sn
elpwid el2v a1i incom disjdif ssdifin0 unen syl22anc eqbrtrrd rexlimddv
@@ -202241,7 +202255,7 @@ is always a subcategory (and it is full, meaning that all morphisms of
wf funcf1 eqidd subcfn subcss1 fssresd simpld rescbas feq2d mpbid wfn cxp
fnmpti eqcomd fndm sqxpeqd fneq12d mpbii wss simprl simprr funcf2 subcss2
resf2nd feq1d mpbird reschom oveq12d feq23d eleq2d biimpar subcid fveq12d
- oveqd eqsstr3d sselda funcid subcidcl 3eqtr4d simp21 simp22 simp23 simp3l
+ oveqd eqsstrrd sselda funcid subcidcl 3eqtr4d simp21 simp22 simp23 simp3l
simp3r subccocl funcco rescco opeq12d fveq1d oveq123d isfuncd df-br sylib
w3a eqeltrd ) ADEUETZDUFIZEUGZUGZJZYPUHIZUIZBEUJTZCUKTZAYPYTHYRHULZDUHIZI
ZUUEEIZJZUMZUIZUUBAHDEBCUKTZBUPIZFGUNZAUUAUUJYTAUUAUUKUHIZUUJAYPUUKUHUUNU
@@ -205326,7 +205340,7 @@ is always a subcategory (and it is full, meaning that all morphisms of
( cfunc co wcel chomf cfv ccat ccomf wceq eqid vf cv cxp cres cresc eqidd
cress cbs setccat syl adantr wss setcbas sseqtrd fullresc simpld resssetc
wa eqtr3d simprd funcrcl adantl fullsubc subccat funcpropd csubc funcres2
- eqsstr3d simpr sseldd ex ssrdv ) AUAECLMZEBLMZAUAUBZVMNZVOVNNAVPURZVMVNVO
+ eqsstrrd simpr sseldd ex ssrdv ) AUAECLMZEBLMZAUAUBZVMNZVOVNNAVPURZVMVNVO
VQVMEBBOPZFFUCUDZUEMZLMZVNVQEEVTCQVQEOPUFVQERPUFVQBFUGMZOPZVTOPZCOPZVQWCW
DSZWBRPZVTRPZSZVQBUHPZBWBFVTVRWJTZVRTZABQNZVPADGNWMJBDGHUIUJUKZAFWJULVPAF
DWJKABDGHJUMUNUKZWBTVTTZUOZUPVQWCWESZWGCRPZSZAWRWTURVPABCDFGHIJKUQUKZUPUS
@@ -213207,7 +213221,7 @@ net proof size (existence part)? $)
cv acsmred simplr elin1d elpwid mrissmrid ralrimiva dfss3 sylibr csn cdif
cun elfpw sylib simpld difss2d snssd unssd simprd snfi unfi sylanbrc wceq
sylancl ad4antr simpllr snidg 3syl eleqtrrd ismri2dad ad3antrrr neldifsnd
- elun2 ssneldd difsnb ssun1 syl5sseqr ssdifd eqsstr3d sstrd eqsstrd mrcssd
+ elun2 ssneldd difsnb ssun1 sseqtrrid ssdifd eqsstrrd sstrd eqsstrd mrcssd
wi ssdifssd sseld mtod rspcimdv syl5bi impancom ralrimiv acsficl2d notbid
ex wb wrex ralnex syl6bbr mpbird an32s ismri2dd impbida ) ACDMZCUBZNUCZDO
ZAXHPZKUGZDMZKXJUDZXKXLXNKXJXLXMXJMZPZBCXMDEFABFUEQMZXHXPABFGUHZUFHIAXHXP
@@ -214161,7 +214175,7 @@ net proof size (existence part)? $)
( vy vz cdir wcel cv wbr wa wex cuni eleq2d wral ccom wss wceq wrex dirdm
cdm syl5eq anbi12d cxp ccnv wrel cid cres isdir simprrd codir sylib breq1
eqid ibi anbi1d exbidv anbi2d rspc2v syl5com sylbid crn reldir relelrn ex
- sylan cun ssun2 dmrnssfld sstri syl5sseqr sseld syld adantrd ancrd eximdv
+ sylan cun ssun2 dmrnssfld sstri sseqtrrid sseld syld adantrd ancrd eximdv
df-rex syl6ibr 3impib ) DIJZBEJZCEJZBAKZDLZCWEDLZMZAEUAZWBWCWDMZWHANZWIWB
WJBDOOZJZCWLJZMZWKWBWCWMWDWNWBEWLBWBEDUCZWLFDUBUDZPWBEWLCWQPUEWBGKZWEDLZH
KZWEDLZMZANZHWLQGWLQZWOWKWBWLWLUFDUGDRSZXDWBDUHZUIWLUJDSMZDDRDSZXEWBXGXHX
@@ -214175,7 +214189,7 @@ net proof size (existence part)? $)
25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.) $)
tsrdir $p |- ( A e. TosetRel -> A e. DirRel ) $=
( ctsr wcel cdir wrel cid cuni cres wss ccom cxp ccnv cps syl jca wceq eqid
- wa syl5eqssr eqsstr3d tsrps psrel cin psref2 inss1 syl6eqssr cdm crn psdmrn
+ wa syl5eqssr eqsstrrd tsrps psrel cin psref2 inss1 syl6eqssr cdm crn psdmrn
pstr2 simpld sqxpeqd cun istsr simprbi relcoi2 cnvresid cnvss coss1 relcoi1
relcnv ax-mp relcnvfld reseq2d coss2 unssd sstrd isdir mpbir2and ) ABCZADCA
EZFAGGZHZAIZRAAJAIZVLVLKZALZAJZIZRVJVKVNVJAMCZVKAUAZAUBNZVJVTVNWAVTVMAVQUCA
@@ -216266,7 +216280,7 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is
resmhm2 $p |- ( ( F e. ( S MndHom U ) /\ X e. ( SubMnd ` T ) ) ->
F e. ( S MndHom T ) ) $=
( vx vy cmhm co wcel cfv wa cmnd cbs wf cplusg wceq c0g eqid csubmnd wral
- cv w3a mhmrcl1 submrcl anim12i mhmf submbas submss eqsstr3d syl2an mhmlin
+ cv w3a mhmrcl1 submrcl anim12i mhmf submbas submss eqsstrrd syl2an mhmlin
wss 3expb adantlr ressplusg ad2antlr oveqd eqtr4d ralrimivva adantr subm0
fss mhm0 adantl 3jca ismhm sylanbrc ) DACIJKZEBUALZKZMZANKZBNKZMAOLZBOLZD
PZGUCZHUCZAQLZJDLZVSDLZVTDLZBQLZJZRZHVPUBGVPUBZASLZDLZBSLZRZUDDABIJKVJVNV
@@ -222599,7 +222613,7 @@ permutation associated with the composition of these two elements (in
$( The center is a subset of the base field. (Contributed by Thierry
Arnoux, 21-Aug-2023.) $)
cntrss $p |- ( Cntr ` M ) C_ B $=
- ( ccntr cfv ccntz eqid cntrval cntzssv eqsstr3i ) BDEABFEZEAABKCKGZHAABKC
+ ( ccntr cfv ccntz eqid cntrval cntzssv eqsstrri ) BDEABFEZEAABKCKGZHAABKC
LIJ $.
$}
@@ -223183,7 +223197,7 @@ operation is a permutation group (group consisting of permutations), see
SO, 9-Jul-2018.) $)
symgbasfi $p |- ( A e. Fin -> B e. Fin ) $=
( vf cfn wcel cmap co mapfi anidms cv wf wf1o symgbas f1of ss2abi eqsstri
- cab wceq mapvalg syl5sseqr ssfid ) AGHZAAIJZBUEUFGHAAKLUEAAFMZNZFTZBUFBAA
+ cab wceq mapvalg sseqtrrid ssfid ) AGHZAAIJZBUEUFGHAAKLUEAAFMZNZFTZBUFBAA
UGOZFTUIFABCDEPUJUHFAAUGQRSUEUFUIUAAAGGFUBLUCUD $.
$( The function value of a permutation. (Contributed by AV,
@@ -224185,7 +224199,7 @@ operation is a permutation group (group consisting of permutations), see
-. dom ( ( F o. G ) \ _I ) C_ X ) $=
( wf1o cid cdif cdm wss ccom wn wa ccnv wceq coass f1of syl difeq1d dmeqd
wf wxo wo excxor f1ococnv1 coeq1d fcoi2 sylan9eq syl5eqr adantr cun mvdco
- cres f1omvdcnv ad2antrr simprl eqsstrd simprr unssd syl5ss eqsstr3d con3d
+ cres f1omvdcnv ad2antrr simprl eqsstrd simprr unssd syl5ss eqsstrrd con3d
expr expimpd f1ococnv2 coeq2d fcoi1 sylan9eqr syl5eq ad2antlr jaod syl5bi
ancomsd 3impia ) AABEZAACEZBFGZHZDIZCFGZHZDIZUAZBCJZFGHZDIZKZWBVRWAKZLZVR
KZWALZUBVNVOLZWFVRWAUCWKWHWFWJWKVRWGWFWKVRLWEWAWKVRWEWAWKVRWELZLZVTBMZWCJ
@@ -227213,7 +227227,7 @@ P pGrp ( G |`s S ) ) $=
ssel syld impr sylan2b en1b fveq2d cvv vuniex ax-mp syl6eq sumeq2dv ssexg
hashsng sylancr eqeltrd sseldi syl2anc wb mpbird oveq1d 3eqtr4rd wral cdm
ssrab3 mulid1d rabexd inss2 crn cxp wrel cpr wrex relopabi relssdmrn erdm
- pwexd errn xpexd ecelqsg eqeltrrd snelpwi adantl elind eqsstr3d snss sneq
+ pwexd errn xpexd ecelqsg eqeltrrd snelpwi adantl elind eqsstrrd snss sneq
eqeq2d syl5ibrcom adantrl unieq unisn syl6req impbid1 hashen eqtr3d diffi
en3d eldifi sylan2 fsumcl subaddd breqtrrd ) AELGUCZMUDZUEZUAUFZUGUHZUAUI
ZLUGUHZMUGUHZUJUKZULABCUADEFGHIJKLMNOPQRSTUMAUWJUWGUPUWIUWGUNUKZUWHUPAUWB
@@ -227719,7 +227733,7 @@ same as the index (number of cosets) of the normalizer of the Sylow
csg cgrp ad2antrr nmzsubg subgslw syl3anc ssnmz cvv cplusg fvexi rabex2
wb ressplusg ax-mp sylow2 cnsg conjnsg sylan eqeq2 syl5ibrcom rexlimdva
nmznsg mpd eqtrd cz eqeltrd impbida 3bitr3d rabsn fveq2d hashsng oveq2d
- eqsstr3d rabbidva breqtrd cn cprime prmnn cn0 hashcl nn0zd 1zzd moddvds
+ eqsstrrd rabbidva breqtrd cn cprime prmnn cn0 hashcl nn0zd 1zzd moddvds
mpbird c2 cuz prmuz2 clt eluz2b2 cr nnre 1mod sylbi 3syl ) AEHUFUGZUHUI
ZEUJUGZUKEUJUGZUKAUVLUVMULZEUVKUKUMUGZUNUOZAEUVKUCUSZMUSZGUGZUVRULZUCHI
UPUGZUTUIZUQZMUVJURZUHUIZUMUGUVOUNADUDMEGDUSZUDUSZVAUVJVBUEUSUWFGUGUWGU
@@ -229420,7 +229434,7 @@ an extension of the previous (inserting an element and its inverse at
efgredlemf $p |- ( ph -> ( ( A ` K ) e. W /\ ( B ` L ) e. W ) ) $=
( vi cfv wcel cc0 chash cfzo co cword wf c0 csn cdm cdif cv cmin wral
c1 crn efgsdm simp1bi syl eldifad wrdf cfz fzossfz cz cn0 lencl nn0zd
- wceq fzoval syl5sseqr cn efgredlema simpld syl5eqel ffvelrnd simpl2im
+ wceq fzoval sseqtrrid cn efgredlema simpld syl5eqel ffvelrnd simpl2im
fzo0end sseldd jca ) ARHURUAUSSIURUAUSAUTHVAURZVBVCZUARHAHUAVDZUSZWSU
AHVEAHWTVFVGZAHLVHZUSZHWTXBVIZUSZUKXDXFUTHURJUSUQVJZHURXGVMVKVCZHURMU
RVNUSUQVMWRVBVCVLBCDEFGJKLMUQNOPHQTUAUDUEUFUGUHUIVOVPVQVRZUAHVSVQAUTW
@@ -229579,7 +229593,7 @@ an extension of the previous (inserting an element and its inverse at
efgredlemd $p |- ( ph -> ( A ` 0 ) = ( B ` 0 ) ) $=
( vi cc0 chash cfv c1 cmin co cfzo cres cs1 cconcat cword wcel wceq
c0 cv crn wral efgsdm simp1bi syl eldifad wf wrdf cfz fzossfz lencl
- cz nn0zd fzoval syl5sseqr cn simpld fzo0end syl5eqel ffvelrnd s1cld
+ cz nn0zd fzoval sseqtrrid cn simpld fzo0end syl5eqel ffvelrnd s1cld
cn0 sseldd wa lbfzo0 sylibr ccatval1 syl3anc simprd eqtr4d c2 caddc
clt wbr cr sseldi nn0red 2rp ltaddrp sylancl cle lem1d wb mpbir2and
fznn syl2anc fveq2d fvresd fveq2i wi fveq1 imbi12d eqeq2d csn sylbi
@@ -232657,7 +232671,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
wf inss1 ax-mp syl6eq eqtr3d oveq2d 3eqtr4d ex cplusg ccom cseq crn wfo
f1ofo ad2antll eqsstrd cores seqeq3d fveq1d fssres sylancl feq1d biimpa
forn eqid syldan resss rnssi cntzidss simprl f1of1 suppssdm fssdm ssind
- wf1 f1ss wi fex ressuppss syl5ibr adantl impcom gsumval3 syl5sseqr expr
+ wf1 f1ss wi fex ressuppss syl5ibr adantl impcom gsumval3 sseqtrrid expr
sseq2 exlimdv expimpd cfsupp wbr cfn wfun fsuppimp simprd fz1f1o mpjaod
wo ) ADHUCUDZUEUFZEDGUGZUHUDZEDUHUDZUFZYRUIUJZUKULZUMUUDUOUDZYRUAUPZUNZ
UAUQZVIZAYSUUCAYSVIZEUBBGURZHUSZUHUDZEUBBHUSZUHUDZUUAUUBAUUNUUPUFYSAUUN
@@ -232691,7 +232705,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
( co wcel wa adantr vf vk vx csupp c0 wceq cgsu chash cfv cn c1 cv wf1o
cfz wex cmpt cvv c0g fvexi a1i ssidd gsumcllem cmnd gsumz syl2anc eqtrd
oveq2d mndidcl syl eqeltrd ex cplusg ccom cseq eqid wf crn simprl f1of1
- wss wf1 ad2antll suppssdm fssdm f1ss ssid f1ofo forn syl5sseqr gsumval3
+ wss wf1 ad2antll suppssdm fssdm f1ss ssid f1ofo forn sseqtrrid gsumval3
wfo cuz nnuz syl6eleq f1f fco ffvelrnda mndcl 3expb sylan seqcl exlimdv
expr expimpd cfn wo fz1f1o mpjaod ) ADGUDQZUEUFZEDUGQZCRZXIUHUIZUJRZUKX
MUNQZXIUAULZUMZUAUOZSZAXJXLAXJSZXKGCXTXKEUBBGUPZUGQZGXTDYAEUGABCUQUBDFX
@@ -232720,7 +232734,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
wa a1i ssidd gsumcllem oveq2d f1of ffvelrnda feqmptd eqidd fmptco 3eqtr4d
wf cplusg cseq ccnv coass cid cres f1ococnv2 coeq1d wss ad2antll suppssdm
ex fssdm f1ss f1f fcoi2 3syl eqtrd syl5reqr coeq2d syl6eqr seqeq3d fveq1d
- eqid crn simprl ssid wfo f1ofo forn syl5sseqr gsumval3 fco rncoss sylancl
+ eqid crn simprl ssid wfo f1ofo forn sseqtrrid gsumval3 fco rncoss sylancl
cntzidss f1ocnv f1co csn cdif cima suppimacnv eqcomd 3sstr4d imass2 cnvco
fex imaeq1i imaco eqtri rnco2 3sstr4g f1oexrnex coexg sseq1d mpbird expr
wb exlimdv expimpd cfsupp wbr cfn wo wfun fsuppimp simprd fz1f1o mpjaod )
@@ -232995,7 +233009,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
gsumadd $p |- ( ph ->
( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) $=
( cfv wcel eqid ccmn cmnd cmnmnd syl csubmnd submid ssid wss wceq cntzcmn
- ccntz sylancl syl5sseqr gsumzadd ) ABCDCEFGHIFULSZJKLUPUAZAFUBTZFUCTZMFUD
+ ccntz sylancl sseqtrrid gsumzadd ) ABCDCEFGHIFULSZJKLUPUAZAFUBTZFUCTZMFUD
UEZNQRAUSCFUFSTUTCFJUGUEACCCUPSZCUHZAURCCUIVACUJMVBCCFUPJUQUKUMUNOPUO $.
$}
@@ -233073,7 +233087,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
wceq wn wi cin c0 noel eleq2 mtbiri elin sylnib imnan sylibr imp iffalsed
oveq12d ffvelrnda mndrid syl2an2r eqtrd con2d mndlid wo cun eleq2d syl6bb
elun biimpa mpjaodan mpteq2dva eqtr4d mndidcl eqidd offval2 reseq1d ssun1
- oveq2d syl5sseqr mpteq2ia resmpt 3eqtr4a cntzidss eldifn suppss2 gsumzres
+ oveq2d sseqtrrid mpteq2ia resmpt 3eqtr4a cntzidss eldifn suppss2 gsumzres
cdif ssun2 3eqtr4d ) AHUCBUCUDZDUEZUUAGUFZJUGZUHZUCBUUAEUEZUUCJUGZUHZFUIU
JZUKUJHUUEUKUJZHUUHUKUJZFUJHGUKUJHGDULZUKUJZHGEULZUKUJZFUJABCFGUMZHUNUFZU
OUFZUFZUUEHUUHIJKLMNOPQABCDUCGIUPJRQJUPUEAJHUQMURUSZTUTZABCEUCGIUPJRQUUTT
@@ -233135,7 +233149,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements
19-Dec-2014.) (Revised by AV, 5-Jun-2019.) $)
gsumsplit2 $p |- ( ph -> ( G gsum ( k e. A |-> X ) ) =
( ( G gsum ( k e. C |-> X ) ) .+ ( G gsum ( k e. D |-> X ) ) ) ) $=
- ( cmpt cgsu co cres fmpttd gsumsplit ssun1 syl5sseqr resmptd oveq2d ssun2
+ ( cmpt cgsu co cres fmpttd gsumsplit ssun1 sseqtrrid resmptd oveq2d ssun2
cun oveq12d eqtrd ) AHGBJUAZUBUCHUODUDZUBUCZHUOEUDZUBUCZFUCHGDJUAZUBUCZHG
EJUAZUBUCZFUCABCDEFUOHIKLMNOPAGBJCQUERSTUFAUQVAUSVCFAUPUTHUBAGBDJADEULZDB
DEUGTUHUIUJAURVBHUBAGBEJAVDEBEDUKTUHUIUJUMUN $.
@@ -235235,7 +235249,7 @@ mapping the (infinite, but finitely supported) cartesian product of
dprdspan $p |- ( G dom DProd S -> ( G DProd S ) = ( K ` U. ran S ) ) $=
( vk cdprd cdm wbr co crn cuni cfv id eqidd csubg wcel wss syl wral iunss
cbs cmre cgrp cacs dprdgrp eqid subgacs acsmre 3syl cv ciun wfn wceq ffnd
- dprdf fniunfv simpl simpr dprdub ralrimiva sylibr eqsstr3d dprdssv syl6ss
+ dprdf fniunfv simpl simpr dprdub ralrimiva sylibr eqsstrrd dprdssv syl6ss
mrccl syl2anc eqimss sylib r19.21bi mrcssidd adantr sstrd dprdlub mrcsscl
wa dprdsubg syl3anc eqssd ) BAFGHZBAFIZAJKZCLZVSAWBEBAGZVSMVSWCNVSBOLZBUA
LZUBLPZWAWEQWBWDPVSBUCPWDWEUDLPWFABUEWEBWEUFZUGWDWEUHUIZVSWAVTWEVSWAEWCEU
@@ -235631,7 +235645,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
( wcel cfv co wceq wbr wa c2nd c1st csn cima cv cmpt cdprd cop df-ov wrel
1st2nd sylan simpr eqeltrrd df-br sylibr wb adantr elrelimasn mpbird eqid
syl oveq2 ovex fvmpt3i fveq2d 3eqtr4a sneq imaeq2d oveq1 mpteq12dv breq2d
- cdm wral ralrimiva wss 1stdm sseldd rspcdva dmmpti a1i dprdub eqsstr3d )
+ cdm wral ralrimiva wss 1stdm sseldd rspcdva dmmpti a1i dprdub eqsstrrd )
AIBPZUAZICQZIUBQZEBIUCQZUDZUEZWIEUFZCRZUGZQZFWNUHRWFWIWHCRZWIWHUIZCQWOWGW
IWHCUJWFWHWKPZWOWPSWFWRWIWHBTZWFWQBPWSWFIWQBABUKZWEIWQSJIBULUMZAWEUNUOWIW
HBUPUQWFWTWRWSURAWTWEJUSWIWHBUTVCVAZEWHWMWPWKWNWLWHWICVDWNVBZWIWLCVEZVFVC
@@ -235650,7 +235664,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
wceq ffun funiunfv wral c1st resss sseli dprd2dlem2 sylan2 cvv c2nd cop
wa wrel 1st2nd syl2an simpr eqeltrrd fvex opelresi simplbi ovex imaeq2d
oveq1 mpteq12dv oveq2d elrnmpt1s sylancl elssuni sstrd ralrimiva sylibr
- sneq iunss eqsstr3d cpw sselda syldan dmmpti a1i imassrn mresspw syl5ss
+ sneq iunss eqsstrrd cpw sselda syldan dmmpti a1i imassrn mresspw syl5ss
frnd sspwuni sylib mrccl syl2anc adantr oveq2 fvmpt3i adantl df-ov ffnd
wfn ad2antrr simplr wb elrelimasn biimpa df-br sylanbrc fnfvima syl3anc
syl5eqel mrcssidd eqsstrd dprdlub elpw fmpttd mrcssvd mrcssd mrcsscl
@@ -235688,7 +235702,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
c2nd sneq oveq1 mpteq12dv breq2d adantr 1stdm sylan rspcdva 3ad2antr1 a1i
sseldd cop 1st2nd simpr df-br sylibr elrelimasn mpbird sneqd eqeq12d fvex
wb opth necon3bid mpbid dprdcntz df-ov oveq2 fvmpt3i fveq2d 3eqtr4a eqtrd
- oveq1d oveq2d sseqtr4d ad2antrr eqsstr3d eqsstrd sstrd cdif cin cun snssd
+ oveq1d oveq2d sseqtr4d ad2antrr eqsstrrd eqsstrd sstrd cdif cin cun snssd
cuni sylib syl6eq eleq1d simplbi crn imassrn frnd syl5ss sspwuni mrcssidd
mrccl difss ax-mp elrnmpt1s unissd mrcssd syl6eqr syl3anc sseqtrd subg0cl
c0 dprdspan ccntz cgrp dprdgrp resiun2 iunid reseq2i eqtr3i wrel relssres
@@ -235697,7 +235711,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
dprdub subgacs acsmre ssequn1 syl5req reseq2d resundi difeq1d difundir wn
undif2 eqtr3d neirr brresi sylbir syl6bi mtoi disjsn disj3 eqcomd imaundi
eldifsni uneq2d unieqd uniun cpw mresspw unss12 lsmunss imass2 mpan2 rgen
- sseqtri ralbidv ralsn wfun ffund resss eqsstr3i fdmd syl5sseqr funimassov
+ sseqtri ralbidv ralsn wfun ffund resss eqsstrri fdmd sseqtrrid funimassov
wf biimpa ffvelrnd fmpttd wfn fnmpti fnressn opeq2d dprdsn simprd disjdif
dprdsubg dprdcntz2 adantlr dprd2dlem1 resmpt oveq2i lsmsubg mrcsscl sslin
lsmlub mpbi2and dprdres simpld df-ima unieqi fveq2i eqimss ss2in dprddisj
@@ -235877,9 +235891,9 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
( ( Y e. I -> ( X =/= Y -> ( S ` X ) C_ ( Z ` ( S ` Y ) ) ) ) /\
( ( S ` X ) i^i ( K ` U. ( S " ( I \ { X } ) ) ) ) C_ { .0. } ) ) $=
( vy wcel wa wne cfv wss wi csn cdif cima cuni cin wo adantr eleq2d
- cun wceq elun syl6bb cdprd cdm wbr ad2antrr csubg syl5sseqr fssresd
+ cun wceq elun syl6bb cdprd cdm wbr ad2antrr csubg sseqtrrid fssresd
cres ssun1 fdmd simplr simprl simprr dprdcntz fvres ad2antlr fveq2d
- ad2antrl 3sstr3d exp32 co dprdub eqsstr3d cbs dprdssv ssun2 cntz2ss
+ ad2antrl 3sstr3d exp32 co dprdub eqsstrrd cbs dprdssv ssun2 cntz2ss
eqid sylancr jaod sylbid clsm cmre cgrp cacs dprdgrp subgacs acsmre
sstrd syl difundir difeq1d c0 simpr snssd sslin incom syl5eqr sseq0
3syl syl2anc disj3 sylib uneq2d 3eqtr4a imaeq2d syl6eq unieqd uniun
@@ -235943,7 +235957,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.) $)
dmdprdsplit2 $p |- ( ph -> G dom DProd S ) $=
( cfv cvv wcel vx csubg cmrc eqid cres cdprd cdm wbr cgrp dprdgrp syl
- vy cun ssun1 syl5sseqr fssresd dprddomcld ssun2 unexg syl2anc eqeltrd
+ vy cun ssun1 sseqtrrid fssresd dprddomcld ssun2 unexg syl2anc eqeltrd
fdmd cv wne wss wo wi eleq2d elun syl6bb wa cdif cima dmdprdsplit2lem
csn cin c0 incom syl5eqr uncom syl6eq dprdsubg cntzrecd jaodan simpld
cuni co ex sylbid 3imp2 biimpa simprd syldan dmdprdd ) AUAULDEFEUBRZU
@@ -235964,7 +235978,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
( G DProd ( S |` C ) ) C_ ( Z ` ( G DProd ( S |` D ) ) ) /\
( ( G DProd ( S |` C ) ) i^i ( G DProd ( S |` D ) ) ) = { .0. } ) ) ) $=
( cdprd wbr wa co wss wceq adantr cdm cres cfv cin csn simpr csubg fdmd
- w3a cun ssun1 syl5sseqr dprdres simpld ssun2 jca c0 dprdcntz2 dprddisj2
+ w3a cun ssun1 sseqtrrid dprdres simpld ssun2 jca c0 dprdcntz2 dprddisj2
3jca wf simpr1l simpr1r simpr2 simpr3 dmdprdsplit2 impbida ) AEDNUAZOZE
DBUBZVHOZEDCUBZVHOZPZEVJNQZEVLNQZHUCRZVOVPUDGUESZUIZAVIPZVNVQVRVTVKVMVT
VKVOEDNQZRVTBDEFAVIUFZADUAFSVIAFEUGUCZDIUHTZVTBCUJZBFBCUKAFWESZVIKTZULZ
@@ -235981,10 +235995,10 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
dprdsplit $p |- ( ph -> ( G DProd S ) =
( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) $=
( cdprd co cfv wcel wss wa wceq adantr vx cres csubg fdmd ccntz cdm ssun1
- wbr cun syl5sseqr dprdres simpld dprdsubg syl cin c0g csn w3a dmdprdsplit
+ wbr cun sseqtrrid dprdres simpld dprdsubg syl cin c0g csn w3a dmdprdsplit
ssun2 eqid mpbid simp2d lsmsubg syl3anc cv wo eleq2d syl6bb biimpa adantl
elun fvres fssresd simpr dprdub lsmub1 syl2anc sstrd lsmub2 jaodan syldan
- eqsstr3d dprdlub simprd wb lsmlub mpbi2and eqssd ) AFEMNZFEBUBZMNZFECUBZM
+ eqsstrrd dprdlub simprd wb lsmlub mpbi2and eqssd ) AFEMNZFEBUBZMNZFECUBZM
NZDNZAEWOUAFGLAGFUCOZEHUDZAWLWPPZWNWPPZWLWNFUEOZOQZWOWPPAFWKMUFZUHZWRAXCW
LWJQZABEFGLWQABCUIZBGBCUGJUJZUKZULZWKFUMUNZAFWMXBUHZWSAXJWNWJQZACEFGLWQAX
ECGCBUTJUJZUKZULZWMFUMUNZAXCXJRZXAWLWNUOFUPOZUQSZAFEXBUHZXPXAXRURLABCEFGX
@@ -236067,7 +236081,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $.
( S ` X ) C_ ( Z ` ( G DProd ( S |` ( I \ { X } ) ) ) ) ) $=
( cfv csn cres cdprd co wbr wss cin wceq cun dpjlem cdm wa c0g dprdf2
cdif w3a c0 disjdif a1i undif2 snssd ssequn1 syl5req eqid dmdprdsplit
- sylib mpbid simp2d eqsstr3d ) AEBKCBELZMZNOZCBDVAUFZMZNOZFKZABCDEGHIU
+ sylib mpbid simp2d eqsstrrd ) AEBKCBELZMZNOZCBDVAUFZMZNOZFKZABCDEGHIU
AACVBNUBZPCVEVHPUCZVCVGQZVCVFRCUDKZLSZACBVHPVIVJVLUGGAVAVDBCDVKFABCDG
HUEVAVDRUHSAVADUIUJAVAVDTVADTZDVADUKAVADQVMDSAEDIULVADUMUQUNJVKUOUPUR
USUT $.
@@ -236616,7 +236630,7 @@ factorization into prime power factors (even if the exponents are
odcl hashcl ssrabdv oveq12d breq2d rabbidv fvmpt3i sseqtr4d nnnn0d pcdvds
oveq1 ne0d sneq difeq2d breq12d notbid wral cmpt ssrab3 ssidd clsm fssres
difss eldif iddvdsexp sylan eqeltrrd breq1 elrab2 sylanbrc ex con3d elnn0
- impr wo ord nncnd exp0d fvex hashsng ax-mp syl6reqr snfi sylancr eqsstr3d
+ impr wo ord nncnd exp0d fvex hashsng ax-mp syl6reqr snfi sylancr eqsstrrd
sylan2b eqsstrd dprdlub lsmss2 undif2 ssequn1 syl5req jca eqtrd cfz fzfid
w3a 3ad2ant2 dvdsle syl2anr 3ad2ant1 fznn mpbir2and syl5eqss ablfac1eulem
3impia rabssdv ralrimiva rspcdva coprm rpexp1i syl31anc ccntz inss1 inass
@@ -238672,7 +238686,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be
.1. = ( 0g ` M ) ) $=
( vy wcel cbs cfv eqid wceq co cvv oveqd 3ad2antl1 eqtr3d syldan crg cmgp
wss w3a cplusg c0g simp3 mgpbas ressbas2 3ad2ant2 eleqtrd cv simp2 sselda
- eqsstr3d wa fvex syl6eqel mgpplusg ressplusg syl adantr ringlidm ringridm
+ eqsstrrd wa fvex syl6eqel mgpplusg ressplusg syl adantr ringlidm ringridm
cmulr ismgmid2 ) CUAJZABUCZDAJZUDZIEKLZEUELZDEEUFLZVKMVMMVLMVJDAVKVGVHVIU
GVHVGAVKNVIABECUBLZFBCVNVNMZGUHUIUJZUKVJIULZVKJZVQBJZDVQVLOZVQNVJVKBVQVJV
KABVPVGVHVIUMUOUNZVJVSUPZDVQCVELZOZVTVQWBWCVLDVQVJWCVLNZVSVJAPJWEVJAVKPVP
@@ -240479,7 +240493,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be
(Revised by Mario Carneiro, 5-Oct-2015.) $)
invrpropd $p |- ( ph -> ( invr ` K ) = ( invr ` L ) ) $=
( cmgp cfv cui cress co cbs wceq eqid wcel wa cvv cminusg cinvr unitpropd
- unitgrpbas a1i syl6eq cmulr cplusg unitss syl5sseqr sselda anim12dan fvex
+ unitgrpbas a1i syl6eq cmulr cplusg unitss sseqtrrid sselda anim12dan fvex
cv syldan mgpplusg ressplusg ax-mp 3eqtr3g grpinvpropd invrfval 3eqtr4g
oveqi ) AEJKZELKZMNZUAKFJKZFLKZMNZUAKEUBKZFUBKZABCVEVFVIVEVFOKPAEVEVFVEQZ
VFQZUDUEAVEVHVIOKABCDEFGHIUCFVHVIVHQZVIQZUDUFABUNZVERZCUNZVERZSZSVPVREUGK
@@ -241741,7 +241755,7 @@ nonzero elements form a group under multiplication (from which it
Stefan O'Rear, 27-Nov-2014.) $)
subrg1 $p |- ( A e. ( SubRing ` R ) -> .1. = ( 1r ` S ) ) $=
( vx cfv wcel cur cbs cmulr co wceq wa eqid crg oveqd eqeq1d syldan sylan
- csubrg cv wral subrg1cl subrgbas eleqtrd subrgss eqsstr3d sselda subrgrcl
+ csubrg cv wral subrg1cl subrgbas eleqtrd subrgss eqsstrrd sselda subrgrcl
ringidmlem ressmulr biimpa ralrimiva subrgring isringid mpbi2and syl6reqr
anbi12d wb syl ) ABUBHZIZCJHZBJHZDVDVFCKHZIZVFGUCZCLHZMZVINZVIVFVJMZVINZO
ZGVGUDZVEVFNZVDVFAVGABVFVFPZUEABCEUFZUGVDVOGVGVDVIVGIVIBKHZIZVOVDVGVTVIVD
@@ -241796,7 +241810,7 @@ nonzero elements form a group under multiplication (from which it
4-Dec-2014.) $)
subrgdvds $p |- ( A e. ( SubRing ` R ) -> E C_ .|| ) $=
( vx vy vz cfv wcel cv wbr cbs cmulr co wrex eqid csubrg reldvdsr a1i cop
- wrel wa subrgbas subrgss eqsstr3d sseld ressmulr oveqd eqeq1d rexbidv wss
+ wrel wa subrgbas subrgss eqsstrrd sseld ressmulr oveqd eqeq1d rexbidv wss
wceq wi ssrexv syl sylbird anim12d dvdsr 3imtr4g df-br 3imtr3g relssdv )
ACUALZMZIJEBEUEVHEDHUBUCVHINZJNZEOZVIVJBOZVIVJUDZEMVMBMVHVIDPLZMZKNZVIDQL
ZRZVJUPZKVNSZUFVICPLZMZVPVICQLZRZVJUPZKWASZUFVKVLVHVOWBVTWFVHVNWAVIVHVNAW
@@ -241815,7 +241829,7 @@ nonzero elements form a group under multiplication (from which it
subrguss $p |- ( A e. ( SubRing ` R ) -> V C_ U ) $=
( vx cfv wcel wa cur cdsr wbr eqid wceq adantr cmulr co coppr simpr sylib
csubrg cv isunit simpld subrg1 breqtrrd wss subrgdvds ssbrd mpd cinvr cbs
- subrgbas subrgss eqsstr3d unitcl adantl subrgring ringinvcl sylan opprbas
+ subrgbas subrgss eqsstrrd unitcl adantl subrgring ringinvcl sylan opprbas
sseldd crg dvdsrmul syl2anc opprmul unitrinv oveqd 3eqtr4d syl5eq breqtrd
ressmulr sylanbrc ex ssrdv ) ABUDJZKZIEDVTIUEZEKZWADKZVTWBLZWABMJZBNJZOZW
AWEBUAJZNJZOWCWDWAWECNJZOWGWDWACMJZWEWJWDWAWKWJOZWAWKCUAJZNJZOZWDWBWLWOLV
@@ -241838,7 +241852,7 @@ nonzero elements form a group under multiplication (from which it
subrginv $p |- ( ( A e. ( SubRing ` R ) /\ X e. U ) ->
( I ` X ) = ( J ` X ) ) $=
( cfv wcel cur cmulr co wceq adantr eqid syl2an2r csubrg crg cbs subrgrcl
- wa wss subrgbas subrgss eqsstr3d subrgring ringinvcl sseldd unitcl adantl
+ wa wss subrgbas subrgss eqsstrrd subrgring ringinvcl sseldd unitcl adantl
sylan cui subrguss sselda ringass syl13anc unitlinv ressmulr oveqd subrg1
3eqtr4d oveq1d unitrinv oveq2d 3eqtr3d ringlidm ringridm ) ABUALZMZGDMZUE
ZBNLZGELZBOLZPZGFLZVPVRPZVQVTVOVTGVRPZVQVRPZVTGVQVRPZVRPZVSWAVOBUBMZVTBUC
@@ -242012,7 +242026,7 @@ nonzero elements form a group under multiplication (from which it
( cdr wcel cfv wa cdif syl wceq wb eqid ad2antlr wss csubrg cv wral cinvr
csn cbs crg cui simpllr subrgring c0g simpr eldifsn sylib simpld subrgbas
eleqtrd simprd subrg0 neeqtrd drngunit mpbir2and syl2anc subrginv 3eltr4d
- wne ringinvcl ralrimiva subrguss isdrng simprbi ad2antrr unitss syl5sseqr
+ wne ringinvcl ralrimiva subrguss isdrng simprbi ad2antrr unitss sseqtrrid
cin sseqtrd ssind subrgss difin2 simprl sseldd simprr subrgunit mpbir3and
sseqtr4d w3a expr ralimdva imp dfss3 sylibr eqssd sneqd difeq12d sylanbrc
eqtrd impbida ) CJKZBCUALKZMZDJKZAUBZELZBKZABFUEZNZUCZWTXAMZXDAXFXHXBXFKZ
@@ -242395,7 +242409,7 @@ nonzero elements form a group under multiplication (from which it
difssd subrgss ressbas2 3syl sseqtr4d ressabs eqtr3d ciin intiin syl5reqr
cv difeq1d oveq2d cmpt crn vex difexi dfiin3 iindif1 syl5eqr difss mgpbas
cvv ax-mp fvexi ciun iinssiun csubg subrgsubg ssriv syl6ss subgint adantr
- subg0 sneqd difeq2d sselda ssdifd eqsstr3d iunssd sstrd syl5eqssr sylancr
+ subg0 sneqd difeq2d sselda ssdifd eqsstrrd iunssd sstrd syl5eqssr sylancr
wral drngmgp syl6sseq sylan eqcomd oveq12d simpr subrg0 difeq12d eqeltrrd
eqeltrd issubg syl3anbrc ralrimiva rnmptss cdm dmmptg difexg mprg eqnetrd
wa a1i dm0rn0 necon3bii sylib subggrp syl5eqel isdrng2 sylanbrc ) ADUAKZD
@@ -242448,7 +242462,7 @@ nonzero elements form a group under multiplication (from which it
( vs vx vy cdr wcel cfv wss cv co cbs eqid cmulr wceq wral syl wa syl2anc
ccrg cfield csdrg id csubrg cress issdrg simp2bi ssriv a1i sdrgid simp3bi
ne0d adantl subdrgint crg drngring cmgp ccntr ccntz ssidd cntzsdrg intss1
- cint mgpbas syl6sseq cntrss syl6ss ressbas2 eqsstr3d adantr simprl sseldd
+ cint mgpbas syl6sseq cntrss syl6ss ressbas2 eqsstrrd adantr simprl sseldd
cntrval simprr mgpplusg cntri cvv ssexd oveqdr 3eqtr3d ralrimivva iscrng2
ressmulr sylanbrc isfld ) BGHZAGHZAUAHZAUBHWGBBUCIZADCWGUDZWJBUEIZJWGDWJW
LDKZWJHZWGWMWLHZBWMUFLGHZBWMUGZUHUIUJWGWJBMIZWRBWRNZUKZUMWNWPWGWNWGWOWPWQ
@@ -244880,7 +244894,7 @@ Absolute value (abstract algebra)
lmodring adantr csubg cgrp lsssubg subggrp cv lssvscl 3impb simpll simpr1
crg w3a ad2antlr simpr2 sseldd simpr3 lmodvsdi lmodvsdir lmodvsass sselda
syl13anc lmodvs1 adantlr syldan islmodd jca simprl fvex syl6eqel a1i clss
- eqcomd eqsstr3d c0 wne lmodgrp ad2antll grpbn0 lsscl sylan islssd eqeltrd
+ eqcomd eqsstrrd c0 wne lmodgrp ad2antll grpbn0 lsscl sylan islssd eqeltrd
lss1 impbida ) DUBJZBAJZBCUCZEUBJZKZXAXBKZXCXDXBXCXAABCDGHUDZLZXFIUAUEDUF
MZNMZDUGMZXIUGMZDUHMZXIUIMZXIUJMZXIBEXBXAXCBENMZOZXGXCXQXABCEDFGUKZLULXBX
KEUGMZOZXABXKDEAFXKPZUMLXBXIEUFMZOZXABXIDEAFXIPZUNLXBXMEUHMZOZXABXMDEAFXM
@@ -245283,7 +245297,7 @@ Absolute value (abstract algebra)
(Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro,
19-Jun-2014.) $)
lspssid $p |- ( ( W e. LMod /\ U C_ V ) -> U C_ ( N ` U ) ) $=
- ( vt clmod wcel wss wa clss cfv crab cint ssintub eqid lspval syl5sseqr
+ ( vt clmod wcel wss wa clss cfv crab cint ssintub eqid lspval sseqtrrid
cv ) DHIACJKAGTJGDLMZNOAABMGAUAPGUAABCDEUAQFRS $.
$( The span of a set of vectors is idempotent. (Contributed by NM,
@@ -245556,7 +245570,7 @@ Absolute value (abstract algebra)
( wcel cfv csn co eqid lmodvneg1 sneqd fveq2d wss cgrp lspsnvsi syl3anc
clmod wa csca cur cminusg cvsca cbs simpl lmodfgrp lmod1cl syl2anc adantr
grpinvcl simpr wceq lmodvnegcl syldan lmodgrp grpinvinv sylan eqtrd eqssd
- eqsstr3d ) DUAIZECIZUBZEAJZKZBJZEKZBJZVFVIDUCJZUDJZVLUEJZJZEDUFJZLZKZBJZV
+ eqsstrrd ) DUAIZECIZUBZEAJZKZBJZEKZBJZVFVIDUCJZUDJZVLUEJZJZEDUFJZLZKZBJZV
KVFVRVHBVFVQVGVPVMVLVNACDEFGVLMZVPMZVMMZVNMZNOPVFVDVOVLUGJZIZVEVSVKQVDVEU
HZVDWEVEVDVLRIVMWDIWEVLDVTUIVMVLWDDVTWDMZWBUJWDVLVNVMWGWCUMUKULZVDVEUNVOV
PVLWDBCDEVTWGFWAHSTVCVFVKVOVGVPLZKZBJZVIVFWJVJBVFWIEVFWIVGAJZEVDVEVGCIZWI
@@ -246382,7 +246396,7 @@ Absolute value (abstract algebra)
E* g e. ( S LMHom T ) ( g |` X ) = F ) $=
( vh wss cfv wceq wa cv cres wral wcel cdm ad2antrl weq wi clmhm co eqtr3
wrmo cin inss1 dmss ax-mp wfn cbs wf eqid lmhmf ffnd adantrr syl syl5sseq
- fndm simplr clmod lmhmlmod1 adantr lmhmeql simprr lspssp syl3anc eqsstr3d
+ fndm simplr clmod lmhmlmod1 adantr lmhmeql simprr lspssp syl3anc eqsstrrd
clss eqssd expr wb ffn simpll fnreseql fneqeql syl2anc 3imtr4d ralrimivva
3syl syl5 reseq1 eqeq1d rmo4 sylibr ) GAKZGFLZAMZNZDOZGPZEMZJOZGPZEMZNZDJ
UAZUBZJBCUCUDZQDWTQWMDWTUFWJWSDJWTWTWQWLWOMZWJWKWTRZWNWTRZNZNZWRWLWOEUEXE
@@ -247409,7 +247423,7 @@ division ring is an Abelian group (vectors) together with a division ring
( wcel co csn cfv syl3anc clvec wne wa w3a clmod lveclmod 3ad2ant1 simp2l
wss simp3 lspsnvsi cinvr cmulr wceq lvecdrng simp2r eqid drnginvrl oveq1d
cur drnginvrcl lmodvsass syl13anc lmodvs1 syl2anc 3eqtr3d fveq2d lmodvscl
- cdr sneqd eqsstr3d eqssd ) GUAPZADPZAIUBZUCZHFPZUDZAHBQZRESZHRZESZVRGUEPZ
+ cdr sneqd eqsstrrd eqssd ) GUAPZADPZAIUBZUCZHFPZUDZAHBQZRESZHRZESZVRGUEPZ
VNVQVTWBUIVMVPWCVQGUFUGZVMVNVOVQUHZVMVPVQUJZABCDEFGHKMJLOUKTVRWBACULSZSZV
SBQZRZESZVTVRWJWAEVRWIHVRWHACUMSZQZHBQZCUTSZHBQZWIHVRWMWOHBVRCVIPZVNVOWMW
OUNVMVPWQVQCGKUOUGZWEVMVNVOVQUPZDCWLWOWGAIMNWLUQZWOUQZWGUQZURTUSVRWCWHDPZ
@@ -248061,7 +248075,7 @@ division ring is an Abelian group (vectors) together with a division ring
mpbid simp2l wi oveq2 eqeq2d biimpac ad2antrr lssel lmod0vrid biimpd syl7
ex exp4a 3imp eleq1 biimparc syl6an necon3bd mpd csg cabl lmodabl simp1l1
simp2r ablpncan2 syl3anc simp1rl eqeltrrd simp1l2 sselda syl22anc simp12r
- lssvsubcl lspsneleq lspsnel5a eqsstr3d mpd3an23 rexlimdvv exlimddv lsmlub
+ lssvsubcl lspsneleq lspsnel5a eqsstrrd mpd3an23 rexlimdvv exlimddv lsmlub
3exp mpbi2and eqssd ) ADEUOZEDIUDFUEZBUFZUGZUHZEYCAYAYDUIYEDEUGZYBEUGZYCE
UGZYEYAYFAYAYDUJZDEUKZRYEUAULZESZYKDSZUMZUNZYGUAYEYAYOUAVGYIUADEUPRYEYOUN
ZYKUBULZUCULZHUQUEZUFZURZUCYBUSUBDUSZYGYPYKYCSZUUBYPEYCYKAYAYDYOUTYEYLYNV
@@ -248450,7 +248464,7 @@ U C_ ( N ` { X } ) ) -> ( U = ( N ` { X } ) \/ U = { .0. } ) ) $=
jca alrimiv 3jca csn cdif wral simpr1 simpr2 simplr1 ssdifssd fvexi ssexg
wn cbs sylancl simplr3 difssd simpr neldifsn nelne1 necomd psseq1 psseq1d
fveq2 imbi12d spcgv syl3c simprbi simplr2 clss ad2antrr adantrr lspcl cun
- dfpss3 ssun1 undif1 sseqtr4i lspssid simprr snssd syl5ss syl3anc eqsstr3d
+ dfpss3 ssun1 undif1 sseqtr4i lspssid simprr snssd syl5ss syl3anc eqsstrrd
unssd lspssp expr mtod ralrimiva wb islbs2 adantr mpbir3and impbida ) EUA
KZABKZADLZACMZDUBZFUCZANZYGCMZDNZUDZFUFZOZYBYCPZYDYFYLYCYDYBABDEGHUEZUGYC
YFYBABCDEGHIUHUGYNYKFYBYCYHYJYBYCYHOZYIDLZYIDQZYJYPEUIKZYGDLZYQYBYCYSYHEU
@@ -250311,7 +250325,7 @@ zero ring (at least if its operations are internal binary operations).
29-Mar-2015.) $)
drngdomn $p |- ( R e. DivRing -> R e. Domn ) $=
( cdr wcel cnzr cbs cfv c0g csn cdif crlreg wss cdomn drngnzr cui wceq eqid
- crg isdrng simprbi drngring unitrrg syl eqsstr3d isdomn2 sylanbrc ) ABCZADC
+ crg isdrng simprbi drngring unitrrg syl eqsstrrd isdomn2 sylanbrc ) ABCZADC
AEFZAGFZHIZAJFZKALCAMUFUIANFZUJUFAQCZUKUIOUGAUKUHUGPZUKPZUHPZRSUFULUKUJKATA
UKUJUJPZUNUAUBUCUGAUJUHUMUPUOUDUE $.
@@ -250776,7 +250790,7 @@ such that every prime ideal contains a prime element (this is a
(Contributed by Mario Carneiro, 7-Jan-2015.) $)
aspssid $p |- ( ( W e. AssAlg /\ S C_ V ) -> S C_ ( A ` S ) ) $=
( vt casa wcel wss wa cv csubrg cfv clss cin crab cint ssintub eqid
- aspval syl5sseqr ) DHIBCJKBGLJGDMNDONZPZQRBBANGBUDSGABUCCDEFUCTUAUB $.
+ aspval sseqtrrid ) DHIBCJKBGLJGDMNDONZPZQRBBANGBUDSGABUCCDEFUCTUAUB $.
$}
${
@@ -251385,7 +251399,7 @@ such that every prime ideal contains a prime element (this is a
cofr 3ad2antr2 simpr2 cv cdif cfv eldifi wral simpr3 ffnd 3ad2antr1 simpl
psrbagf inidm eqidd ofrfval mpbid r19.21bi sylan2 cvv wss jca eqimss 3syl
c0ex a1i suppssr breqtrd ffvelrn syl2an nn0ge0d nn0red 0re letri3 sylancl
- cr wb mpbir2and suppss eqsstr3d ) EFIZCAIZEJDKZDCLUDMZUAZNZDOPQZDRSUBZCOP
+ cr wb mpbir2and suppss eqsstrrd ) EFIZCAIZEJDKZDCLUDMZUAZNZDOPQZDRSUBZCOP
QZWDWEWFWKWJTWGDEFUCUEWIEJHDWLRWDWEWFWGUFZWIHUGZEWLUHIZNZWNDUIZRTZWQRLMZR
WQLMZWPWQWNCUIZRLWOWIWNEIZWQXALMZWNEWLUJZWIXCHEWIWGXCHEUKWDWEWFWGULWIHEEW
QXALEDCFFWIEJDWMUMWIEJCWDWFWEEJCKZWGABCEFGUPUNZUMWDWHUOZXGEUQWIXBNZWQURXH
@@ -252519,7 +252533,7 @@ such that every prime ideal contains a prime element (this is a
by Mario Carneiro, 3-Jul-2015.) $)
resspsrbas $p |- ( ph -> B = ( Base ` P ) ) $=
( vf cbs cfv cvv c0 wss wceq wcel wa cv ccnv cn cima cfn cn0 cmap co crab
- fvex csubrg subrgbas syl eqid subrgss eqsstr3d adantr mapss sylancr simpr
+ fvex csubrg subrgbas syl eqid subrgss eqsstrrd adantr mapss sylancr simpr
psrbas 3sstr4d wn cmps reldmpsr ovprc1 syl5eq adantl fveq2d base0 3eqtr4g
0ss syl6eqss pm2.61dan ressbas2 ) ABEQRZUAZBCQRUBAISUCZWAAWBUDZHQRZPUEUFU
GUHUIUCPUJIUKULUMZUKULZDQRZWEUKULZBVTWCWGSUCWDWGUAZWFWHUADQUNAWIWBAWDFWGA
@@ -252532,7 +252546,7 @@ such that every prime ideal contains a prime element (this is a
resspsradd $p |- ( ( ph /\ ( X e. B /\ Y e. B ) ) ->
( X ( +g ` U ) Y ) = ( X ( +g ` P ) Y ) ) $=
( wcel cfv co vf wa cplusg cof eqid simprl simprr psradd cbs cv ccnv cima
- cn cfn cn0 cmap crab wss cvv fvex csubrg wceq subrgbas syl eqsstr3d mapss
+ cn cfn cn0 cmap crab wss cvv fvex csubrg wceq subrgbas syl eqsstrrd mapss
subrgss sylancr adantr cmps reldmpsr elbasov simpld psrbas 3sstr4d sseldd
ad2antrl ressplusg ofeq oveqd eqtrd fvexi mp1i 3eqtr2d ) AJBRZKBRZUBZUBZJ
KGUCSZTJKHUCSZUDZTZJKEUCSZTZJKCUCSZTWHBWJWIHGIJKNOWJUEWIUEAWEWFUFZAWEWFUG
@@ -252552,7 +252566,7 @@ such that every prime ideal contains a prime element (this is a
psrbaglefi sylan csubmnd csubrg subrgsubg syl subgsubm ad2antrr ad3antrrr
csubg cbs wf simprl psrelbas adantr elrabi ffvelrn wceq subrgbas eleqtrrd
syl2an simprr ssrab2 simplr simpr syl3anc sseldi ffvelrnd subrgmcl fmpttd
- psrbagconcl gsumsubm ressmulr oveqd mpteq2dva oveq2d wss subrgss eqsstr3d
+ psrbagconcl gsumsubm ressmulr oveqd mpteq2dva oveq2d wss subrgss eqsstrrd
eqtrd fvex mapss sylancr psrbas 3sstr4d sseldd psrmulfval 3eqtr4rd fvexi
mp1i ) AJBRZKBRZUEZUEZJKGUFSZTZJKEUFSZTZJKCUFSZTYBUAUBUGUHUIUJUKRUBULIUMT
UNZDUCUDUGUAUGZUPUOUQZUDYHUNZUCUGZJSZYIYLURUSTZKSZDUFSZTZUTZVATZUTUAYHHUC
@@ -253942,7 +253956,7 @@ series in the subring which are also polynomials (in the parent ring).
A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } ) $=
( cvv wcel wa cv cpr wss cfv wbr wceq wi wral wrex wo copab cple cnx csts
cop co eqid simprl simprr cxp adantr opsrval fveq2d cmps ovexi fvexi xpex
- cbs vex anbi1i opabbii opabssxp eqsstr3i ssexi pleid setsid mp2an 3eqtr4g
+ cbs vex anbi1i opabbii opabssxp eqsstrri ssexi pleid setsid mp2an 3eqtr4g
prss wn c0 copws reldmopsr ovprc adantl fveq1d syl5eq 0fv syl6eq reldmpsr
str0 base0 xpeq2d xp0 sseq0 sylancr eqtr4d pm2.61dan ) ANUEUFZIUEUFZUGZOB
UHZCUHZUIFUJZDUHZXIUKXLXJUKKULEUHZXLGULXMXIUKXMXJUKUMUNEHUOUGDHUPXIXJUMUQ
@@ -254929,7 +254943,7 @@ This function maps an element of the formal polynomial algebra (with
syl6 cmrc cmps casp cascl casa psrassa mvrf frnd aspval2 mplbas2 mplsubrg
mplval2 subsubrg2 fveq2d ressascl syl6reqr rneqd cmre subrgmre 3syl unssd
cpw syl3anc ad2antrr wfn wb rhmf ffn adantr uneq1d assaring submrc eqtr2d
- fveq12d eqsstr3d 3eqtr3d simpr ad2antlr mrcsscl eqsstrd fnreseql fneqeql2
+ fveq12d eqsstrrd 3eqtr3d simpr ad2antlr mrcsscl eqsstrd fnreseql fneqeql2
rhmeql simprl simprr 3imtr4d syl5 ralrimivva reseq1 rmo4 sylibr sylc reu5
rmoim sylanbrc ) AGUEZBUFZHUGZUVGKUFZIUGZUHZGDFUIUJZUKZUVLGUVMULZUVLGUVMU
MAUADUNUOZFUBUCUEUPVMUQURUSUCUTJVAUJVBZUBUEZUAUEUOHUOFVCUOZUVRIUVSVDUOZVE
@@ -263639,7 +263653,7 @@ S C_ ( ._|_ ` ( ._|_ ` S ) ) ) $=
(Contributed by Mario Carneiro, 16-Oct-2015.) $)
pjcss $p |- ( W e. PreHil -> dom K C_ C ) $=
( vx cphl wcel cdm cv wa clsm cfv cocv cbs eqid simpl clss wss co wceq ex
- pjdm2 simprbda lssss syl ocvss simplbda syl5sseqr lsmcss ssrdv ) CGHZFBIZ
+ pjdm2 simprbda lssss syl ocvss simplbda sseqtrrid lsmcss ssrdv ) CGHZFBIZ
AULFJZUMHZUNAHULUOKZACLMZUNCNMZCOMZCEUSPZURPZUQPZULUOQUPUNCRMZHZUNUSSULUO
VDUNUNURMZUQTZUSUAZUQUNBVCURUSCUTVCPZVAVBDUCZUDVCUNUSCUTVHUEUFUPUSVEURMVF
VEURUSCUTVAUGULUOVDVGVIUHUIUJUBUK $.
@@ -265886,7 +265900,7 @@ each element of the range ( ~ islindf5 ). (Contributed by Stefan
c0g wral 3adant3 f1f 3ad2ant3 fco ffdmd wne simpl2 eleq2d biimpa ffvelrnd
fdmd adantrr eldifi ad2antll eldifsni lindfind syl22anc wi wfn f1fn fvco2
oveq2d eleq1d simpl1 imassrn frnd syl5ss imaco difeq1d imaeq2d ccnv df-f1
- crn simprbi imadif syl eqtrd fnsnfv sylan difeq2d ssdifd eqsstr3d eqsstrd
+ crn simprbi imadif syl eqtrd fnsnfv sylan difeq2d ssdifd eqsstrrd eqsstrd
wfun imass2 syl5eqss lspss syl3anc syldan sseld sylbid mtod ralrimivva wb
simp1 rellindf brrelex1i 3ad2ant2 simp3 dmexd fex coexg islindf mpbir2and
f1dmex ) DGHZADIUAZCAUBZBUCZUDZABUEZDIUAZYHUBZDUFJZYHKZEUGZFUGZYHJZDUHJZU
@@ -279936,7 +279950,7 @@ _Introduction to General Topology_ (1983), p. 114) and it is convenient
istopon $p |- ( J e. ( TopOn ` B ) <-> ( J e. Top /\ B = U. J ) ) $=
( vj vb ctopon cfv wcel cvv ctop cuni wceq elfvex uniexg eleq1 syl5ibrcom
wa cv crab cpw wss imp eqeq1 rabbidv df-topon vpwex pwex rabss pwuni pweq
- wi syl5sseqr selpw sylibr mprgbir ssexi fvmpt3i eleq2d unieq eqeq2d elrab
+ wi sseqtrrid selpw sylibr mprgbir ssexi fvmpt3i eleq2d unieq eqeq2d elrab
a1i syl6bb pm5.21nii ) BAEFZGZAHGZBIGZABJZKZPZBAELVGVIVFVGVFVIVHHGBIMAVHH
NOUAVFVEBACQZJZKZCIRZGVJVFVDVNBDADQZVLKZCIRZVNHEVOAKVPVMCIVOAVLUBUCCDUDVQ
VOSZSZVRDUEUFVQVSTVPVKVSGZUJZCIVPCIVSUGWAVKIGVPVKVRTVTVPVLSVKVRVKUHVOVLUI
@@ -280965,7 +280979,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
{ x e. ~P A | ( P e. x -> x = A ) } e. ( TopOn ` A ) ) $=
( vy vz wcel wa cv wceq wi wss wral eleq2 eqeq1 imbi12d adantr syl ex cvv
cpw crab ctop ctopon cfv wal cin ssrab simprl sspwuni sylib vuniex sylibr
- cuni elpw wrex eluni2 r19.29 simpr impr elssuni eqsstr3d rexlimiva syl5bi
+ cuni elpw wrex eluni2 r19.29 simpr impr elssuni eqsstrrd rexlimiva syl5bi
ad2antll jctild eqss syl6ibr elrabd alrimiv inss1 elpwid syl5ss vex inex1
simprll elin simprlr simprrr anim12d ineq12 inidm syl6eq syl6 jca anbi12i
elrab 3imtr4g ralrimivv wb pwexg rabexg istopg mpbir2and pwidg a1d ssrab2
@@ -281442,7 +281456,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
$( A subset of a topology's underlying set is included in its closure.
(Contributed by NM, 22-Feb-2007.) $)
sscls $p |- ( ( J e. Top /\ S C_ X ) -> S C_ ( ( cls ` J ) ` S ) ) $=
- ( vx ctop wcel wss wa cv ccld cfv crab cint ccl ssintub clsval syl5sseqr
+ ( vx ctop wcel wss wa cv ccld cfv crab cint ccl ssintub clsval sseqtrrid
) BFGACHIAEJHEBKLZMNAABOLLEASPEABCDQR $.
$( A subset includes its interior. (Contributed by NM, 3-Oct-2007.)
@@ -281790,7 +281804,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
( vx c0 cpr ccld cfv cv wcel cid wss cdif wa ctop wb indistop wceq difeq2
ax-mp syl6eq syl6eqel indisuni iscld simpl dfss4 sylib wo simpr syl6eleqr
indislem elpri dif0 fvex prid2 eleqtri difid 0ex prid1 jaoi 3syl eqeltrrd
- sylbi ssriv 0cld topcld prssi mp2an eqsstr3i eqssi ) CADZEFZVIBVJVIBGZVJH
+ sylbi ssriv 0cld topcld prssi mp2an eqsstrri eqssi ) CADZEFZVIBVJVIBGZVJH
ZVKAIFZJZVMVKKZVIHZLZVKVIHVIMHZVLVQNAOZVKVIVMAUAZUBRVQVMVOKZVKVIVQVNWAVKP
VNVPUCVKVMUDUEVQVOCVMDZHVOCPZVOVMPZUFWAVIHZVQVOVIWBVNVPUGAUIZUHVOCVMUJWCW
EWDWCWAVMVIWCWAVMCKVMVOCVMQVMUKSVMWBVICVMAIULUMWFUNTWDWACVIWDWAVMVMKCVOVM
@@ -282983,7 +282997,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
( vc vd va vb ve ctop wcel wss cv wa wrex cin wceq syl2anc cvv crest cnei
w3a cfv cuni wex nfv nfre1 nfan simpl anim2i cdif cun simp-5r simp1 simp2
co restuni ad5antr sseqtr4d sstrd eltopss ssdifssd unssd simpr1l 3anassrs
- simplr simpr sseqtrd inss1 inundif simpr1r eqsstr3d unss1 syl5eqssr sseq2
+ simplr simpr sseqtrd inss1 inundif simpr1r eqsstrrd unss1 syl5eqssr sseq2
syl6ss sseq1 anbi12d rspcev syl12anc c0 indir incom disjdif eqtr3i uneq2i
syl un0 3eqtri df-ss biimpi syl5req vex difexi unex anbi2d rexbidv eqeq2d
ineq1 spcev syl21anc ad3antrrr uniexg ssexd elrest biimpa r19.29a sylanl1
@@ -283265,10 +283279,10 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
( vm vn ctsr wcel cun cvv wss vz csn cfi cfv cv cin wceq wrex w3o wb snex
ssun2 cuni ordtuni cdm dmexg syl5eqel eqeltrrd uniexb sylibr ssexg elfiun
sylancr wi ssun1 eqsstri sseli a1i ordtbas2 syl6eqss sseld wa cpw fipwuni
- fisn elpwid ad2antll unissi syl5sseqr adantr simprl syl6eleq syl sseqtr4d
+ fisn elpwid ad2antll unissi sseqtrrid adantr simprl syl6eleq syl sseqtr4d
sstrd elsni sseqin2 sylib sselda eleq1 syl5ibrcom rexlimdvva 3jaod sylbid
adantrl eqeltrd ssrdv ssfii unssad fiss sylancl unssd eqssd unass syl6eqr
- eqsstr3d ) FPQZGUBZCDRZRZUCUDZXHXIERZRZXJERXGXKXMXGUAXKXMXGUAUEZXKQZXNXHU
+ eqsstrrd ) FPQZGUBZCDRZRZUCUDZXHXIERZRZXJERXGXKXMXGUAXKXMXGUAUEZXKQZXNXHU
CUDZQZXNXIUCUDZQZXNNUEZOUEZUFZUGZOXRUHNXPUHZUIZXNXMQZXGXHSQXISQZXOYEUJGUK
XGXIXJTZXJSQZYGXIXHULZXGXJUMZSQYIXGGYKSABCDFPGJKLUNZXGGFUOSJFPUPUQURXJUSU
TZXIXJSVAVCNOXNXHXISSVBVCXGXQYFXSYDXQYFVDXGXPXMXNXPXHXMGVOZXHXLVEVFVGVHXG
@@ -283460,7 +283474,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
fibas ctop inex1g ordttop cuni ordtuni uniexb sylibr restval wral sseqin2
eqeltrrd wf sylib ctopon ordttopon psssdm fveq2d eleqtrd toponmax eqeltrd
elsni ineq1d eleq1d syl5ibrcom ralrimiv ordtrest2lem df-rn cnvtsr sseqtrd
- ccnv psrn wa adantr rabeqdv vex brcnv anbi12ci rabbii eqsstr3d ancom2s wb
+ ccnv psrn wa adantr rabeqdv vex brcnv anbi12ci rabbii eqsstrrd ancom2s wb
bicomi a1i notbid rabeqbidv mpteq12dv rneqd cnvin cnvxp ineq2i eqtri psss
ordtcnv syl5reqr eleq2d raleqbidv mpbird ralunb sylanbrc fmpt frnd tgfiss
eqsstrd eqssd ) AFEEUAZUBZUCNZFUCNZEUDUEZAFUFOZEPOZUUAUUCUGAFQOZUUDIFUIRZ
@@ -283546,7 +283560,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on
fmpti cico icossxr unssi ssexi unex ssun2 fiss fvex cuni ovex unipr mnfxr
frn cpr cicc 0xr pnfxr w3a wbr mnflt0 0lepnf df-icc df-ioc xrltnle xrletr
wa xrlttr wi xrltle 3adant2 syld ixxun mpanr12 mp3an 0lt1 df-ico xrlelttr
- 1xr ixxss2 unss1 eqsstr3i iccmax uncom 3sstr3i eqssi eqtri ssun1 rspceeqv
+ 1xr ixxss2 unss1 eqsstrri iccmax uncom 3sstr3i eqssi eqtri ssun1 rspceeqv
oveq1 elrnmpti mpbir eleqtrri sselii oveq2 prssi ssfii sstri eltg3i snssi
wrex eqeltrri bastg ctb ctop fibas tgcl fitop mp2b sseqtri 2basgen eqtr4i
) GUCHZIUDZBCUEZUEZUFHZUGHZUUCUFHZUGHZGUHJUUAUUFKUIAFBCGUHIUJAFBDUKAFBCDE
@@ -283688,7 +283702,7 @@ information ensuring that it is not too fine (and of course ~ iooordt
tg2 wi wceq cvv elrnmpt elv cif mnfxr a1i simprl 0xr sylancl pnfxr xrmax1
sylancr ge0gtmnf syl2anc simpll simprr eleqtrd elioc1 mpbid simp2d 0ltpnf
ifcl w3a breq1 ifboth xrre2 syl32anc xrmax2 df-ioc xrlelttr ixxss1 simplr
- eqsstr3d sstrd oveq1 sseq1d rspcev rexlimdvaa com12 wn pnfnlt elico1 mpan
+ eqsstrrd sstrd oveq1 sseq1d rspcev rexlimdvaa com12 wn pnfnlt elico1 mpan
sylbi simp3 syl6bi mtod notbid syl5ibrcom rexlimiv pm2.21d adantrd pnfnre
eleq2 jaoi neli cuni elssuni unirnioo syl6sseqr sseld mtoi syl sylanb ) B
EUDUEZFBCGCUFZHIJZUGZUHZCGKYEUIJZUGZUHZUJZUKUHZUJZULUEZFZHBFZAUFZHIJZBLZA
@@ -283718,7 +283732,7 @@ information ensuring that it is not too fine (and of course ~ iooordt
elv w3a mtod eleq2 notbid syl5ibrcom rexlimiv pm2.21d adantrd sylbi mnfxr
cif a1i 0xr simprl mnflt0 simpll simprr eleqtrd elico1 mpbid simp3d breq2
ifcl ifboth xrmin1 0re ltpnf mp1i xrlelttrd syl32anc xrmin2 df-ico ixxss2
- xrre2 xrltletr simplr eqsstr3d sstrd oveq2 sseq1d rspcev rexlimdvaa com12
+ xrre2 xrltletr simplr eqsstrrd sstrd oveq2 sseq1d rspcev rexlimdvaa com12
syl2anc jaoi mnfnre neli cuni elssuni unirnioo syl6sseqr sseld syl sylanb
mtoi ) BEUDUEZFBCGCUFZHUGIZUHZUIZCGJYGKIZUHZUIZUJZUKUIZUJZULUEZFZJBFZJAUF
ZKIZBLZAMNZYFYQBCYJYMYOYJTYMTYOTUMUNYRYSOJDUFZFZUUDBLZOZDYPNUUCDBYPJUPUUG
@@ -283745,7 +283759,7 @@ information ensuring that it is not too fine (and of course ~ iooordt
ordtrestixx $p |- ( ( ordTop ` <_ ) |`t A ) =
( ordTop ` ( <_ i^i ( A X. A ) ) ) $=
( vz cle cordt cfv co wceq wtru cxr wcel a1i wss cv wa wbr sseli cxp ledm
- crest ctsr letsr crab cicc iccval syl2an eqsstr3d adantl ordtrest2 eqcomd
+ crest ctsr letsr crab cicc iccval syl2an eqsstrrd adantl ordtrest2 eqcomd
cin mptru ) GHICUCJZGCCUAUNHIZKLUQUPLABFCGMUBGUDNLUEOCMPLDOAQZCNZBQZCNZRZ
URFQZGSVCUTGSRFMUFZCPLVBVDURUTUGJZCUSURMNUTMNVEVDKVACMURDTCMUTDTFURUTUHUI
EUJUKULUMUO $.
@@ -284472,7 +284486,7 @@ converges to zero (in the standard topology on the reals) with this
( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) ) ) $=
( ctopon cfv wcel wa cima ccl wss cpw wral cuni wi wceq toponuni ad2antlr
ccn co wf ccnv cv cnf2 3expia elpwi adantl sseqtrd eqid cncls2i ralrimdva
- expcom syl jcad ccld cldss2 pweqd syl5sseqr sseld imim1d ad2antll imaeq2d
+ expcom syl jcad ccld cldss2 pweqd sseqtrrid sseld imim1d ad2antll imaeq2d
cldcls sseq2d ctop topontop ad2antrr cdm cnvimass ad2antrl eqtrd syl5sseq
fdm iscld4 syl2anc bitr4d expr pm5.74d sylibd ralimdv2 imdistanda sylibrd
wb iscncl impbid ) CEGHIZDFGHIZJZBCDUAUBIZEFBUCZBUDZAUEZKZCLHHZWMWNDLHHZK
@@ -285908,7 +285922,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) ->
F : X --> { P } ) $=
( ct1 wcel ccn co wa ccnv wss cfv wf ccld syl2anc csn cima ccl w3a simplr
- wceq wfn cnf ffn 3syl simpr3 simpll simpr1 t1sncld simpr2 clsss2 eqsstr3d
+ wceq wfn cnf ffn 3syl simpr3 simpll simpr1 t1sncld simpr2 clsss2 eqsstrrd
cnclima fconst3 sylanbrc ) EJKZCDELMKZNZBGKZACOBUAZUBZPZADUCQQZFUFZUDZNZC
FUGZFVFPFVECRVKVBFGCRVLVAVBVJUEZCDEFGHIUHFGCUIUJVKFVHVFVCVDVGVIUKVKVFDSQK
ZVGVHVFPVKVBVEESQKZVNVMVKVAVDVOVAVBVJULVCVDVGVIUMBEGIUNTVECDEURTVCVDVGVIU
@@ -286345,7 +286359,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
eqid ralbidv bastg adantr sspwb sylib ssralv sylbid syl5 elpwi simprr wel
simprl sselda adantrr tg2 syl2anc reximdva eluni2 elunirab r19.42v rexbii
crab expr rexcom 3bitr2i 3imtr4g ssrdv eqsstrd ssrab2 unissi simplr eqssd
- syl5sseqr wb elpw2g ad2antrr mpbiri unieq eqeq2d pweq ineq1d rexeqdv mpid
+ sseqtrrid wb elpw2g ad2antrr mpbiri unieq eqeq2d pweq ineq1d rexeqdv mpid
rspcv wex elfpw ad2antrl simplbi ssrab sseq2 ac6sfi crn frn wfo wfn dffn4
wf ffn fofi syl2an sylanbrc simplrr ciun uniiun syl5eqss ad2antll fniunfv
ss2iun sseqtrd sseqtr4d rspceeqv exlimddv rexlimdvaa syld com23 ralrimdva
@@ -286385,7 +286399,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
( J |`t S ) e. Comp ) $=
( vs vt vu vw ccmp wcel wa cv cuni wss cpw cfn wi wceq cun 3ad2ant1 sylib
cvv vv ccld cfv crest co cin wrex wral selpw w3a cdif simp1l simp2 cldopn
- csn adantl snssd unssd simp3 uniss 3ad2ant2 sstrd undif 3ad2ant3 eqsstr3d
+ csn adantl snssd unssd simp3 uniss 3ad2ant2 sstrd undif 3ad2ant3 eqsstrrd
eqid unss1 difss unss sylanblc uniexg ad2antrr 3adant3 difexg unisng 3syl
eqssd uneq2d eqtr4d uniun syl6eqr cmpcov syl3anc elfpw simp2l uncom diffi
syl6sseq ssundif ad2antll sylanbrc wex sseqtrd sselda eluni a1i wn elndif
@@ -286461,7 +286475,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
vex csbeq1 iunxsn eqtri simpl3 nfv nfel1 cbvral sylib ssun2 simprl sylibr
nfov snss rspcdva syl5eqel unexg syl2anc resttop eqid restin unieqd inss2
sseqtr4i restuni sylancl eqtr4d uneq2i ineq1i indir inss1 restabs syl3anc
- sstri a1i eqeltrd eqsstr3i uncmp syl22anc exp32 a2d syl5 findcard2 mpcom
+ sstri a1i eqeltrd eqsstrri uncmp syl22anc exp32 a2d syl5 findcard2 mpcom
a2i mpi ) DHIZBUBIZDCJKZLIZABUDZUEZBBMZDABCNZJKZLIZBUFYHYLYMYPOZYGYHYKUGY
LGUHZBMZDAYRCNZJKZLIZOZOYLPBMZDPJKZLIZOZOYLUAUHZBMZDAUUHCNZJKZLIZOZOZYLUU
HUCUHZUIZUJZBMZDAUUQCNZJKZLIZOZOZYLYQOGUAUCBYRPQZUUCUUGYLUVDYSUUDUUBUUFYR
@@ -286612,7 +286626,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
0ss 0fin elfpw mpbir2an simprr simprl unieqd eqtrd unieq rspceeqv sylancr
wn expr iineq1 0iin syl6eq eqeq1d necon3bbid mpbiri pm2.21d 2thd ad2antlr
vn0 uniss eqss eqcom ssdif0 3bitr3g iindif2 uniiun difeq2i syl6eqr bitr4d
- baib ciun imassrn cres df-ima resmpt rneqd syl5eq ad2antrr syl5sseqr wfun
+ baib ciun imassrn cres df-ima resmpt rneqd syl5eq ad2antrr sseqtrrid wfun
crn funmpt elinel2 imafi sylanbrc topopn difexg ralrimivw fnmpt ad3antrrr
wfn eqid simpr sylib simpld sseqtrd simprd fipreima syl3anc rexbii inteqd
eqeq2d rexxfrd 0ex wfo wf1o ccnv opncldf1 f1ofo forn sylibr eldifsn bitri
@@ -287261,7 +287275,7 @@ require the space to be Hausdorff (which would make it the same as T_3),
syl mpd simpl reximi eleq2w cbvrexv sylib rabn0 sylibr rabex 0sdom ssrab2
vex cvv ssdomg mp2 cen simprrl nnenom ensymi domentr sylancl domtr fodomr
sylancr syl2anc cfz cima cint cmpt cfn ad3antrrr crn imassrn forn simprll
- ad2antll elpwid syl5ss eqsstrd cdm cin fz1ssnn fof fdmd syl5sseqr sseqin2
+ ad2antll elpwid syl5ss eqsstrd cdm cin fz1ssnn fof fdmd sseqtrrid sseqin2
adantr elfz1end ne0i adantl sylan2b eqnetrd syl2an2r imp wb oveq2 imaeq2d
weq inteqd ralrimiva eleq1d rspccva sylan fvmptd3 sseq1d fveq1 ralbidv id
imadisj necon3bii cres fzfid ffund fores fiinopn syl13anc fmpttd syl5sseq
@@ -289132,7 +289146,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
vj vs crest co ccmp cpw cin wrex wi wral simprr ssun2 wb ctopon clm wbr
lmcl snssg mpbiri adantr sseldd eluni2 sylib nnuz 1zzd ad2antrr simplrl
cuz c1 elpwid simprl lmcvg cima imassrn ssun1 sstri id syl5ss ctop frnd
- cfz resttopon topontop cres wfo fzfid wfun ffund fz1ssnn fdmd syl5sseqr
+ cfz resttopon topontop cres wfo fzfid wfun ffund fz1ssnn fdmd sseqtrrid
cdm fores fofi pwfi restsspw ssfi sylancl elind fincmp toponuni sseqtrd
wceq eqid cmpsub mpbid r19.21bi syl5 impr simprll snssd unssd vex elpw2
elin1d sylibr elin2d snfi unfi wfn ffnd ad3antrrr simprrr fveq2 rspccva
@@ -289639,7 +289653,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
( wcel wa wceq wss cvv vs vh vm ctop wf csn cv cfv cmpt ccnv cima crn cun
cmpo cfi wfn wral cuni cdif cfn wrex w3a cixp wex elpt df-3an cin ciin c0
simprr disjdif2 raleqdv biimpac ixpeq2 syl weq fveq2 unieqd cbvixpv eqtri
- syl6eqr sylan ssv iineq1 0iin syl6eq syl5sseqr adantl df-ss sylib wne cif
+ syl6eqr sylan ssv iineq1 0iin syl6eq sseqtrrid adantl df-ss sylib wne cif
eqtr4d simplll sseldi eleq12d ad2antrr rspcdva ptpjpre1 syl12anc iineq2dv
inss1 simpr adantlr cdm cnvimass eqid dmmptss sstri sseqtri rgenw sylancl
r19.2z iinss wi wn elssuni syl5ibrcom sylibr eqtrd eqssd eleq1 pm2.61dane
@@ -290623,7 +290637,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
ffn simp-4l simpllr simplr ffvelrnd toponss elin1d sseqtr4d inss1 mpsyl
syl6bb fnfvelrn intss1 syl5ss r19.26 sylan biimpd ralimia sylbir syl6an
wi sylbid ralimdaa impr eldifi syl2an ralbidv cun inundif raleqi ralunb
- bitr3i ralcom mptexg syl2anr ralbidva syl5bb ralrimivw syl5sseqr rspcev
+ bitr3i ralcom mptexg syl2anr ralbidva syl5bb ralrimivw sseqtrrid rspcev
bitrd syl12anc exlimddv ) ABUKZJNULZKUHUMZUNZFGUMZUWQUOZUPZCOEUQZUWTURZ
UWSIUOZUSZUKZGUWPUTZUKZFDUMZUPZCOGJEUQZUQZUXIURZGJUXDVAZUSZUKZDKVBZUHUW
OUWPVCUPZFUIUMZUPZUXBUXSURZUXDUSZUKZUIKVBZGUWPUTUXHUHVEUWONVCUPUWPNUSUX
@@ -290913,7 +290927,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
( vy cfv ctopn wss wceq cvv wcel vg vz vx cts ccom cpt cbs cpw cdm fnex
wfn syl2anc eqid eqidd prdstset cuni cv wral cdif cfn wrex w3a cixp wex
wa cab ctg wf topnfn dffn2 sylib fnfco sylancr ptval unieqd fvco2 sylan
- crest co topnval restsspw eqsstr3i syl6eqss fvex syl6ib ralimdva simpl2
+ crest co topnval restsspw eqsstrri syl6eqss fvex syl6ib ralimdva simpl2
sseld elpw impel ss2ixp simprr prdsbas2 adantr 3sstr4d ex exlimdv selpw
syl6ibr abssdv pwex ssex unitg 3syl eqtrd sspwuni eqsstrd sylibr topnid
syl syl6eqr eqtr3d ) AHUDOZEPBUEZUFOZAXMHPOZEAXMHUGOZUHZQXMXPRAXMXOXRAX
@@ -291020,7 +291034,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
( vx vy vz vw c0 cpr cid cfv cxp cv wcel wceq wo wn wss wa ctop wne co wi
ctx wex neq0 wrex wral wb indistop eltx mp2an sylbi cuni elssuni indisuni
rsp txunii syl6sseqr ad2antrr ne0i ad2antrl sylibr simpld neneqd indislem
- xpnz simpll syl6eleqr elpri syl ord simprd simplr xpeq12d simprr eqsstr3d
+ xpnz simpll syl6eleqr elpri syl ord simprd simplr xpeq12d simprr eqsstrrd
mpd adantll eqssd ex rexlimdvva syld exlimdv syl5bi orrd vex ssriv ctopon
elpr toptopon txtopon topgele ax-mp simpli eqssi txindislem preq2i 3eqtri
cpw mpbi ) GAHZGBHZUCUAZGAIJZBIJZKZHZGABKZIJZHGXHHXCXGCXCXGCLZXCMZXJGNZXJ
@@ -291265,7 +291279,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
frnd elin2d wfn ffnd dffn4 fofi w3a fiinopn imp elssuni syl6sseqr sseqin2
syl13anc pm2.61dane ad2antrr ciin simprrr simpl ralimi eliin mpbird elind
fniinfv eleqtrd ciun simprl uniiun syl6eq xpeq1d xpiundir wi inss2 iinss2
- eqsstr3d syl5ss xpss2 sstr2 3syl ralimdva sylc sylibr eqsstrd rspcev expr
+ eqsstrrd syl5ss xpss2 sstr2 3syl ralimdva sylc sylibr eqsstrd rspcev expr
iunss syl12anc exlimdv expimpd rexlimdva mpd ) AGUAUEZUFZUGZUUMEUBUEZUHZC
BUEZUUPUIZQZUURUUSUJZFRZSZBUUMUOZSZUBUKZSZUADULZUMUNZTZCUURQZGUURUJZFRZSZ
BETZADUPQZUCUEZUURQZCPUEZQZUURUVSUJZFRZSZPETSZBDTZUCGUOUVJKAUWEUCGAUVQGQZ
@@ -291789,7 +291803,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
ffnd rexnal df-ne simpllr ffvelrnd hausnei syl13anc expr syl5bir csn cima
adantr crab simp-4l ad4antlr simplr snssd crest cpw ccmp ctopon toptopon2
sylib restsn2 cfn snfi discmp mpbi syl6eqel simprll xkoopn simprlr imaeq1
- sseq1d ad2antrr fnsnfv simprr1 eqsstr3d elrabd simprr2 inrab cdm eleqtrrd
+ sseq1d ad2antrr fnsnfv simprr1 eqsstrrd elrabd simprr2 inrab cdm eleqtrrd
fdmd adantl simprr3 sseq0 expcom imadisj disjsn bitri syl6ib mt2d sylnibr
ralrimiva rabeq0 sylibr syl5eq eleq2 ineq1 eqeq1d 3anbi13d ineq2 3anbi23d
ssin rspc2ev syl113anc rexlimdvva syld rexlimdva sylbid sylbird ralrimivv
@@ -291902,7 +291916,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually
ex ad2antrr ixpconstg eqsstrd eqtrd sseqtrd sseqin2 eqtr3d elin2d simplrr
sylib topopn ad3antrrr fvconst2g ad4ant14 eleqtrrd eldifn iffalsed eldifi
ifcld cdif unieqd eqtr4d ptopn eqeltrd eleq1 syl5ibrcom rexlimdvva abssdv
- tgfiss ptuniconst oveq2d restid xkoptsub eqsstr3d eqssd ) BHIZACIZJZBAUBZ
+ tgfiss ptuniconst oveq2d restid xkoptsub eqsstrrd eqssd ) BHIZACIZJZBAUBZ
UCKZABUDUEZUFLZUVNUVPDEUVOFMZNKZUGIZFUVOUHZBGMZDMZUIEMZOZGUVOBUJKZUHZUKZU
LZUMLUNLZUVRUVMUVOHIZUVLUVLUVPUWKPACURZUVLUVMUOZFEUVOBUWIGDUWBAUVOUPAAUQU
SZUWBQUWIQUTVAUVNUVRHIZUWJUVROUWKUVROUVNUVMAHUVQUAZUWPUVLUVMVBUVLUWQUVMAB
@@ -292059,7 +292073,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $=
sseq2d oveq2 eleq1d cmpcovf wi eqeq1d biimpar cntop2 cntop1 sspwuni wfn
frnd ffn fniunfv oveq2d simplr elin2d simpr fiuncmp ntropn iunopn txopn
eqeltrrd imaiun imaeq2d syl5eqr simpl1 3ad2ant2 simp3 r19.21bi ralbidva
- 3bitr4g mpbird eqsstr3d uniiun syl6eq simpl opelxpd anbi12i coeq1 coeq2
+ 3bitr4g mpbird eqsstrrd uniiun syl6eq simpl opelxpd anbi12i coeq1 coeq2
simprll ovmpo ntrss2 sseqtrd imass2 simprlr syl5eqss eqeltrd ralrimivva
cnco sylan2b mpofun xpss12 dmmpo sseqtr4i funimassov sseq1 expr exlimdv
sylibr syldan expimpd rexlimdva ) AFDLUKZULUMZUNZUBUOZUNZUPZUYKCUQMUKZU
@@ -292729,7 +292743,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $=
weq bitri 3bitrd rabbidva sneq xpeq2d elrab ctop ad2antrr topontop adantl
cin txtop syl2anc ad3antrrr toponmax xpexg simprr elrestr txrest syl22anc
syl3anc oveq2d restid eqtrd eleqtrd resttopon xpeq1d simprl xpss2 mpbird
- ciun snssd ssind eqsstr3d txtube toponss ssrab baib biantrud iunid xpeq2i
+ ciun snssd ssind eqsstrrd txtube toponss ssrab baib biantrud iunid xpeq2i
xpiundi eqtr3i sseq1i iunss ssin 3bitr3g anbi2d rexbidva ralrimiva eltop2
sylan2b 3syl eqeltrd imaeq2 mptpreima syl6eq syl5ibrcom rexlimdvva syl5bi
expimpd ralrimiv simpl ovex xkotf frn ax-mp ssexi cfi ctg xkoval xkotopon
@@ -292943,7 +292957,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $=
( J qTop F ) = { s e. ~P Y | ( `' F " s ) e. J } ) $=
( wcel wfo wss w3a cqtop cima cpw crab cvv wceq 3ad2ant2 syl2anc ccnv cin
co cv simp1 fof cuni uniexg 3ad2ant1 syl5eqel simp3 ssexd fex qtopval crn
- wf imassrn forn syl5sseq foima imass2 eqsstr3d eqssd pweqd cnvimass fssdm
+ wf imassrn forn syl5sseq foima imass2 eqsstrrd eqssd pweqd cnvimass fssdm
syl sstrd df-ss sylib eleq1d rabeqbidv eqtrd ) BCIZFEAJZFDKZLZBAMUCZAUAGU
DZNZDUBZBIZGADNZOZPZVTBIZGEOZPVQVNAQIZVRWERVNVOVPUEVQFEAUPZFQIWHVOVNWIVPF
EAUFSZVQFDQVQDBUGZQHVNVOWKQIVPBCUHUIUJVNVOVPUKZULFEQAUMTABCQDGHUNTVQWBWFG
@@ -294550,7 +294564,7 @@ Kolmogorov quotient is regular Hausdorff (T_3). (Contributed by Mario
`' G = ( z e. U. J |-> <. ( z |` A ) , ( z |` B ) >. ) ) $=
( vw vk cxp cuni wf1o ccnv cv cres cop cmpt wceq c1st cfv c2nd cun op1std
cmpo vex op2ndd uneq12d mpompt eqtr4i wcel wa cixp cdif wss adantl ixpeq2
- xp1st fvres unieqd mprg cvv ctop wf ssun1 syl5sseqr ssexd fssresd syl2anc
+ xp1st fvres unieqd mprg cvv ctop wf ssun1 sseqtrrid ssexd fssresd syl2anc
ptuni syl5eqr syl6eqr adantr eleqtrrd xp2nd eqcomd cin c0 uneqdifeq mpbid
wb ixpeq1d ssun2 eqtrd syl3anc eleqtrd eleq2d biimpar resixp opelxpd eqop
undifixp ad2antrl wfn adantrl ixpfn fnresdm reseq2d eqtr3d resundi syl6eq
@@ -294591,7 +294605,7 @@ Kolmogorov quotient is regular Hausdorff (T_3). (Contributed by Mario
ptunhmeo $p |- ( ph -> G e. ( ( K tX L ) Homeo J ) ) $=
( vz vk vn vf ctx ccn wcel ccnv chmeo cxp c1st cfv c2nd cun cmpt cmpo cop
co cv wceq vex op1std op2ndd uneq12d mpompt eqtr4i wfn cuni cixp cdif wss
- wa xp1st adantl cres ixpeq2 fvres unieqd mprg cvv ctop wf ssun1 syl5sseqr
+ wa xp1st adantl cres ixpeq2 fvres unieqd mprg cvv ctop wf ssun1 sseqtrrid
ssexd fssresd ptuni syl2anc syl5eqr syl6eqr adantr eleqtrrd eqcomd cin c0
xp2nd wb uneqdifeq mpbid ixpeq1d ssun2 eqtrd undifixp syl3anc ixpfn dffn5
syl sylib mpteq2dva syl5eq ctopon cpt syl5eqel toptopon txtopon wo eleq2d
@@ -295911,7 +295925,7 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro,
fdm expimpd eqcom imadisj notbii syl6ibr syld ralrimiv eleq2i 0ex elrnmpt
bitri eqid df-nel ralnex 3bitr4i sylibr imaeq2 cbvmptv elv anbi12i reeanv
ax-mp bitr4i fbasssin 3expb 3ad2antl1 adantrr rspceeqv mpan2 ad2antrl vex
- funimaex syl5bb ad2antrr mpbird imass2 inss1 inss2 ssini ineq12 syl5sseqr
+ funimaex syl5bb ad2antrr mpbird imass2 inss1 inss2 ssini ineq12 sseqtrrid
ad2antll ad2antlr sstrd sseq1 rspcev syl2anc adantlrl rexlimddv rexlimdvv
exp32 syl5bi ralrimivv 3jca isfbas2 3ad2ant3 mpbir2and ) BFUIUEKZFGDVBZGE
KZUFZCGUIUEKZCGUGZLZCMUHZMCUJZUANZUBNZUCNZUKZLZUACOZUCCULUBCULZUFZUUPCABD
@@ -296451,7 +296465,7 @@ contains the choice as a hypothesis (in the assumption that ` ~P ~P X `
( vf vg vu cufl wcel wss wa cv cufil cfv wrex co cfbas cpw syl2anc adantr
cfil syl wral cfg simpll filfbas adantl simplr sspwb sylib sstrd fbasweak
filsspw syl3anc fgcl ufli crest ssfg simprr filtop ad2antlr sseldd simprl
- wb trufil mpbird restid2 ssrest eqsstr3d sseq2 rspcev rexlimddv ralrimiva
+ wb trufil mpbird restid2 ssrest eqsstrrd sseq2 rspcev rexlimddv ralrimiva
wceq cvv ssexg ancoms isufl ) AFGZBAHZIZBFGZCJZDJZHZDBKLZMZCBSLZUAZVSWECW
FVSWAWFGZIZAWAUBNZEJZHZWEEAKLZWIVQWJASLGZWLEWMMVQVRWHUCZWIWAAOLGZWNWIWABO
LGZWAAPZHVQWPWHWQVSWABUDUEWIWABPZWRWHWAWSHZVSWABUKUEZWIVRWSWRHVQVRWHUFZBA
@@ -297310,7 +297324,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
co cdom wbr cen wex domeng wf1o bren biimpi ssufl wi cufil wrex cfil wral
simplr cfbas wf adantl f1of ad2antrr fmfil syl3anc ufli syl2anc cdm f1odm
ccnv vex dmex syl6eqelr simprl f1ocnv ad3antrrr fmufil ccom cid f1ococnv1
- cres oveq2d fveq1d fmco syl32anc fmid 3eqtr3d ufilfil 3syl eqsstr3d sseq2
+ cres oveq2d fveq1d fmco syl32anc fmid 3eqtr3d ufilfil 3syl eqsstrrd sseq2
simprr fmss rspcev rexlimddv ralrimiva wb isufl mpbird exlimiv imp syl2an
ex an12s exlimdv sylbid ) AHIZBAUAUBZBHIZXDXEBCJZUCUBZXGAKZLZCUDXFCBAHUEX
DXJXFCXDXJXFXHXDXIXFXHBXGDJZUFZDUDZXGHIZXFXDXILXHXMBXGDUGUHAXGUIXMXNXFXLX
@@ -298025,7 +298039,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
-> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. f e. ( Fil ` X )
( F " ( J fLim f ) ) C_ ( ( K fLimf f ) ` F ) ) ) ) $=
( vx ctopon cfv wcel wa ccn co wf cv cflf cflim wral wss wceq cfil cdm wb
- cima cnflf wfun ffun cuni flimelbas ssriv fdm toponuni ad2antrr syl5sseqr
+ cima cnflf wfun ffun cuni flimelbas ssriv fdm toponuni ad2antrr sseqtrrid
eqid adantl eqtrd funimass4 syl2an2 ralbidv pm5.32da bitr4d ) CEHIJZDFHIJ
ZKZBCDLMJEFBNZGOZBIBDAOZPMIZJGCVHQMZRZAEUAIZRZKVFBVJUDVISZAVLRZKGABCDEFUE
VEVFVOVMVEVFKZVNVKAVLVFBUFVEVJBUBZSVNVKUCEFBUGVPCUHZVJVQGVJVRVGVHCVRVRUOU
@@ -298571,7 +298585,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
( vx vy cfv wcel cv cfcls co c0 wne cfil wral wi wa wss wceq cvv syl ccmp
ctopon cuni eqid fclscmpi ralrimiva toponuni fveq2d raleqdv cfi cint ccld
syl5ibr wn cpw elpwi vn0 simpr inteqd int0 syl6eq neeq1d mpbiri a1d ssfii
- cfg ccl elv cfbas simplrl ad2antrr pweqd syl5sseqr sstrd simplrr toponmax
+ cfg ccl elv cfbas simplrl ad2antrr pweqd sseqtrrid sstrd simplrr toponmax
cldss2 w3a wb fsubbas mpbir3and ssfg syl5ss sselda fclssscls cldcls ssint
sylibr fgcl oveq2 rspcv 3syl ssn0 syl6an pm2.61dane expr sylan2 ralrimdva
sseqtrd com23 ctop topontop cmpfi sylibrd impbid ) BCUBFGZBUAGZBAHZIJZKLZ
@@ -298855,7 +298869,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
eleq2 eleq1 rexlimdva sylbid imp32 adantrrr elssuni fibas tgtopon ax-mp
ctopon ctb syl6eqel fiuni eqtrd fveq2d eleqtrrd toponuni sseqtr4d filss
simprrr rexlimddv ralrimiva imdistanda syl5bi flimopn sylibrd ralrimivw
- difexg unieq syl6eqr eqnetrd necon3bii sylib ciun sseq0 difss syl5sseqr
+ difexg unieq syl6eqr eqnetrd necon3bii sylib ciun sseq0 difss sseqtrrid
ssdif0 unissi eqssd jctil sseq1 eqeq2d anbi12d anbi2d pweq ineq1d mpcom
rexeqdv vtoclg mpdan uni0 syl6eq neeq2d ciin cmpt crn difssd riinn0 cab
sylan dfiin2g eqid rnmpt inteqi eldifbd difss2d ufilb mpbid fmpttd frnd
@@ -299612,7 +299626,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
syld biimpa impel syl12anc snid syl5eleqr flfnei simprd r19.21bi ad4ant13
cfil ad3antrrr 3an1rs adantllr creg simprl regsep expcom ad2antll syl5bir
reximi 3expib anim1d eltopss 3expa cfm cflim flfval cfbas uniexg syl5eqel
- elrestr cfg filfbas fgfil eleqtrrd eqid imaelfm flimclsi eqsstrd eqsstr3d
+ elrestr cfg filfbas fgfil eleqtrrd eqid imaelfm flimclsi eqsstrd eqsstrrd
sylibr adantlr expl wfun cdm cnextf ffund fdmd sseqtr4d funimass4 biimprd
reximdva cnnei mpbird ) AFGHUIUJRZGHUKUJSZUXQUAULZUMUBULZUNZUABULZUQZGUOR
ZRZUPZUBUYBUXQRZUQZHUORRZURZBEURZAUYJBEAUYBESZTZUYFUBUYIUYMUXTUYISZTZAUCU
@@ -300103,7 +300117,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
anbi12d sylib wi crn cint cin toponuni ad2antrr simplrl simplrr fvconst2g
ineq1d eleq2d ralbidva mpbid ffnfv sylanbrc frnd wfo dffn4 exlimdv simprl
fofi rintopn mptelixpg ralrn elrint inex1 ixpconstg inss2 fnfvelrn intss1
- sylancl syl5ss ralrimiva ss2ixp eqsstr3d ssrab ad2antll ssralv sylc oveq1
+ sylancl syl5ss ralrimiva ss2ixp eqsstrrd ssrab ad2antll ssralv sylc oveq1
simprbi raleqdv rspcev syl12anc ex 3adantr3 imbi1d syl5ibrcom expimpd mpd
impd ) AUAURZCUGZUBURZUVMUHZUVOCIUIUJZUHZTZUBCUKZUVPUVRULUMUBCUCURZUNUKUC
UOUPZUQZUDURZUBCUVPUSZUMZUTZUAVAZCJUIUJZUWDTZUWDHGURZVBVCZFTZGDCVDVCZVEZV
@@ -300390,7 +300404,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
cxp cima opelxpi ctx txcls syl22anc txtopon cnvimass cgrp wf tgpgrp fssdm
grpsubf subgsubcl 3expb ralrimivva fveq2 syl6eqr eleq1d ralxp sylibr wfun
cdm ffund xpss12 fdmd funimass5 mpbird clsss syl3anc ccn tgpsubcn cncls2i
- wb sstrd eqsstr3d sselda sylan2 wfn ad2antrr elpreima 4syl mpbid syl5eqel
+ wb sstrd eqsstrrd sselda sylan2 wfn ad2antrr elpreima 4syl mpbid syl5eqel
ffn simprd w3a issubg4 mpbir3and ) BUAHZABUBIZHZJZACUCIIZXPHZXSBUDIZKZXSU
EUFZELZFLZBUGIZMZXSHZFXSNEXSNZXRXSCUHZYAXRCUIHZAYJKZXSYJKXRCYAUJIHZYKXOYM
XQBCYADYAOZUKPZYACULQZXRAYAYJXQAYAKZXOYAABYNUNRZXRYMYAYJSYOYACUMQZUOZACYJ
@@ -300417,7 +300431,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
cbs ctopon tgptopon topontop ad2antlr subgss wceq toponuni sseqtrd clsss3
syl2anc sseqtr4d resmptd rneqd syl5eq ccn tgptmd simpr cnmptc cnmpt1plusg
ctmd cnmptid tgpsubcn cnmpt12f cnclsi nsgconj ad4ant234 fmpttd frnd clsss
- ctx eqsstrd syl3anc sstrd eqsstr3d wfn ovex fnmpti df-f mpbiran ralrimiva
+ ctx eqsstrd syl3anc sstrd eqsstrrd wfn ovex fnmpti df-f mpbiran ralrimiva
sylibr fmpt isnsg3 sylanbrc ) BUAGZABUBHZGZIZACUCHZHZBUDHZGZEJZFJZBUEHZKZ
XGBUFHZKZXDGFXDLZEBUNHZLXDWTGXAWSAXEGZXFABUGZABCDUHUIXBXMEXNXBXGXNGZIZXDX
DFXDXLMZUJZXMXRXSNZXDOZXTXRYAFXNXLMZXDPZXDXRYDYCXDQZNYAYCXDUKXRYEXSXRFXNX
@@ -300599,7 +300613,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
ghmgrp2 simprrr eqeltrd elrabd cnpimaex ssrab simprbi wfn wb ffnd toponss
sylan ralima syl5ib simpl1 ad2antrr fveq2 oveq2d rspccv grpcl c0g cminusg
ghmlin grpinvsub grprinv eqtr3d grpass syl13anc 3eqtr3d adantlr ralrimiva
- grplid fveq2d sylibd ralrab2 sylibr wfun ffund ssrab2 syl5sseqr funimass4
+ grplid fveq2d sylibd ralrab2 sylibr wfun ffund ssrab2 sseqtrrid funimass4
cdm fdmd mpbird eleq2 expr mpbir2and imaeq2 sseq1d anbi12d rspcev sylan2d
syl12anc rexlimdva mpd anassrs iscnp ex jcad cncnpi ancoms impbid1 eleq2d
cncnp anbi1d bitr4d ) CUELZDUELZBCDUHMLZUFZBAEFUIMZNLZAEUJZLZBEFUKMLZOZAG
@@ -301296,7 +301310,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the
( vz wcel wa ad2antrr vx vu vy ctsu co cgsu csn ccl cfv cv wss cres cpw
wi cfn cin wral wrex cuni wne ctopon wceq ctps istps sylib toponuni syl
eleq2d csupp cun elfpw simplbi adantl suppssdm fssdm elinel2 cfsupp wbr
- c0 unssd fsuppimpd syl2anc sylanbrc wb ssun1 id syl5sseqr reseq2 oveq2d
+ c0 unssd fsuppimpd syl2anc sylanbrc wb ssun1 id sseqtrrid reseq2 oveq2d
unfi pm5.5 eleq1d bitrd rspcv ccmn wf ssun2 a1i sylibd rexlimdva simprr
gsumres simplrr eqeltrd expr ralrimiva rspceaimv syl2an2r impbid disjsn
sseq1 necon2abii syl6bb imbi2d ralbidva anbi12d eqid eltsms ctop gsumcl
@@ -301419,7 +301433,7 @@ discrete topology (which is Hausdorff), this theorem can be used to turn
wa resabs1 syl6eq oveq2d eleq1d rspcv syl ad2antlr syl6ss biantrud resres
imbi12d oveq2i ccmn ad2antrr wf fssresd csupp cvv syl2anc fvexi ressuppss
fex c0g sstrd a1i fdmfifsupp gsumres syl5reqr sylibrd ralrimdva rspceaimv
- sseq1 syl6an rexlimdva cun unssd syl2an wb ssun1 id syl5sseqr pm5.5 bitrd
+ sseq1 syl6an rexlimdva cun unssd syl2an wb ssun1 id sseqtrrid pm5.5 bitrd
unfi adantrr jctir indir df-ss sylib uneq2d simprr ssequn1 syl5eq reseq2d
eqtrd resabs1d eqtr4d eqtr3d biimpd expr eqid eltsms com23 impbid ralbidv
syld imbi2d anbi2d inex1g fssres wfn ffn fnresdm 3syl reseq1d feq1d mpbid
@@ -301697,7 +301711,7 @@ discrete topology (which is Hausdorff), this theorem can be used to turn
Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.) $)
tsmssplit $p |- ( ph -> ( X .+ Y ) e. ( G tsums F ) ) $=
( vk co cv wcel cfv c0g cif cmpt cof ctsu ffvelrnda cmnd ccmn cmnmnd eqid
- wa syl mndidcl adantr ifcld fmpttd cres feqmptd reseq1d wss cun syl5sseqr
+ wa syl mndidcl adantr ifcld fmpttd cres feqmptd reseq1d wss cun sseqtrrid
wceq ssun1 iftrue mpteq2ia resmpt 3eqtr4a eqtr4d oveq2d ctmd ctps cdif wn
tmdtps eldifn adantl iffalsed suppss2 tsmsres eqtrd eleqtrd ssun2 tsmsadd
wi cin c0 noel eleq2 mtbiri elin sylnib imnan sylibr imp oveq12d syl2an2r
@@ -301763,7 +301777,7 @@ discrete topology (which is Hausdorff), this theorem can be used to turn
reseq2d grpinvhmeo tgplacthmeo hmeoco hmeoima grpsubid1 sylancl tsmsi
elrnmpt1s ralrimiva sseq1 imbi1d ac6sfi cuni cun inss1 sspwuni rnxpss
frn rnss unssd wfo wfn ffn dffn4 fofi inss2 elinel2 rnfi unfi adantrr
- unifi ssun2 wb fvssunirn ssun1 sstri syl5sseqr pm5.5 bitrd rspcv cmnd
+ unifi ssun2 wb fvssunirn ssun1 sstri sseqtrrid pm5.5 bitrd rspcv cmnd
id reseq2 cmnmnd simplr jca ovexd fvexi fsuppmptdm gsumcl velsn ovres
c0g sylanbr oveq1 eqtrd gsumsn syl3anc snfi snssd xpss12 fssresd xpfi
sylancr fdmfifsupp gsumxp 3eqtr4rd elrnmpti cabl isabl ablnncan simpr
@@ -302536,7 +302550,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed
composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.) $)
ustssco $p |- ( ( U e. ( UnifOn ` X ) /\ V e. U ) -> V C_ ( V o. V ) ) $=
( cust cfv wcel wa cid cres cun ccom ssun1 coires1 wrel cdm wss wceq ustrel
- cxp ustssxp dmss syl dmxpid syl6sseq syl2anc syl5eq uneq1d syl5sseqr coundi
+ cxp ustssxp dmss syl dmxpid syl6sseq syl2anc syl5eq uneq1d sseqtrrid coundi
relssres syl6sseqr ustdiag ssequn1 sylib coeq2d sseqtrd ) ACDEFBAFGZBBHCIZB
JZKZBBKZUQBBURKZVAJZUTUQBVAJBVCBVALUQVBBVAUQVBBCIZBBCMUQBNBOZCPVDBQABCRUQVE
CCSZOZCUQBVFPVEVGPABCTBVFUAUBCUCUDBCUJUEUFUGUHBURBUIUKUQUSBBUQURBPUSBQABCUL
@@ -302650,7 +302664,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed
17-Nov-2017.) $)
ustund $p |- ( ph -> ( ( A u. B ) X. ( A u. B ) ) C_ ( V o. V ) ) $=
( cun cxp cin ccom wne xpindir inss1 syl5ss syl5eqss inss2 unssd xpindi
- c0 wceq xpco syl xpundi xpundir coss12d eqsstr3d ) ABCHZUHIZBCJZUHIZUHUJI
+ c0 wceq xpco syl xpundi xpundir coss12d eqsstrrd ) ABCHZUHIZBCJZUHIZUHUJI
ZKZDDKAUJTLUMUIUAGUHUJUHUBUCAUKDULDAUKUJBIZUJCIZHDUJBCUDAUNUODAUNBBIZCBIZ
JZDBCBMAURUPDUPUQNEOPAUOBCIZCCIZJZDBCCMAVAUTDUSUTQFOPRPAULBUJIZCUJIZHDBCU
JUEAVBVCDAVBUPUSJZDBBCSAVDUPDUPUSNEOPAVCUQUTJZDCBCSAVEUTDUQUTQFOPRPUFUG
@@ -302672,8 +302686,8 @@ unit group (that is, the nonzero numbers) to the field. (Contributed
18-Nov-2017.) $)
ustneism $p |- ( ( V C_ ( X X. X ) /\ A e. X )
-> ( ( V " { A } ) X. ( V " { A } ) ) C_ ( V o. `' V ) ) $=
- ( cxp wss wcel wa csn cima ccom ccnv c0 wne wceq snnzg adantl xpco eqsstr3i
- syl mp1i cnvxp ressn cnveqi resss cnvss ax-mp coss2 coss1 sstrd eqsstr3d
+ ( cxp wss wcel wa csn cima ccom ccnv c0 wne wceq snnzg adantl xpco eqsstrri
+ syl mp1i cnvxp ressn cnveqi resss cnvss ax-mp coss2 coss1 sstrd eqsstrrd
cres ) BCCDEZACFZGZBAHZIZUPDZUOUPDZUPUODZJZBBKZJZUNUOLMZUTUQNUMVCULACOPUPUO
UPQSUNUTURVAJZVBUSVAEUTVDEUNUSURKZVAUOUPUAVEBUOUKZKZVAVFURBAUBZUCVFBEVGVAEB
UOUDZVFBUEUFRRUSVAURUGTURBEVDVBEUNURVFBVHVIRURBVAUHTUIUJ $.
@@ -302743,7 +302757,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed
cvv vy cust cfv cxp crest co cpw wral cres ccnv w3a restsspw inxp sseqin2
a1i biimpi sqxpeqd syl5eq adantl simpl elfvex adantr ssexd xpexd ustbasel
wi elrestr syl3anc eqeltrrd cun simplr simp-4r elpwid sstrd ustssxp unssd
- xpss12 ssun2 ustssel mpi df-ss sylib uneq1d simpllr eqsstr3d eqtr2d indir
+ xpss12 ssun2 ustssel mpi df-ss sylib uneq1d simpllr eqsstrrd eqtr2d indir
ssequn2 syl6eqr rspceeqv elrest biimpar syl21anc syldanl ad2antrr r19.29a
biimpa ex ralrimiva ustincl simprl simprr ineq12d inindir reeanv sylanbrc
ineq1 r19.29vva simp-4l ustdiag cdm inss1 resss sstri iss wbr ssel2 equid
@@ -303893,7 +303907,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed
wf cvv cfilufbas syl2anc cucn co wa wbr wi isucn simprbda syl21anc fbasrn
elfvexd syl3anc cop cmpo ccnv cima cmpt wceq simplr eqid rspceeqv sylancl
crn imaeq2 wb imaexg elrnmpt 3syl ad3antrrr cbvmptv rneqi eqtri syl6eleqr
- mpbird wfn ffnd fbelss sylan ad4ant13 fmucndlem mpofun funimass2 eqsstr3d
+ mpbird wfn ffnd fbelss sylan ad4ant13 fmucndlem mpofun funimass2 eqsstrrd
wfun mpan adantl sqxpeqd sseq1d rspcev adantr simpr nfcv weq simpl fveq2d
opeq12d cbvmpo ucnprima cfiluexsm r19.29a ralrimiva iscfilu syl mpbir2and
id ) ACFUFRSZCHUGRSZUATZYCUHZOTZUIZUACUJZOFUKZABGUGRSZGHEUMZHUNSYBADGULRS
@@ -305137,7 +305151,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric
anbi1d eliun eleq2i elv elrab2 simprbi simp2d wf1 f1of1 simplr elfz1end
sylib nnuz syl6eleq eluzfz1 xrleidd fcompt reseq1d 1st2nd2 simp1d xp1st
opeq1d eqtr2d syl5eq 3eqtr4d a1d simprl simprr peano2fzr expr cun fzsuc
- imim1d elfzuz3 ad2antll fzss2 eqsstr3d unssad fssresd xleadd1a syl13anc
+ imim1d elfzuz3 ad2antll fzss2 eqsstrrd unssad fssresd xleadd1a syl13anc
ex xmettri fvoveq1 simp3d nnz ad2antrl eluzp1m1 elfzuzb rspcdva xaddcld
eqtr4d xrletr mpand syld cplusg xrex difexi ressplusg fzelp1 sylan2 cin
wn fzp1disj disjsn gsumunsn reseq12d oveq1d sylibrd animpimp2impd nnind
@@ -306277,7 +306291,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric
the base set. (Contributed by NM, 3-Sep-2006.) (Revised by Mario
Carneiro, 12-Nov-2013.) $)
mopnfss $p |- ( D e. ( *Met ` X ) -> J C_ ~P X ) $=
- ( cxmet cfv wcel cuni cpw pwuni mopnuni pweqd syl5sseqr ) ACEFGZBHZIBCIBJ
+ ( cxmet cfv wcel cuni cpw pwuni mopnuni pweqd sseqtrrid ) ACEFGZBHZIBCIBJ
NCOABCDKLM $.
$( The base set of a metric space is open. Part of Theorem T1 of
@@ -306586,7 +306600,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric
( vx cmopn cfv cpw wss wceq cdm wcel eqid syl ctopn setsmstset cxmet cuni
cts cbs crn cv cbl df-mopn dmmptss sseli wa simpr xmetunirn sylib mopnuni
ctg cxp cds cin cres dmeqd dmres syl6eq inss1 syl6eqss dmss dmxpid adantr
- syl6sseq eqsstr3d sspwuni sylibr ex wn ndmfv 0ss pm2.61d1 setsmsbas pweqd
+ syl6sseq eqsstrrd sspwuni sylibr ex wn ndmfv 0ss pm2.61d1 setsmsbas pweqd
syl5 c0 3sstr3d topnid eqtrd ) ABLMZCUEMZCUAMZABCDEFGHIJUBZAWHCUFMZNZOWHW
IPAWGFNZWHWLABLQZRZWGWMOZWOBUCUGUDZRZAWPWNWQBKWQKUHUIMUGURMLKUJUKULAWRWPA
WRUMZWGUDZFOWPWSWTBQZQZFWSBXBUCMRZXBWTPWSWRXCAWRUNBUOUPBWGXBWGSUQTAXBFOWR
@@ -307088,7 +307102,7 @@ S C_ ( P ( ball ` D ) T ) ) $=
( A. x e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\
( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> J = K ) ) $=
( cfv wcel wa cv cbl co crp wrex wral wss cle wbr wceq wi simprrr simplll
- cxmet cxr simplr simprlr rpxrd simprll simprrl syl221anc eqsstr3d simpllr
+ cxmet cxr simplr simprlr rpxrd simprll simprrl syl221anc eqsstrrd simpllr
ssbl eqsstrd jca expr anassrs reximdva r19.40 syl6 ralimdva r19.26 syl6ib
metequiv sylibrd ) BFUGKZLZCVJLZMZGNZHNZUAUBZANZVNBOKZPZVQVNCOKZPZUCZMZGQ
RZHQSZAFSWAVQVOVRPZTZGQRZHQSVSVQVOVTPZTZGQRZHQSMZAFSDEUCVMWEWLAFVMVQFLZMZ
@@ -308208,7 +308222,7 @@ the half element (corresponding to half the distance) is also in this
filsspw filtop ad3antrrr simpllr elpwid filss syl13anc ralrimiva ad2antrr
3syl ex filin syl3anc metustid ad5ant24 sstrd biimpa simprd sylan r19.29a
elfg adantr ssfg sseldd simpld cnvss cnvxp syl6sseq wceq metustsym adantl
- eqsstr3d metustexhalf ad4ant13 r19.41v sstr reximi sylbir sylancom ssrexv
+ eqsstrrd metustexhalf ad4ant13 r19.41v sstr reximi sylbir sylancom ssrexv
sylc 3jca cvv wb elfvex isust mpbir3and ) CUCUAZACUDIJZKZCCUBZBUEUFZCUGIJ
ZXIXHUHZLZXHXIJZFMZGMZLZXOXIJZUIZGXKNZXNXOUJXIJZGXINZUKCULZXNLZXNOZXIJZXO
XOUMZXNLZGXIPZQZQZFXINZXGBXHUNIJZXIXHUOIJZXLABCDEUPZBXHUQZXIXHURVGXGYLYMX
@@ -308482,7 +308496,7 @@ the half element (corresponding to half the distance) is also in this
nfv nfne ineq1 cxr wf wfun psmetf ffun respreima 4syl ad6antr eqtr4d rspe
simp2 inex1 eqid elrnmpt ax-mp simpllr ssinss1 pweq eleq2d syl6bb syl6bbr
a1i ssin ad5antlr mpbir2and inelcm elv sylib r19.29af2 ssn0 ancoms metuel
- 3adant2 simplbda adantr r19.29a r19.29an jca cun elpwid rspceeqv eqsstr3d
+ 3adant2 simplbda adantr r19.29a r19.29an jca cun elpwid rspceeqv eqsstrrd
cdif unssd 3bitri wex simprl simpl3 xpss12 difssd ad4antr cnvexg cnvimass
sstrd eqidd imaexg mpbird fssdm ssdif0 0ss syl6eqss simpr ssundif difdif2
difcom sseq1i rspcev anbi1i ancom exbii n0 df-rex 3bitr4i biimpi ad2antll
@@ -309610,7 +309624,7 @@ definition of norm (which itself uses the metric). (Contributed by
ccom adantl dmeqd dmcoss cminusg cplusg grpsubfval ovex sseqtri syl6eqssr
co dmmpo adantr dmss dmxpid syl6sseq simpr xmetunirn sylib mopnuni tngbas
ad2antlr 3sstr3d sspwuni sylibr ex syl5 wn c0 ndmfv 0ss syl6eqss pm2.61d1
- syl5eqss eqsstr3d topnid eqtrd ) CFMZEGMZUAZDBUBNZBUCNZABCDEFGHIJUDZWTXAB
+ syl5eqss eqsstrrd topnid eqtrd ) CFMZEGMZUAZDBUBNZBUCNZABCDEFGHIJUDZWTXAB
UENZUFZOXAXBPWTXADXEXCWTDAQNZXEJWTAQRZMZXFXEOZXHAUGUHUIZMZWTXIXGXJAKXJKUJ
ZUKNUHULNQKUMUNUOWTXKXIWTXKUAZXFUIZXDOXIXMARZRZCUENZXNXDXMXPXQXQUPZRZXQXM
XOXROZXPXSOWTXTXKWTXOECUQNZVAZRZXRWTYBAWSYBAPWRWSYBBURNABCYAEGHYASZUSIUTV
@@ -314045,7 +314059,7 @@ Normed space homomorphisms (bounded linear operators)
if ( x <_ B , D , E ) ) e. ( ( O tX J ) Cn K ) ) $=
( cicc co cxp cv cle wbr cif cmpo ctx cuni eqid ccld cfv crest cr iccssre
wcel wss syl2anc cioo crn ctg sseldd icccld fveq2i syl6eleqr cun iccsplit
- ssun1 wceq syl3anc syl5sseqr uniretop unieqi eqtr4i restcldi toponuni syl
+ ssun1 wceq syl3anc sseqtrrid uniretop unieqi eqtr4i restcldi toponuni syl
ctopon ctop topcld 3syl eqeltrd txcld ssun2 xpeq1d xpundir syl6eq retopon
topontop eqeltri resttopon sylancr syl5eqel txtopon eqtr3d wral ccn sstrd
cntop2 toptopon2 sylib w3a elicc2 biimpa simp3d 3adant3 iftrued mpoeq3dva
@@ -315170,7 +315184,7 @@ topological space to the reals is bounded (above). (Boundedness below
( vy vz cv co c1 crp wcel vr cle wbr cif cmpo cbl cfv wss wrex wral cmopn
eqid cxmet cmet 1rp stdbdmet sylancl ccmp cxr cc0 clt wceq rpxr mp1i 0lt1
a1i stdbdmopn syl3anc eqeltrrd sseqtrd lebnum wa simpr wi ad2antrr adantr
- ifcl syl cr rpre ad2antlr 1re min2 stdbdbl syl33anc metxmet min1 eqsstr3d
+ ifcl syl cr rpre ad2antlr 1re min2 stdbdbl syl33anc metxmet min1 eqsstrrd
ssbl syl221anc sstr2 reximdv ralimdva oveq2 sseq1d rexbidv ralbidv rspcev
syl6an rexlimdva mpd ) ABPZUAPZNOGGNPOPDQZRUBUCXDRUDUEZUFUGZQZCPZUHZCEUIZ
BGUJZUASUIXBHPZDUFUGZQZXHUHZCEUIZBGUJZHSUIZABCXEEXEUKUGZGUAXSULADGUMUGZTZ
@@ -317032,7 +317046,7 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` )
1m1e0 simp1d cbvmptv ctopon wf iitopon cnf2 mp3an2i feqmptd iirev cc cr
ax-1cn unitssre sseli recnd nncan sylancr eqtrd mpteq2ia syl6eqr oveq1d
eceq1d opeq2d mpteq2dv rneqd syl5eq cnveqd a1i unieqd mpteq12dv 3sstr4d
- oveq2d cnvss eqsstr3d eqssd pi1xfr eqeltrd jca ) AIUDZJUEYNEDUFUGZUHAYN
+ oveq2d cnvss eqsstrrd eqssd pi1xfr eqeltrd jca ) AIUDZJUEYNEDUFUGZUHAYN
JABCDEFGHIJKLMNOPQRSTUAUIAJJUDZUDZYNAJUJZYQJUEAJUKUKULUMYRAGGUPZLUNUOZU
QZHYSKLURUOZUGZUUBUGZYTUQZUKUKJEVIUOZUSZUAYTUKUHZUUAUKUHAYSUUGUHUTZLUNV
AZYSUKYTVBVCUUHUUEUKUHUUIUUJUUDUKYTVBVCVDJVEVFJVGVHAYPIUMYQYNUMAGUUGUUA
@@ -320228,7 +320242,7 @@ is an accumulation point (limit point) of subset ` y ` ". @)
copab cms
cne wceq fveq2 syl6eqr sseq2d eleq2d oveqd sseq1d anbi2d rexbidv
3anbi123d opabbidv df-neiOLD cpw cxp cvv fvex eqeltri pwex xpex df-3an
- vex elpw 3anbi1i bitr3i opabbii opabssxp eqsstr3i ssexi
+ vex elpw 3anbi1i bitr3i opabbii opabssxp eqsstrri ssexi
fvopab4 ) GHDAIZ
GIZJKZLZBIZVQMZNCIZOPZVSWAVPQKZRZVOLZSZCTUAZUBZABUCVOELZVSEMZWBVSWADQKZRZ
VOLZSZCTUAZUBZABUCZUDUEVPDUFZWHWPABWRVRWIVTWJWGWOWRVQEVOWRVQDJKZEVPDJUGFU
@@ -320245,7 +320259,7 @@ is an accumulation point (limit point) of subset ` y ` ". @)
( vz vw cv c1st cfv wcel wss cne wbr wa wceq wn wrex wi wal w3a copab
cms cacOLD fveq2 syl6eqr eleq2d sseq2d wb breq syl anbi12d imbi1d albidv
3anbi123d opabbidv df-accOLD cpw cxp cvv fvex eqeltri pwex xpex df-3an
- vex elpw 3anbi2i bitr3i opabbii opabssxp eqsstr3i ssexi
+ vex elpw 3anbi2i bitr3i opabbii opabssxp eqsstrri ssexi
fvopab4 ) HIEAJZ
HJZKLZMZBJZVSNZCJZVSNZWCVQVROLZPZQZDJZWAMWHVQRSQDWCTZUAZCUBZUCZABUDVQFMZW
AFNZWCFNZWCVQEOLZPZQZWIUAZCUBZUCZABUDZUEUFVRERZWLXAABXCVTWMWBWNWKWTXCVSFV
@@ -320263,7 +320277,7 @@ is an accumulation point (limit point) of subset ` y ` ". @)
( vz vw cv c1st cfv wcel wss cc0 clt wbr wceq wn wa cblOLD co wrex wi cr
wral w3a copab cms cac fveq2 syl6eqr eleq2d sseq2d wb oveqd rexeq syl
imbi2d ralbidv 3anbi123d opabbidv df-acc cpw cxp cvv fvex eqeltri pwex
- xpex df-3an vex elpw 3anbi2i bitr3i opabbii opabssxp eqsstr3i ssexi
+ xpex df-3an vex elpw 3anbi2i bitr3i opabbii opabssxp eqsstrri ssexi
fvopab4 ) HIEAJZHJZKLZMZBJZWCNZOCJZPQZDJZWEMWIWARSTZDWAWGWBUALZUBZUCZUDZC
UEUFZUGZABUHWAFMZWEFNZWHWJDWAWGEUALZUBZUCZUDZCUEUFZUGZABUHZUIUJWBERZWPXDA
BXFWDWQWFWRWOXCXFWCFWAXFWCEKLZFWBEKUKGULZUMXFWCFWEXHUNXFWNXBCUEXFWMXAWHXF
@@ -327601,7 +327615,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically
cxr cop 1st2nd2 fveq2d df-ov syl6eqr eqtrd df-pr preq12d syl5eqr 3eqtr4rd
uneq12d iuneq2dv cpw ffn inss2 rexpssxrxp sstri fss sylancl fnfco sylancr
iccf fniunfv iunun ioof wfun cdm wfo fo1st fofn ssv fndm eqimss2 dfimafn2
- fnfun 3syl syl2anc fnima eqtr3d rnco2 syl6eq fo2nd 3eqtr3d syl5sseqr cdom
+ fnfun 3syl syl2anc fnima eqtr3d rnco2 syl6eq fo2nd 3eqtr3d sseqtrrid cdom
syl5eq ovolficcss ssdifssd cen ccrd cres con0 omelon nnenom ensymi isnumi
mp2an fofun fof fdmi sseqtr4i fores ffnd dffn4 sylib fodomnum mpsyl unctb
foco domentr ctex ssid syl5sseq ssundif ssdomg sylc domtr ovolctb2 jca )
@@ -327842,7 +327856,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically
cr ccom crn cuni cc0 wceq simpld cn cle cxp wf sstrd ovolsscl syl3anc
syl difssd c1 cfz cv wral wa c1st elfznn syl2an sseldi fveq2d syl6eqr
c2nd ralrimiva sylibr readdcld cuz cima cun wfn cxr sylancl ffnd cmin
- nnuz peano2nnd syl6eleq syl5eq cc nncnd ax-1cn eqtrd syl6eq syl5sseqr
+ nnuz peano2nnd syl6eleq syl5eq cc nncnd ax-1cn eqtrd syl6eq sseqtrrid
ovollb unss1 ax-mp sseqtrd unss2 eqsstri syl6ss ovolss syl2anc ovolun
csup wbr syl22anc letrd eqid resubcld nnaddcl cz adantr adantl wb csu
nnzd mpbird simp2d simp1d recnd fsumser ad2antrr oveq12d lelttrd cicc
@@ -327960,7 +327974,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically
wa c0 wdisj mpbid cmul eqsstri csup uniioombllem1 ssid ovollecl ssun2
ovollb peano2nnd syl6eleq uzsplit ax-1cn pncan oveq2d iuneq1d cpw wfn
iunxun rexpssxrxp sstri ffn fniunfv sylan indi 3eqtr4ri eqtr3i uneq2d
- 3eqtr4d syl5sseqr syl6ss rpred ovolun syl22anc eqbrtrd iunss r19.21bi
+ 3eqtr4d sseqtrrid syl6ss rpred ovolun syl22anc eqbrtrd iunss r19.21bi
fsumrecl cfn jca ovolfiniun cdiv nndivred cdm c1st cop ffvelrn elin2d
c2nd 1st2nd2 df-ov ioombl syl6eqel adantlr finiunmbl syl5eqss ioossre
inmbl syl6eqss ovolfcl ovolioo simp2d resubcld mblsplit 3sstr4i sslin
@@ -330764,7 +330778,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
opelxpd eqcomd rspceeqv ralrimiva ssabral rnmpt eldifi simprr inss1 wbr
eldifn eliniseg brelrn sylbi nsyl pm2.21d ssrdv covol 0mbl mblvol ovol0
elv ss0 eqtri mul01d expr cif iftrue c0ex ovmpoa mp2an 0cn mul01i f1ofo
- pm2.61d2 an32s dfin4 eqsstr3i sseli syl6eqel mul02d sylan2 3eqtr2d
+ pm2.61d2 an32s dfin4 eqsstrri sseli syl6eqel mul02d sylan2 3eqtr2d
elsni ) AGHNUDOZUEUFZHUGZDUGZPUHZUIZUAUKZVUECUKZUJOZVUFIOZULOZUAUMZCUMZ
VUAGUGZBUKZVUFNOZVUMVUFIOZULOZBUMZCUMVULVUAVUPCUMBUMAUYTVUDVUEUYSUNVUEU
HUOZUPUFZULOZUAUMZVUDVUAVUICUMZUAUMVUKAUYSUEUQZQVUDURQZUYSUGZVUCUIVUDUS
@@ -330837,7 +330851,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
inmbl ssdifssd eldifsni ad2antlr necon3ai itg1addlem3 syl21anc eqeltrrd
wne simpl itg1addlem1 iunin2 mblss iunid imaeq2i imaiun cnvimarndm fdmd
3eqtr3i syl5eq sseqtr4d df-ss sylib syl5req fveq2d 3eqtr4d oveq2d dfin4
- fsummulc2 difssd eqsstr3i sseli elsni oveq1d 0re syl6eqel mul02d cuz wo
+ fsummulc2 difssd eqsstrri sseli elsni oveq1d 0re syl6eqel mul02d cuz wo
olcd sumz sylan2 fsumss 3eqtrd inss1 simpr incom iuneq2i eqtri cnvimass
syl5sseq cc anasss fsumcom oveq12d ) AEUAZFUAZLUBZUUAMUBZGNZUCNZMOZYTUU
BUUCUCNZMOZUDNZLOZYSUUELOZYSUUGLOZUDNEFUDUENUIUFZEUIUFZFUIUFZUDNAYSUUEU
@@ -332217,7 +332231,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
eqidd i1fadd eqeltrrd cdif i1ff eldifi ffvelrn syl2an leidd iftrue adantl
wf wn eldifn elin sylnib imnan sylibr iffalse oveq12d recnd addid1d eqtrd
imp breqtrrd ad2antrr syl6bb ffnd adantlr 0e0iccpnf ifclda fmptd r19.21bi
- cc eldif nfcv cid fveq2d breqtrd syldan pm2.61dan eleq1w syl5sseqr sselda
+ cc eldif nfcv cid fveq2d breqtrd syldan pm2.61dan eleq1w sseqtrrid sselda
ifbieq1d nfv nfan ofrfval2 iftrued 3brtr4d 0le0 a1d ralrimi mpbird itg2ub
wb syl3anc wo cun eleq2d elun notbid ioran biimpar simprr cpnf cicc inidm
wfn ofrfval mpbid sylan2 nfmpt1 nfcxfr nffv nfeq1 fveqeq2 fvmpt2i 0cn fvi
@@ -332285,7 +332299,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
cof simprll simprrl itg1add cin i1fadd inss1 cvol mblss syl5ss covol wceq
wss cdif nfv nfcv cmpt nfmpt1 nfcxfr nfbr nfan eldifi cvv i1ff ffnd inidm
reex eqidd ofval sylan2 ffvelrn syl2an iccssxr sseldi 0red simprrr wfn wb
- fvexd syl5sseqr sselda syldan simpr dffn5 sylib ofrfval2 r19.21bi breqtrd
+ fvexd sseqtrrid sselda syldan simpr dffn5 sylib ofrfval2 r19.21bi breqtrd
mpbid adantl recnd syl2anc breqtrrd xrletrd fveq2 expr ralrimiva resubcld
itg2leub mpbird cun ssun2 eldifn elin sylnib sylibr imp iffalsed leadd2dd
imnan addid1d simprlr ssun1 iftrued fvmpt2 iftrue 3eqtr4d iffalse addid2d
@@ -334700,7 +334714,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
by Mario Carneiro, 11-Aug-2014.) $)
itgsplit $p |- ( ph -> S. U C _d x = ( S. A C _d x + S. B C _d x ) ) $=
( vk cc0 co cr wcel wa cif adantr c3 cfz ci cv cexp cdiv cre cfv cle cmpt
- wbr citg2 cmul csu caddc citg cvol cdm cibl cmbf iblmbf syl cun syl5sseqr
+ wbr citg2 cmul csu caddc citg cvol cdm cibl cmbf iblmbf syl cun sseqtrrid
ssun1 sselda syldan mbfdm2 ssun2 cin covol wceq cxr cpnf cicc eleq2d elun
cc wo syl6bb biimpa mbfmptcl jaodan adantlr ax-icn elfznn0 adantl sylancr
cn0 expcl wne ine0 elfzelz expne0i mp3an12i divcld recld 0re ifcl sylancl
@@ -335326,7 +335340,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
limcdif $p |- ( ph ->
( F limCC B ) = ( ( F |` ( A \ { B } ) ) limCC B ) ) $=
( vx vz co cc wss wcel wa wceq adantr wf ex cfv cmpt eqid sylbi climc csn
- cdif cres cv cdm w3a limcrcl adantl simp2d eqsstr3d simp3d jca cun undif1
+ cdif cres cv cdm w3a limcrcl adantl simp2d eqsstrrd simp3d jca cun undif1
fdmd difss fssres sylancl snssd unssd syl5eqssr unssad wb cif ctopn crest
ccnfld ccnp simprl simprr ellimc eqcomi oveq2i mpteq1i wo velsn orbi2i wn
elun pm5.61 fvres ifeq2da mpteq2ia eqtr4i ssdifssd bitr4d pm5.21ndd eqrdv
@@ -335360,7 +335374,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
sseq1d sylancl ralrimivw fnmpt fmpt df-f simplrr elinel2 sylbi syl5ibrcom
baib 3syl ralrimiv undif1 ineq2i indi eqtr3i raleqi inss2 3bitrd eldifsni
ralunb bitr3d wne ifnefalse ralbiia syl6bb cres df-ima resmpt syl5eq wfun
- rneqd ffund difss sstri fdmd syl5sseqr funimass4 syl2anc 3bitr4d rexbidva
+ rneqd ffund difss sstri fdmd sseqtrrid funimass4 syl2anc 3bitr4d rexbidva
cdm pm5.74da bitrd ralbidva pm5.32da syl5bb ) FGEUBUCZNZFONZUVJPAUVKFCUJZ
NZEBUJZNZGUVNDEUDZUEZUFZUGUVLQZPZBHUHZUKZCHRZPUVJUVKUVIOFEGUIULUMAUVKUVJU
WCAUVKPZUVJMDUVPUNZMUJZEUOZFUWFGUPZUQZURZEHUWEVDUCZHUSUCUPNZEUWJUPZUVLNZE
@@ -335429,7 +335443,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
imass2 reximdv syl5com impr rexlimiva ralrimdva r19.29r ad3antrrr imaeq2d
syl6 rpxrd ineq1 sseq1d anbi12d rspcev expr syl2anc rexlimdva anim2d syl9
impd syl5 expd expdimp com23 impbid wb wfun wf ad2antrr ffund inss2 difss
- cdm fdmd syl5sseqr syl5ss funimass4 a1i simplrr sselda syl22anc cnmetdval
+ cdm fdmd sseqtrrid syl5ss funimass4 a1i simplrr sselda syl22anc cnmetdval
elbl3 breq1d bitrd simplrl simpllr eldifi ffvelrn ralbidva imbi1i ralbii2
impexp syl2an elin biancomi bitr2i eldifsn imbi2i 3bitr4i 3bitr3g anassrs
rexbidva pm5.32da ) AGHFUBMNGONZGUAUCZNZFLUCZNZHUUOEFUDZUEZUIZUFZUUMPZQZL
@@ -335717,7 +335731,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
wf wss cnfldtopon resttopon sylancr syl5eqel cnpf2 syl3anc wa cuni cnprcl
a1i eqid syl toponuni eleqtrrd ad2antrr wn elun elsni orim2i sylbi adantl
wo orcomd orcanai ffvelrnd ifclda fvco3 syl2anc ifeq2da syl6eqr mpteq2dva
- cofmpt fvif eqtr4d fssd cdm fdmd w3a limcrcl simp2d eqsstr3d simp3d mpbid
+ cofmpt fvif eqtr4d fssd cdm fdmd w3a limcrcl simp2d eqsstrrd simp3d mpbid
ellimc cnfldtop fmpttd snssd unssd feq2d toponunii cnprest2 oveq2i fveq1i
ctop wb syl6eleqr iftrue ssun2 snssg mpbiri fvmptd3 fveq2d cnpco eqeltrrd
fco mpbird ) ADGQZGFUAZCUBRSPBCUCZUDZPUEZCUFZXTYDYAQZUGZUHZCIYCUIRZIUJRQZ
@@ -335759,7 +335773,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
mp2an resttopon sylancr syl5eqel toponuni eleqtrrd opelxp simpld ad2antrr
sylib wn simpll simpr elun ord elsni syl6 con1d imp ifclda simprd opelxpd
wo eqidd wf cnpf2 syl3anc feqmptd fveq2 df-ov ovif12 eqtr3i syl6eq fmptco
- a1i cdm dmmptd w3a limcrcl simp2d eqsstr3d simp3d snssd unssd ssun2 snssg
+ a1i cdm dmmptd w3a limcrcl simp2d eqsstrrd simp3d snssd unssd ssun2 snssg
mpbiri adantr sseldd limcmpt mpbid txcnp topontopi fmpttd feq2d toponunii
wb ctop cnprest2 oveq2i fveq1i syl6eleqr iftrue opeq12d opex fvmpt fveq2d
cnpco eqeltrrd fovrnd mpbird ) AEFIUDZBCGHIUDZUEDUFUDUGBCDUHZUIZBUSZDUJZU
@@ -335801,7 +335815,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
limcco $p |- ( ph -> D e. ( ( x e. A |-> T ) limCC X ) ) $=
( co wcel cc csn cun cv wceq cif cmpt ccom climc ccnfld ctopn crest wa wo
cfv wn wne expr necon1bd wb limccl sseldi adantr elsn2g sylibrd orrd elun
- syl sylibr fmpttd cdm eqid dmmptd wf wss w3a limcrcl eqsstr3d snssd unssd
+ syl sylibr fmpttd cdm eqid dmmptd wf wss w3a limcrcl eqsstrrd snssd unssd
simp2d ccnp limcmpt mpbid limccnp iftrue ssun2 snssg mpbiri fvmptd3 eqidd
eqeq1 ifbieq2d fmptco ifid anassrs ifeq1da mpteq2dva eqtrd oveq1d 3eltr3d
syl5reqr ) AFCEFUAZUBZCUCZFUDZGIUEZUFZUNXGBDHUFZUGZKUHRGBDJUFZKUHRADKFXCX
@@ -336516,7 +336530,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( cc wcel co cc0 cdvn cfv cdm cn0 3ad2ant3 cuz syl22anc nn0cnd wss cvv wf
syl3an3 cr cpr cpm cfz w3a cmin caddc wceq simp1 elfznn0 elfzuz3 uznn0sub
simp2 dvnadd elfzuz2 nn0uz syl6eleqr pncan3d fveq2d eqtrd dmeqd cnex dvnf
- syl a1i dvnbss wa elpmi 3ad2ant2 simprd sstrd elpm2r syl3anc eqsstr3d ) A
+ syl a1i dvnbss wa elpmi 3ad2ant2 simprd sstrd elpm2r syl3anc eqsstrrd ) A
UAEUBZFZBEAUCGZFZCHDUDGFZUEZDABIGZJZKDCUFGZACWAJZIGJZKZWDKZVTWEWBVTWECWCU
GGZWAJZWBVTVPVRCLFZWCLFZWEWIUHVPVRVSUIZVPVRVSUMVSVPWJVRCDUJZMZVTDCNJFZWKV
SVPWOVRCHDUKMCDULVDZABCWCUNOVTWHDWAVTCDVTCWNPVTDVTDHNJZLVSVPDWQFVRCHDUOMU
@@ -336594,7 +336608,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( cc wcel cn0 cfv wss wa cv wi co wceq fveq2 sseq1d imbi2d cdm wf syl3anc
syl vn vm vf cr cpr cuz ccpn c1 caddc ssid 2a1i cpm cdvn ccncf simprl cdv
recnprss ad2antrr adantr simplll eluznn0 adantll dvnf dvnbss dvnp1 simprr
- cz eqeltrrd cncff fdmd wb cnex elpm2g sylancr mpbid simprd sstrd eqsstr3d
+ cz eqeltrrd cncff fdmd wb cnex elpm2g sylancr mpbid simprd sstrd eqsstrrd
cvv dvbss eqssd feq2d dvcn syl31anc peano2nn0 elcpn syl2anc 3imtr4d ssrdv
jca ex sstr2 expcom a2d uzind4 com12 3impia ) AUDDUEZEZBFEZCBUFGZEZCAUGGZ
GZBXCGZHZXBWSWTIZXFXGUAJZXCGZXEHZKXGXEXEHZKXGUBJZXCGZXEHZKXGXLUHUILZXCGZX
@@ -336916,7 +336930,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( ( S X. { A } ) oF x. ( S _D F ) ) ) $=
( vx cmul co cdv cc wf syl cc0 cfv wceq cmpt csn cxp cof caddc wcel snssd
fconstg fssd c0ex fconst ccnfld ctopn crest cnt wss cr cpr recnprss ssidd
- cres cdm dvbsss a1i eqsstr3d eqid dvres resmptd fconstmpt reseq1i 3eqtr4g
+ cres cdm dvbsss a1i eqsstrrd eqid dvres resmptd fconstmpt reseq1i 3eqtr4g
syl22anc oveq2d dvconst dmeqd fdmi syl6eq sseqtr4d dvres3 xpssres reseq1d
eqtrd 3eqtr3d ctop cnfldtopon resttopon sylancr topontop toponuni sseqtrd
ctopon ntrss2 syl2anc dvbssntr eqssd reseq12d 3eqtr4d feq1d mpbiri dvmulf
@@ -337378,7 +337392,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
wf cr dmeqd wral ralrimiva dmmptg eqtrd dvbsss syl6eqssr sstrd syl22anc
dvres resmptd oveq2d reseq1d reseq2d ctop cuni crest cnfldtopon sylancr
ctopon resttopon syl5eqel topontop toponuni sseqtrd eqid ntrss2 syl2anc
- eqsstr3d 3eqtrd 3eqtr3d ) AEBICUAZKUBZUCUDZEWHUCUDZKFUEUFUFZUBZEBKCUAZU
+ eqsstrrd 3eqtrd 3eqtr3d ) AEBICUAZKUBZUCUDZEWHUCUDZKFUEUFUFZUBZEBKCUAZU
CUDBJDUAZAETUGZITWHUOIEUGKEUGWJWMUHAEUPTUIUJWPLEUKULZABICTMUMAIWKUNZEAW
RBIDUAZUNZIAWKWSOUQADHUJZBIURWTIUHAXABINUSBIDHUTULVAEWHVBVCZAKIEPXBVDZI
KEFWHGRQVFVEAWIWNEUCABIKCPVGVHAWMWSWLUBWSJUBWOAWKWSWLOVIAWLJWSSVJABIJDA
@@ -337413,7 +337427,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( cmul co cmpt cc0 cc wcel cfv cdv caddc cv adantr wa 0cnd ccnfld ctopn
crest dvmptc cdm dmeqd wral ralrimiva dmmptg syl eqtrd dvbsss syl6eqssr
wceq eqid cnt ctop cuni wss ctopon cnfldtopon cr cpr recnprss resttopon
- sylancr topontop toponuni sseqtrd ntrss2 fmpttd dvbssntr eqsstr3d eqssd
+ sylancr topontop toponuni sseqtrd ntrss2 fmpttd dvbssntr eqsstrrd eqssd
syl2anc dvmptres2 dvmptmul mul02d oveq1d dvmptcl mulcld addid2d mulcomd
3eqtrd mpteq2dva ) AFBHECNOPUAOBHQCNOZDENOZUBOZPBHEDNOZPABEQCDFRGHIAERS
ZBUCZHSZMUDZAWRUEZUFABEQFUGUHTZFUIOZXARFHHIAWPWQFSZMUDAXCUEUFABEFIMUJAH
@@ -337522,7 +337536,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( S _D ( x e. X |-> A ) ) = ( S _D ( x e. Y |-> A ) ) ) $=
( cdv co cfv wss wceq cc cmpt cres ctop wcel cuni ctopon crest cnfldtopon
cnt resttopon sylancr syl5eqel topontop syl toponuni sseqtrd eqid syl2anc
- ntridm fveq2d eqtr3d reseq2d fmpttd dvres syl22anc eqsstr3d sstrd 3eqtr4d
+ ntridm fveq2d eqtr3d reseq2d fmpttd dvres syl22anc eqsstrrd sstrd 3eqtr4d
wf ntrss2 ssid resmpt mp1i oveq2d resmptd ) ADBGCUAZHUBZOPZDVPOPZDBHCUAZO
PADVPGUBZOPZVRVSAVSGEUIQZQZUBZVSHWCQZUBZWBVRAWDWFVSAWDWCQZWDWFAEUCUDZGEUE
ZRZWHWDSAEDUFQZUDZWIAEFDUGPZWLLAFTUFQUDDTRZWNWLUDFMUHIDFTUJUKULZDEUMUNZAG
@@ -339702,7 +339716,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( ( F ` z ) / ( G ` z ) ) ) limCC B ) ) $=
( vr cv cmin cr cres cfv co wss crp cdiv cmpt climc wcel cioo crn syl3anc
eqid a1i wa wceq syl6sseqr syl sseldd cdif cin adantr rexrd wf cun uneq1i
- cxr clt wbr syl5eq c0 wb syl2anc wn eqtri sylancl mpbid syl5sseqr cdv cdm
+ cxr clt wbr syl5eq c0 wb syl2anc wn eqtri sylancl mpbid sseqtrrid cdv cdm
cc ax-resscn fss sstrd fssresd ioossre dvres syl22anc ctop retop iooretop
cnt isopn3i mp2an reseq2i syl6eq dmeqd ssdmres sylib limcresi sseldi cima
eqtrd df-ima imass2 syl5eqssr ssneldd rneqd syl6eqr eqsstrd resmptd fvres
@@ -339863,7 +339877,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
dvcnvrelem2 $p |- ( ph -> ( ( F ` C ) e. ( ( int ` T ) ` Y ) /\
`' F e. ( ( N CnP M ) ` ( F ` C ) ) ) ) $=
( cfv cnt wcel ccnv ccnp co cmin caddc cicc cima ctop cr wss cioo retop
- crn ctg eqeltri wf1o wfo wceq f1ofo forn 3syl ccncf wf eqsstr3d imassrn
+ crn ctg eqeltri wf1o wfo wceq f1ofo forn 3syl ccncf wf eqsstrrd imassrn
cncff frn syl5sseq cuni uniretop unieqi eqtr4i ntrss dvcnvrelem1 fveq2i
mp3an2i fveq1i syl6eleqr sseldd cres crest wfun f1of ffun funcnvres ccn
f1ocnv cdv cdm dvbsss syl6eqssr ax-resscn syl6ss cncfss syl2anc wf1 syl
@@ -339918,7 +339932,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
( x e. Y |-> ( 1 / ( ( RR _D F ) ` ( `' F ` x ) ) ) ) ) $=
( cr cfv eqid cc wcel wceq wss co sylancr wa wb vr vy ccnfld cioo crn ctg
ctopn tgioo2 cpr reelprrecn a1i cnt ctop retop wf1o f1ofo forn 3syl ccncf
- wfo wf cncff frn eqsstr3d uniretop ntrss2 ccnv f1ocnvfv2 sylan crest ccnp
+ wfo wf cncff frn eqsstrrd uniretop ntrss2 ccnv f1ocnvfv2 sylan crest ccnp
cv cabs cmin ccom cxp cres cbl crp cxmet wrex rexmet cdv dvbsss ax-resscn
cdm syl sylancl dvbssntr eqssd isopn3 mpbird f1ocnv ffvelrnda cmopn tgioo
fss f1of mopni2 mp3an2ani c2 cdiv ad2antrr cc0 wn rphalfcl ad2antrl caddc
@@ -341628,7 +341642,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
simpll wf ffvelrnda nn0cnd cfsupp wbr psrbagfsupp ancoms eqbrtrrd disjdif
cin c0 incom eqtri a1i difsnid eqcomd ad2antrl gsumsplit2 3eqtrd ad2antrr
cun cn difexg csubmnd nn0subm eldifi ffvelrn syl2an fmpttd wfun csupp wss
- mptexd funmpt cres difss resmpt ax-mp resss eqsstr3i funsssuppss mp3an12i
+ mptexd funmpt cres difss resmpt ax-mp resss eqsstrri funsssuppss mp3an12i
mptexg fsuppsssupp syl22anc gsumsubmcl cmnd ringmnd simprl ffvelrnd fveq2
gsumsn syl3anc wo simprr sylib elnn0 orel2 sylc eqeltrd nn0nnaddcl nnne0d
syl2anc eqnetrd expr syl5bir rexlimdva necon4bd wfn wb ffnd 0nn0 fnconstg
@@ -344686,7 +344700,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
adddird eqtrd sumeq2dv wss cuz cz nn0zd ifcld cr nn0red syl2anc syl3anbrc
max1 eluz2 fzss2 syl cdif csn c1 cima wn eldifn adantl cun wo eldifi cmin
nn0uz peano2nn0 syl6eleq uzsplit syl5eq nn0cnd ax-1cn pncan oveq2d uneq1d
- sylancl ad2antrr eleqtrd elun sylib ord mpd wi wfun ffund ssun2 syl5sseqr
+ sylancl ad2antrr eleqtrd elun sylib ord mpd wi wfun ffund ssun2 sseqtrrid
cdm fdmd sseqtr4d funfvima2 elsni mul02d fsumss max2 oveq12d 3eqtr4d
mpteq2dva eqtr4d ) AGHUAUBZUCBUDUEIUFUCZFUGZCUHZBUGZUUEUIUCZUJUCZFUKZUEJU
FUCZUUEDUHZUUHUJUCZFUKZUAUCZULBUDUEIJUMUNZJIUOZUFUCZUUECDUUCUCUHZUUHUJUCZ
@@ -344731,7 +344745,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by
cuz wss nn0cnd ad2antrr adantl addsubd cz fznn0sub nn0uz syl6eleq eluzadd
nn0zd syl2anc eqeltrd addid2d fveq2d eleqtrd fzss2 adantlr cdif csn c1 wn
eldifn cun wo eldifi peano2nn0 uzsplit syl5eq ax-1cn pncan sylancl uneq1d
- cima eqtrd ad3antrrr elun sylib ord mpd wi wfun cdm ffund ssun2 syl5sseqr
+ cima eqtrd ad3antrrr elun sylib ord mpd wi wfun cdm ffund ssun2 sseqtrrid
fdmd sseqtr4d funfvima2 elsni oveq1d simplr syl2an mul02d fsumss sumeq2dv
fzfid eqtr4d mul01d fsum2mul addcomd fsumcl olcd 3eqtr3d simpll fsummulc1
sumz mul4d expaddd ad2antlr pncan3d eqtr3d 3eqtr4rd cbvsumv oveq2i syl6eq
@@ -347564,7 +347578,7 @@ of all kernels (preimages of ` { 0 } ` ) of all polynomials in
cuni wss ccrd cdm wfo omelon nn0ennn nnenom entri ensymi isnumi mp2an wfn
con0 cc0 wceq c0p wne cdgr cle ccoe cabs wral w3a cz cply crab wrex rabex
cnex fnmpti dffn4 mpbi fodomnum domentr wb fvelrnb ax-mp aannenlem1 eleq1
- mp2 syl5ibcom rexlimiv sylbi ssriv wi aasscn eqsstr3i iunfictbso mp3an12i
+ mp2 syl5ibcom rexlimiv sylbi ssriv wi aasscn eqsstrri iunfictbso mp3an12i
soss eqbrtrid cnso exlimiiv cvv ssexi cq nnssq qssaa sstri ssdomg sbth )
JKLMZKJLMZJKUAMJNLMZNKUAMXGOHPZUBZXIHXKJBUCZUEZNLABCDEFGUDZXLNLMZXLQUFXKX
MXJUBZXMNLMXLRLMZRNUAMXORUGUHSZRXLBUIZXQNURSNRUAMXRUJRNRKNUKULUMZUNNRUOUP
@@ -348705,7 +348719,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
mpteq2dv imbi2d weq fveq2 oveq1 cbvmptv syl6eq cuz ccnfld ctopn crest cnt
cdv wbr wa eqid fvmpt syl cn0 nnnn0d nn0uz syl6eleq eluzfz2b cdm eleqtrrd
ovex dvntaylp0 oveq2d cr cpr cpm wf cvv wss cnex a1i elpm2r syl22anc dvnf
- syl3anc ffvelrnd subidd 3eqtrd wfun wb funmpt sylancr mpbid cfzo eqsstr3d
+ syl3anc ffvelrnd subidd 3eqtrd wfun wb funmpt sylancr mpbid cfzo eqsstrrd
dmmpti eqssd feq2d dvnp1 eqtr3d feqmptd sselda ctayl syl2anc eqtrd taylpf
ffvelrnda dmeqd dvntaylp feq1d mpbird syldan dvn0 syl6eqr ctopon subcld
syl6eleqr funbrfvb nnm1nn0 dvnbss fzo0end elfzofz 3syl dvn2bss nncnd 1cnd
@@ -348805,7 +348819,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
( ( x - B ) ^ ( M + 1 ) ) ) ) limCC B ) ) $=
( co cr cc wcel vy vk cc0 cv c1 caddc cmin cdvn cmpt cexp cdiv climc wa
cfv cdm wss cn0 cfz syl sseldi fznn0sub2 elfznn0 syl3anc cpm reelprrecn
- wf dvnfre a1i cvv cnex ax-resscn sylancl dvnbss fssdmd dvn2bss eqsstr3d
+ wf dvnfre a1i cvv cnex ax-resscn sylancl dvnbss fssdmd dvn2bss eqsstrrd
fss eqssd feq2d mpbid ffvelrnda sselda cres adantl ccnfld nnnn0d sseldd
wceq simpr adantr ffvelrnd dvnply2 eqeltrrd syldan resubcld fmpttd 1nn0
cn nn0addcld reexpcld crn retop uniretop sylancr cdv nncnd fveq2d dvnp1
@@ -348817,7 +348831,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
cdgr cfa faccld nndivred taylply2 simpld plyreres cuz elfzouz syl6eleqr
nnuz cioo ctg cnt ctop ntrss2 nppcan2d fzonnsub elfzofz tgioo2 dvbssntr
ctopn isopn3 mpbird difss dvnf fvexd recn cnfldtopon toponmax cin df-ss
- mp1i sylib ssid cmap mapsspm elmap sylibr dmmpti syl5sseqr addcld prid2
+ mp1i sylib ssid cmap mapsspm elmap sylibr dmmpti sseqtrrid addcld prid2
wb mulcld elfznn ovexd dvmptid 0cnd dvmptc 1m0e1 mpteq2i dvexp mpteq2dv
pncand dvmptco mulid1d dvntaylp0 subidd ctx ccn subcn dvcn plycn syl6ss
syl31anc cncfmptid cncfmpt1f cncfmpt2f 0expd wrex ssdifssd wne eldifsni
@@ -349645,7 +349659,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
( wcel wa cfv vy vv vu vr vx vn vm vj vs vw cv cdv wbr ccnfld ctopn crest
co cnt csn cdif cmin cdiv cmpt climc wss wceq biidd ulmdvlem2 cc recnprss
cdm cr cpr adantr cmap wf ffvelrnda elmapi dvbsss syl6eqssr eqid dvbssntr
- syl eqsstr3d ralrimiva cuz cz uzid syl6eleqr rspcdva sselda cabs clt wral
+ syl eqsstrrd ralrimiva cuz cz uzid syl6eleqr rspcdva sselda cabs clt wral
wi crp wrex sylancr cvv ovex syl2anc mpbid uztrn2 weq fveq2 oveq2d fveq1d
c2 oveq12d breq1d ralbidv ad2antrr rphalfcl adantl simplr oveq1d fvoveq1d
fvmpt imbi2d simpllr wb ad3antrrr eldv simprd dvlem fmpttd sseldd ellimc3
@@ -349766,7 +349780,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
wb sylancr eqeltrd ralrimiva ffnfv sylanbrc ad2antrr uztrn2 simprl subcld
ffvelrnd abscld fzfid ssun2 simprr elfzuzb adantlr syldan cr recnd fvmpt2
cun mpan2 sylan9eq oveq12d fsumser fsumcl fsumsplit mvrladdd fveq2d eqidd
- letrd ralimdva fzsplit syl5sseqr sselda fsumrecl resubcld eqeq12d rspcdva
+ letrd ralimdva fzsplit sseqtrrid sselda fsumrecl resubcld eqeq12d rspcdva
serfre sylan2 cin c0 eluzelre ltp1d fzdisj 3eqtr2d fsumabs eqbrtrd simpll
ad4ant14 anass1rs fsumle eqtr3d eqeltrrd cc0 absge0d fsumge0 absidd eqtrd
0red breqtrrd simpllr rpred lelttr syl3anc ralrimdva anassrs reximdva mpd
@@ -350419,7 +350433,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the
cr clt abscld absge0d crp simp2d cxr w3a wb 0re simp1d elico2 sylancr
rpxrd mpbir3and wf wfn ffn elpreima mp2b sylanbrc wceq eqid cnbl0 syl
eleqtrd icossicc imass2 mp1i cpnf iccssxr adantr sseldi simp3d df-ico
- eqsstr3d radcnvcl df-icc xrlelttr ixxss2 syl2anc syl6sseqr 3jca ) AMU
+ eqsstrrd radcnvcl df-icc xrlelttr ixxss2 syl2anc syl6sseqr 3jca ) AMU
EZFTZUFZXMUGKUHUIUJZUKULUMZTXQUHUNZUGKUOUMZUPZUQXTFUQXOXMXRUGKURUMZUP
ZXQXOXMUSTZXMUHULZYATZXMYBTZAFUSXMFUSUQAFXRUGEURUMZUPZUSRYHUHUTUSUHYG
VAUSVJUHVBVCVDVEVFVGZXOYEYDVJTZUGYDVHVIZYDKVKVIZXOXMYIVLXOXMYIVMXOKVN
@@ -352961,7 +352975,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is
wa wb fniniseg mp2b sylanbrc syl6eleqr fmpti wss frn cpi cres df-ima cmpt
c2 reseq1i clt wbr cle cxr w3a 0xr 2re pire remulcli elioc2 mp2an simp1bi
ssriv resmpt rneqi wf1o 0re eqid caddc recni addid2i oveq2i eqcomi efif1o
- f1ofo forn imassrn eqsstr3i eqssi df-fo mpbir2an ) FBCGCFHZCIZBJFBCUAZXDA
+ f1ofo forn imassrn eqsstrri eqssi df-fo mpbir2an ) FBCGCFHZCIZBJFBCUAZXDA
FBKAUBZUDLZUCUEZCDXGFMZXINUFOUGUHZBXJXIPMZXINUEOJZXIXKMZXJXHPMZXLXJKPMXGP
MXOUIXGUJKXGUKULXHUMUNXGUOPFNUANPHXNXLXMURUSUPPFNUQPOXINUTVAVBEVCVDZFBCUQ
QXEBXFXEBVEXPFBCVFQBCRVKVGUDLZSLZUHZXEXSCXRVHZIZBCXRVIYAAXRXIVJZIZBXTYBXT
@@ -361565,7 +361579,7 @@ already know is total (except at ` 0 ` ). There are branch points at
( cpnf wcel wa cc wss ad2antrr vy vz vr vk vw cmpt crli wbr ccnp co wf cv
cfv cima wrex wi wral fmpttd adantr wceq eqid csn ssun2 pnfex snid sselii
cun syl5eleqr eleq1d ralrimiva rspcdva fvmptd3 cabs cmin cbl cxmet cnxmet
- ccom crp cnfldtopn mopni2 mp3an1 cle clt cr ssun1 syl5sseqr ssralv simprl
+ ccom crp cnfldtopn mopni2 mp3an1 cle clt cr ssun1 sseqtrrid ssralv simprl
sylc simplr rlimi cioc cin cordt crest ctop cvv letop cxr a1i syl mp3an2i
pnfxr rexrd syl3anc elind crab wb sylancl syl2anc syld ad3antrrr r19.21bi
sselda breq1d bitrd simplrl eleq1 sylibr mpbird eqtri inss2 sylancr eleq2
@@ -361812,7 +361826,7 @@ exponential function (see also ~ dfef2 ). (Contributed by Mario
wf wceq eqid fmpt sylibr simpr rprecred rpcxpcld rlimi rlimmptrcl adantlr
abscld absge0d rpred rpge0d cxplt2d subid1d fveq2d breq1d abscxp2 syl2anc
ad2antrr cmul rpcnd rpne0d recid2d oveq2d simplr cxpmuld 3eqtr3rd breq12d
- cxp1d 3bitr4d biimpd imim2d ralimdva reximdv cxpcld rlimss eqsstr3d rlim0
+ cxp1d 3bitr4d biimpd imim2d ralimdva reximdv cxpcld rlimss eqsstrrd rlim0
mpd wss mpbird ) AEBCDKLZUBMUCNJUGEUGZUDNZXFUEUFZUAUGZONZUHZEBPZJQUIZUAUJ
PAXNUAUJAXJUJRZUKZXHCMULLZUEUFZXJUMDUNLZKLZONZUHZEBPZJQUIXNXPJEBCMXTSACSR
ZEBPZXOABSEBCUBZVBZYEAYFUOZSYFVBZYGAYFMUCNZYIHMYFUPUQAYHBSYFACFRZEBPYHBVC
@@ -361839,7 +361853,7 @@ exponential function (see also ~ dfef2 ). (Contributed by Mario
o1bdd cmul simpr eqid fvmpt2 oveq1d cvv ovex sylancl eqtr4d nffvmpt1 nfcv
ce nfv nfov fveq2 eqeq12d cbvral sylib r19.21bi ad2ant2r fveq2d ffvelrnda
nfeq ad2antrr simprr 0re ifcl adantr abscld max2 letrd abscxpbnd eqbrtrrd
- expr imim2d ralimdva wss o1mptrcl cxpcld fmpttd o1dm eqsstr3d simprl max1
+ expr imim2d ralimdva wss o1mptrcl cxpcld fmpttd o1dm eqsstrrd simprl max1
sylancr recld recxpcld pire remulcl reefcld remulcld elo12r syl22anc syld
3expia rexlimdvva mpd ) AUAUCZKUCZLMZYCBCDUDZNZUENZUBUCZLMZUFZKCUGZUBOUJU
AOUJZBCDEPUHZUDZUIQZAYEUIQZCRYEUKZYLJAYEULZRYEUKZYQAYPYSJYEUMUNAYRCRYEADF
@@ -362913,7 +362927,7 @@ group sum in the additive group (i.e. the sum of the elements). This is
simprd fsumabs absdivd wceq nnrpd rprege0d absid syl oveq2d eqtrd breqtrd
sumeq2dv cin c0 cn0 rpdivcld flge0nn0 nn0red ltp1d fzdisj cuz cun nn0p1nn
syl6eleq rpred wb jca rpregt0d lediv2 syl211anc mpbid recnd div1d flword2
- nnuz syl3anc syl2anc fsumsplit ssun1 syl5sseqr sselda syldan ssun2 fznnfl
+ nnuz syl3anc syl2anc fsumsplit ssun1 sseqtrrid sselda syldan ssun2 fznnfl
fzsplit2 simplbda adantr lemuldiv2 lemuldivd bitr3d mpd ledivmul2d mpbird
wi ex cc fsumle fsumless letrd nnrecred crp wn cmin rpge0d sylancl eleq2d
divrecd eqeltrrd noel elin syl5bbr mtbiri imnan sylibr con2d baibd syl2an
@@ -366164,7 +366178,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100
cr eluzel2 2z ifcl a1i zred min2 eluz2 syl3anbrc ppisval2 syl2anc eluzelz
2re flid oveq2d sumeq1d clt ltp1d fzdisj inindir 0in 3eqtr3g min1 elfzuzb
cun id sylanbrc fzsplit indir syl6eq wss fzfid inss1 ssfi wa simpr elin2d
- cn prmnn nnrpd relogcld recnd fsumsplit oveq12d mp2an cc syl5sseqr sselda
+ cn prmnn nnrpd relogcld recnd fsumsplit oveq12d mp2an cc sseqtrrid sselda
fzfi ssun1 syldan fsumcl ssun2 pncan2d ) BAUADEZBUBDZAUBDZUCFAGUDHZAGUEZA
IFZJKZCUFZUGDZCLZAUHUIFZBIFZJKZXQCLZUIFZXRUCFYBXIXJYCXKXRUCXIXJUJBUKFJKZX
QCLZYCXIBUOEZXJYEMABULZBCUMNXIYEXMBIFZJKZXQCLYCXIYDYIXQCXIYDXMBUNDZIFZJKZ
@@ -368832,7 +368846,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100
crg ccrg cn0 dchrrcl nnnn0d zncrng crngring eqid unitsubm syl2anc cc0 csn
cn resmhm cdif crn wss wb cnring cnfldbas cnfld0 cndrng drngui ax-mp cima
df-ima cv wral wa wne cbs dchrf unitss sseli ffvelrn syl2an adantr adantl
- wf simpr dchrn0 mpbird eldifsn sylanbrc ralrimiva wfun cdm fdmd syl5sseqr
+ wf simpr dchrn0 mpbird eldifsn sylanbrc ralrimiva wfun cdm fdmd sseqtrrid
ffund funimass4 syl5eqssr resmhm2b sylancr mpbid cgrp wceq unitgrp ablgrp
cabl cnmgpabl ghmmhmb sylancl eleqtrrd ) AHCUBZEFUCUDZEFUEUDZAXOEUFUGQZUC
UDRZXOXPRZAHIUGQZXRUCUDZRCYAUHQRZXSABYBHBDGIJKLUIPUJAIUKRZYCAIULRZYDAGUMR
@@ -372762,7 +372776,7 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and
csu chash oveq2i cin wceq gausslemma2dlem0b nnred rehalfcld flcld syl cc0
c0 wo cun cn0 cle cr nnrpd rpge0d flge0nn0 syl2anc rphalflt cz nnzd mpbid
crp cuz syl6eleq syl6eq oveq1 0p1e1 oveq1d fzfid gausslemma2dlem0a adantr
- wb wa cn elfznn adantl sylancr remulcld oveq2d c1st cdvds wn cc syl5sseqr
+ wb wa cn elfznn adantl sylancr remulcld oveq2d c1st cdvds wn cc sseqtrrid
a1i sselda syldan fsumnn0cl expaddd cfn wss cxp sylancl ssrab2 hashcl csn
crab ssfi wrel bitri wne ad2antrr elfzle2 elfzelz flge 2re 2pos syl112anc
mpbird peano2rem eqbrtrid cprime eldifad prmz mtod syl3anc simpr nnmulcld
@@ -375782,7 +375796,7 @@ to the second component (see, for example, ~ 2sqreunnltb and
( vi crp wcel csb cr wi cpnf cico co cc0 cle wbr wral ralrimiva nfel1
nfcsb1v cv wceq csbeq1a eleq1d rspc syl5com wa cmpt cfl cfv caddc cuz
c1 csn cxp eqid wb nnred elicopnf syl simprbda flcld peano2zd cn nnuz
- cli 1zzd cdm nnrp ssriv dmmptd syl5sseqr rlimclim1 adantr cc 0red clt
+ cli 1zzd cdm nnrp ssriv dmmptd sseqtrrid rlimclim1 adantr cc 0red clt
nngt0d simplbda ltletrd elrpd sylc recnd ssid fvex climconst2 syl2anc
cz cn0 rpge0d flge0nn0 nn0p1nn eluznn sylan nnrpd ad2antrr weq fvmpts
eqeltrd fvconst2g 3expia ralrimivva nfcv nfv nfbr nfral breq2 anbi12d
@@ -375922,7 +375936,7 @@ to the second component (see, for example, ~ 2sqreunnltb and
breq2 imbi12d cseq cin fzodisj eluzp1p1 elfzuzb sylanbrc fzosplit
cun nnz cpnf cico simpld nnrp impel fsumsplit eluzelz fzval3 3syl
nnzd 3eqtr4d elfznn simpr nfov 2fveq3 oveq12d fsumser fzfid ssun2
- c0 fvmptf syl5sseqr sselda pncan2d fveq2d 2re nfel1 subcld impcom
+ c0 fvmptf sseqtrrid sselda pncan2d fveq2d 2re nfel1 subcld impcom
peano2nn syl2an mulcomd nnnn0 nn0uz elfzelz adantlr eqtr2d oveq2d
resubcld fzosump1 sumeq2dv elfzuz eluznn rspccva fsumparts eluzle
syl2an2r telfsumo fsumrecl eqeltrrd eluzelre wb nnred elicopnf ex
@@ -377391,7 +377405,7 @@ a multiplicative function (but not completely multiplicative).
( cv crp wcel wa c1 cfl cfv cfz co caddc c2 cexp cdiv csqrt cabs cseq
csu cmin cmul fzfid cc cun ssun2 cuz wceq cn cn0 cr cc0 cle wbr simpr
rprege0d flge0nn0 syl nn0p1nn adantr syl6eleq dchrisum0lem1a fzsplit2
- nnuz simprd syl2anc syl5sseqr sselda csn cdif ssrab3 sseldi ad3antrrr
+ nnuz simprd syl2anc sseqtrrid sselda csn cdif ssrab3 sseldi ad3antrrr
cz adantl dchrzrhcl elfznn nnrpd rpsqrtcld rpcnd rpne0d divcld syldan
eldifad abscld cmpt weq 2fveq3 fveq2 oveq12d ad2antrr rpregt0d simpld
wb rpred letrd ffvelrnd subcld cpnf rerpdivcld fsumser eqtr3d eqbrtrd
@@ -377465,7 +377479,7 @@ a multiplicative function (but not completely multiplicative).
nnred cn0 reflcl fllep1 eluzle lemul1d ledivmuld mpbird nnre nndivred
sylbid nnge1 pm3.2i lediv2 mp3an2i div1d breqtrd simpl nndivre syl2an
0lt1 sylbird 3bitr3d 3bitr4d ex pm5.21ndd cun syl6eleq dchrisum0lem1a
- ssun2 nnuz fzsplit2 syl5sseqr sselda csn cdif ssrab3 sseldi ad3antrrr
+ ssun2 nnuz fzsplit2 sseqtrrid sselda csn cdif ssrab3 sseldi ad3antrrr
eldifad elfzelz dchrzrhcl wne rpcnne0d anasss fsumcom2 cmin 2cn mulcl
subcld cpnf elrege0 sylib rerpdivcld adddird pncan3d mulassd divcan2d
cico 2cnd 3eqtr3d mulcld wss o1const cdm eqid divsqrsum rlimdmo1 mp1i
@@ -380703,7 +380717,7 @@ a multiplicative function (but not completely multiplicative).
c2 csu clo1 wa crp cr elioore adantl 1rp a1i 1red clt wbr eliooord simpld
ltled rpgecld pntrf ffvelrni abscld relogcld remulcld rplogcld rerpdivcld
syl recnd 2re fzfid adantr cn elfznn nnrpd rpdivcld fsumrecl resubcld cun
- ssun2 pntrlog2bndlem6a syl5sseqr sselda syldan rpne0d divdird subsubd cin
+ ssun2 pntrlog2bndlem6a sseqtrrid sselda syldan rpne0d divdird subsubd cin
subdid ssun1 c0 wceq reflcl ltp1d fzdisj fsumsplit mvrraddd oveq2d eqtr3d
oveq1d mpteq2dva pntrlog2bndlem5 wss ioossre readdcld cle rpge0d nnrecred
rpred 2rp nndivred cc absge0d elfzle2 cz mpbird mpbid lemul2ad ledivmul2d
@@ -381998,7 +382012,7 @@ a multiplicative function (but not completely multiplicative).
flge mpbird jca pntlemn fsumge0 eqbrtrd cfzo eqid pntlemi elfzoelz zred
nncnd cun ssun1 peano2zd 1re ltle mpd peano2uz leexp2ad lediv2d flword2
uzid eluzp1p1 flcld simprd elfzofz pntlemh sylan2 lttrd ltled syl3anbrc
- flwordi eluz2 fzsplit2 syl5sseqr sstrd adantlr ssun2 le2add 1cnd subcld
+ flwordi eluz2 fzsplit2 sseqtrrid sstrd adantlr ssun2 le2add 1cnd subcld
mpand adddid addcomd addsubd mulid1d 3eqtr3d cin c0 reflcl ltp1d fzdisj
zcnd fsumsplit sylibrd expcom a2d fzind2 mpcom fsumless letrd ) AJLVBVC
ZOLVDVEVCZVFVCZVGVDUUCZFVFVCZVHVCZUAVIVJZVDVEVCZVFVCZVFVCZVVOOLVFVCZVKV
@@ -403008,7 +403022,7 @@ other vertices (of the graph), or equivalently, if all other vertices
-> ( ( G NeighbVtx K ) = ( V \ { K } )
-> ( S NeighbVtx K ) = ( V \ { N , K } ) ) ) $=
( vn wcel wa csn cdif cnbgr co wceq cupgr cpr wss nbgrssovtx upgrres1lem2
- cvtx cfv eqid difpr eqcomi a1i difeq1d syl5eq syl5sseqr adantr w3a anim1i
+ cvtx cfv eqid difpr eqcomi a1i difeq1d syl5eq sseqtrrid adantr w3a anim1i
cv simpl df-3an sylibr wne dif32 eqtri eleq2i eldifsn bitri simplbi eleq2
wi syl5ibr adantl imp nbupgrres sylc eqelssd ex ) EUANGHNOZFHGPZQZNZOZEFR
SZHFPZQZTZAFRSZHGFUBQZTWBWFOZMWGWHWBWGWHUCWFWBAUFUGZWDQZWGWHAWJFWJUHUDWBW
@@ -411643,7 +411657,7 @@ is finite (in the graph). (Contributed by Alexander van der Vekens,
umgr2adedgwlk $p |- ( ph -> ( F ( Walks ` G ) P /\ ( # ` F ) = 2
/\ ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) ) ) $=
( cfv cwlks wbr chash c2 wceq cc0 c1 w3a cvtx wne wcel cumgr cpr sylanbrc
- wa 3anass umgr2adedgwlklem syl simprd simpld wss ssid syl5sseqr jca 2wlkd
+ wa 3anass umgr2adedgwlklem syl simprd simpld wss ssid sseqtrrid jca 2wlkd
eqid cs2 fveq2i s2len eqtri a1i s3fv0 s3fv1 s3fv2 3anim123i fveq1i eqeq2i
cs3 eqcom bitri 3anbi123i sylibr 3jca ) AGEHUATUBGUCTZUDUEZBUFETZUEZCUGET
ZUEZDUDETZUEZUHZABCDEGHIJKHUITZONABCUJCDUJUOZBWMUKZCWMUKZDWMUKZUHZAHULUKZ
@@ -411658,7 +411672,7 @@ is finite (in the graph). (Contributed by Alexander van der Vekens,
30-Jan-2021.) $)
umgr2adedgwlkon $p |- ( ph -> F ( A ( WalksOn ` G ) C ) P ) $=
( wcel cvtx cfv wne wa w3a cumgr cpr 3anass sylanbrc umgr2adedgwlklem syl
- simprd simpld wss ssid syl5sseqr jca eqid 2wlkond ) ABCDEGHIJKHUAUBZONABC
+ simprd simpld wss ssid sseqtrrid jca eqid 2wlkond ) ABCDEGHIJKHUAUBZONABC
UCCDUCUDZBUTTCUTTDUTTUEZAHUFTZBCUGZFTZCDUGZFTZUEZVAVBUDAVCVEVGUDVHPQVCVEV
GUHUIBCDFHLUJUKZULAVAVBVIUMAVDJIUBZUNVFKIUBZUNAVDVDVJVDUORUPAVFVFVKVFUOSU
PUQUTURMUS $.
@@ -411688,7 +411702,7 @@ between two (different) vertices. (Contributed by Alexander van der
1-Feb-2018.) (Revised by AV, 29-Jan-2021.) $)
umgr2adedgspth $p |- ( ph -> F ( SPaths ` G ) P ) $=
( cvtx cfv wne wa wcel w3a cumgr cpr 3anass sylanbrc umgr2adedgwlklem syl
- simprd simpld wss ssid syl5sseqr jca eqid wi fveq2 eqcoms eqeq1d eqtr2 ex
+ simprd simpld wss ssid sseqtrrid jca eqid wi fveq2 eqcoms eqeq1d eqtr2 ex
syl6bi com13 eqcom prcom eqeq2i bitri umgrpredgv anim12d preqr1g eqneqall
wceq sylc syl6ci syl5bi syld neqne pm2.61d1 2spthd ) ABCDEGHIJKHUAUBZONAB
CUCCDUCUDZBWDUEZCWDUEZDWDUEZUFZAHUGUEZBCUHZFUEZCDUHZFUEZUFZWEWIUDAWJWLWNU
@@ -417705,7 +417719,7 @@ edge remains odd if it was odd before (regarding the subgraphs induced
( # ` x ) <_ 2 } ) $=
( cvv cword wcel cconcat co wceq chash cfv c2 crab wf w3a cv cpw cc0 cfzo
cle wbr c0 csn cdif konigsbergssiedgwpr wss prprrab wi 2re eqlei2 ss2rabi
- wrdf a1i eqsstr3i fss mpan2 iswrdb sylibr 3syl ) BJKZLCVFLDBCMNOUABAUBZPQ
+ wrdf a1i eqsstrri fss mpan2 iswrdb sylibr 3syl ) BJKZLCVFLDBCMNOUABAUBZPQ
ZROZAFUCZSZKLUDBPQUENZVKBTZBVHRUFUGZAVJUHUIUJZSZKLZABCDEFGHIUKVKBURVMVLVP
BTZVQVMVKVPULVRVKVIAVOSVPAFUMVIVNAVOVIVNUNVGVOLRVHUOUPUSUQUTVLVKVPBVAVBVP
BVCVDVE $.
@@ -424250,7 +424264,7 @@ Below is the final Metamath proof (which reorders some steps).
( F |` B ) = { <. 2 , 6 >. } ) $=
( c2 c6 cop c3 c9 cpr wceq c1 cres csn cun syl6eq c0 2re elexi wcel gtneii
cr simpl df-pr reseq1d resundir wrel cdm wss relsnop dmsnopss snsspr2 simpr
- wa 6re syl5sseqr syl5ss relssres sylancr 1re 1lt3 2lt3 nelpri eleq2d mtbiri
+ wa 6re sseqtrrid syl5ss relssres sylancr 1re 1lt3 2lt3 nelpri eleq2d mtbiri
wn ressnop0 syl uneq12d un0 eqtrd ) BCDEZFGEZHZIZAJCHZIZULZBAKZVJLZAKZVKLZA
KZMZVRVPVQVRVTMZAKWBVPBWCAVPBVLWCVMVOUAVJVKUBNUCVRVTAUDNVPWBVROMVRVPVSVRWAO
VPVRUEVRUFZAUGVSVRICDCTPQDTUMQUHVPWDCLZACDUIVPVNWEAJCUJVMVOUKZUNUOVRAUPUQVP
@@ -428096,7 +428110,7 @@ which is an Abelian group (i.e. the vectors, with the operation of
sspg $p |- ( ( U e. NrmCVec /\ W e. H ) -> F = ( G |` ( Y X. Y ) ) ) $=
( vx vy wcel wa wceq wfn wss cfv eqid syl cnv cxp cres cv co wral w3a cba
wfun nvgf ffund funres adantr sspnv ffnd fnresdm cns cnmcv isssp simplbda
- wf simp1d ssres eqsstr3d 3jca oprssov sylan eqcomd ralrimivva jctil sspba
+ wf simp1d ssres eqsstrrd 3jca oprssov sylan eqcomd ralrimivva jctil sspba
wb xpss12 syl2anc fnssres eqfnov mpbird ) AUAMZEDMZNZBCFFUBZUCZOZWAWAOZKU
DZLUDZBUEZWEWFWBUEZOZLFUFKFUFZNZVTWJWDVTWIKLFFVTWEFMWFFMNZNWHWGVTWBUIZBWA
PZBWBQZUGWLWHWGOVTWMWNWOVRWMVSVRCUIWMVRAUHRZWPUBZWPCACWPWPSZHUJZUKWACULTU
@@ -428128,7 +428142,7 @@ which is an Abelian group (i.e. the vectors, with the operation of
ssps $p |- ( ( U e. NrmCVec /\ W e. H ) -> R = ( S |` ( CC X. Y ) ) ) $=
( vx vy wcel wa cc wceq wss cfv eqid syl cnv cxp cres cv co wral wfun wfn
w3a cba nvsf ffund funres adantr wf sspnv ffnd fnresdm cpv cnmcv simplbda
- isssp simp2d ssres eqsstr3d 3jca oprssov sylan eqcomd ralrimivva jctil wb
+ isssp simp2d ssres eqsstrrd 3jca oprssov sylan eqcomd ralrimivva jctil wb
ssid sspba xpss12 sylancr fnssres syl2anc eqfnov mpbird ) CUAMZEDMZNZABOF
UBZUCZPZWDWDPZKUDZLUDZAUEZWHWIWEUEZPZLFUFKOUFZNZWCWMWGWCWLKLOFWCWHOMWIFMN
ZNWKWJWCWEUGZAWDUHZAWEQZUIWOWKWJPWCWPWQWRWAWPWBWABUGWPWAOCUJRZUBZWSBBCWSW
@@ -428270,7 +428284,7 @@ which is an Abelian group (i.e. the vectors, with the operation of
sspn $p |- ( ( U e. NrmCVec /\ W e. H ) -> M = ( N |` Y ) ) $=
( vx cnv wcel wa cr syl cfv wfn wss eqid cres wf sspnv nvf ffnd cba sspba
adantr fnssres syl2anc cv wfun cdm wceq ffund funres ad2antrr fnresdm cpv
- cns w3a isssp simplbda simp3d ssres eqsstr3d eleq2d biimpar sylan funssfv
+ cns w3a isssp simplbda simp3d ssres eqsstrrd eleq2d biimpar sylan funssfv
fdmd syl3anc eqcomd eqfnfvd ) ALMZEBMZNZKFCDFUAZVQFOCVQELMZFOCUBABEJUCZEC
FGIUDZPUEZVQDAUFQZRZFWCSVRFRVOWDVPVOWCODADWCWCTZHUDZUEUHABEWCFWEGJUGWCFDU
IUJVQKUKZFMZNZWGVRQZWGCQZWIVRULZCVRSZWGCUMZMZWJWKUNVOWLVPWHVODULWLVOWCODW
@@ -442844,7 +442858,7 @@ to a member of the subspace (Definition of complete subspace in [Beran]
$( A subset of Hilbert space is included in its span. (Contributed by NM,
2-Jun-2004.) (New usage is discouraged.) $)
spanss2 $p |- ( A C_ ~H -> A C_ ( span ` A ) ) $=
- ( vx chba wss cv csh crab cint cspn cfv ssintub spanval syl5sseqr ) ACDAB
+ ( vx chba wss cv csh crab cint cspn cfv ssintub spanval sseqtrrid ) ACDAB
EDBFGHAAIJBAFKBALM $.
$( The union of a set of subspaces is smaller than its supremum.
@@ -445221,7 +445235,7 @@ equals the join of their closures (double orthocomplements).
pjoml4i $p |- ( A vH ( B i^i ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) ) =
( A vH B ) $=
( cort cfv chj cin wss inss1 choccli chjcli chincli chlej2i ax-mp chdmm1i
- co chub1i ineq1i incom eqtri oveq2i inss2 pjoml2i eqtr3i chlej1i eqsstr3i
+ co chub1i ineq1i incom eqtri oveq2i inss2 pjoml2i eqtr3i chlej1i eqsstrri
wceq chlubii mp2an eqssi ) ABAEFZBEFZGQZHZGQZABGQZUOBIUPUQIBUNJUOBABUNDUL
UMACKBDKLMZDCNOAUPIBUPIUQUPIAUOCURRBABHZUOGQZUPUSUSEFZBHZGQZUTBVBUOUSGVBU
NBHUOVAUNBABCDPSUNBTUAUBUSBIVCBUHABUCUSBABCDMZDUDOUEUSAIUTUPIABJUSAUOVDCU
@@ -446264,7 +446278,7 @@ Note that the (countable) Axiom of Choice is used for this proof via
vH ( ( ( C vH F ) i^i ( D vH G ) ) i^i
( ( ( C vH R ) i^i ( D vH S ) ) vH
( ( F vH R ) i^i ( G vH S ) ) ) ) ) ) ) ) ) $=
- ( chj co cin cph cort cfv wss osumi ax-mp ineq12i chshii 5oalem7 eqsstr3i
+ ( chj co cin cph cort cfv wss osumi ax-mp ineq12i chshii 5oalem7 eqsstrri
shscli shincli shsleji chsleji ss2in mp2an shjshcli shlej2i shlej1i sstri
wceq chjcli chincli shjcli sslin ) ABUAUBZCDUAUBZUCZGHUAUBZEFUAUBZUCZUCZB
ACACUDUBZBDUDUBZUCZAEUDUBZBFUDUBZUCZCEUDUBZDFUDUBZUCZUDUBZUCZAGUDUBZBHUDU
@@ -446391,7 +446405,7 @@ Note that the (countable) Axiom of Choice is used for this proof via
(New usage is discouraged.) $)
3oai $p |- ( ( B vH R ) i^i ( C vH S ) ) C_
( B vH ( R i^i ( S vH ( ( B vH C ) i^i ( R vH S ) ) ) ) ) $=
- ( chj co cin cph cort cfv cch choccli chjcli chincli 3oalem5 eqsstr3i
+ ( chj co cin cph cort cfv cch choccli chjcli chincli 3oalem5 eqsstrri
eqeltri 3oalem3 3oalem6 sstri ) BDKLCEKLMBDNLCENLMZBDEBCKLDEKLMKLMKLZABCD
EFGHIJUAUGBDEBCNLDENLMNLMNLUHBCDEGHDBOPZBAKLZMQIUIUJBGRBAGFSTUCECOPZCAKLZ
MQJUKULCHRCAHFSTUCUDABCDEFGHIJUEUFUB $.
@@ -451352,7 +451366,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240.
$( Every bounded linear Hilbert space operator has an adjoint. (Contributed
by NM, 19-Feb-2006.) (New usage is discouraged.) $)
bdopssadj $p |- BndLinOp C_ dom adjh $=
- ( cbo clo ccop cin cado cdm lncnbd cnlnssadj eqsstr3i ) ABCDEFGHI $.
+ ( cbo clo ccop cin cado cdm lncnbd cnlnssadj eqsstrri ) ABCDEFGHI $.
$( Every bounded linear Hilbert space operator has an adjoint. (Contributed
by NM, 22-Feb-2006.) (New usage is discouraged.) $)
@@ -454635,7 +454649,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) =
( ( C vH D ) i^i B ) = ( ( C i^i B ) vH ( D i^i B ) ) ) $=
( wa cin wss chj co bicomi chjcli chlubi w3a wceq cch wcel cmd cdmd simpr
ssin anbi12i simpl 3pm3.2i dmdsl3 mpan syl3an chincli chub1i chlej1i mp1i
- wbr eqsstr3d chub2i sylib ssrind ssrin syl6ss adantr inss2 mpbir2an mdsl3
+ wbr eqsstrrd chub2i sylib ssrind ssrin syl6ss adantr inss2 mpbir2an mdsl3
jca a1i sseqtrd 3expb sylan2b lediri eqssd ) ABUAUOZBAUBUOZIZACDJKZCDLMZA
BLMZKZIZIZVQBJZCBJZDBJZLMZVTVOACKZADKZIZCVRKZDVRKZIZIWBWEKZVPWHVSWKWHVPAC
DUDNWKVSCDVRGHABEFOPNUEVOWHWKWLVOWHWKQZWBWEALMZBJZWEWMVQWNBWMCWNKZDWNKZIV
@@ -454656,7 +454670,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) =
( wa cin wss chj co w3a wceq simpr cch wcel chincli 3pm3.2i cmd cdmd ssin
wbr lejdiri bicomi chlubi anbi12i chub2i chlej1i chjcomi syl6sseq ssinss1
a1i ssini syl adantr chjcli dmdsl3 mpan syl3an inss1 ssrin ax-mp syl5sseq
- simpl mdsl3 inss2 ssind eqsstr3d 3expb sylan2b eqssd ) ABUAUDZBAUBUDZIZAB
+ simpl mdsl3 inss2 ssind eqsstrrd 3expb sylan2b eqssd ) ABUAUDZBAUBUDZIZAB
JZCDJZKZCDLMBKZIZIZVRALMZCALMZDALMZJZWCWFKWBCDAGHEUEUNWAVPVQCKZVQDKZIZCBK
ZDBKZIZIWFWCKZVSWIVTWLWIVSVQCDUCUFWLVTCDBGHFUGUFUHVPWIWLWMVPWIWLNZWFWFBJZ
ALMZWCVPVOWIAWFKZWLWFABLMZKZWPWFOZVNVOPWQWIAWDWEACEGUIADEHUIUOUNWJWSWKWJW
@@ -455132,7 +455146,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) =
its generating vector. (Contributed by NM, 24-Nov-2004.)
(New usage is discouraged.) $)
sh1dle $p |- ( ( A e. SH /\ B e. A ) -> ( _|_ ` ( _|_ ` { B } ) ) C_ A ) $=
- ( csh wcel wa csn cort cfv cspn chba wceq shel spansn syl spansnss eqsstr3d
+ ( csh wcel wa csn cort cfv cspn chba wceq shel spansn syl spansnss eqsstrrd
) ACDBADEZBFZGHGHZRIHZAQBJDTSKBALBMNABOP $.
$( A 1-dimensional subspace is less than or equal to any member of ` CH `
@@ -455318,7 +455332,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) =
( vy wss cat crab cuni cspn cfv wcel c0v wceq chba csh spanid 3syl adantr
cv mpan eleq1 wne wa sheli spansnsh spansna sylan spansnss sseq1 sylanbrc
csn elrab elssuni cch atssch chsssh sstri rabss2 uniss mp2b unimax shssii
- ax-mp eqsstri spanss eqsstr3d spansnid syl sseldd spancl sh0 a1i pm2.61ne
+ ax-mp eqsstri spanss eqsstrrd spansnid syl sseldd spancl sh0 a1i pm2.61ne
ssriv mp2an fveq2i eqtri sseqtri eqssi ) BASZBEZAFGZHZIJZDBWDDSZBKZWEWDKL
WDKZWELWELWDUAWFWELUBZUCZWEUKIJZWDWEWIWJWJIJZWDWFWKWJMZWHWFWENKZWJOKWLWEB
CUDZWEUEWJPQRWIWJWBKZWJWCEZWKWDEZWIWJFKZWJBEZWOWFWMWHWRWNWEUFUGWFWSWHBOKZ
@@ -456137,7 +456151,7 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130,
mdsymi $p |- ( A MH B <-> B MH A ) $=
( vx cmd wbr wb cort cfv c0h wne cdmd choccli cch wcel mddmd chba mp3an12
wa cv chj co eqid mdsymlem8 mp2an 3bitr4g wceq chssii fveq2 pjococi choc0
- wss 3eqtr3g syl5sseqr ssmd1 ssmd2 jca pm5.1 3syl pm2.61iine ) ABFGZBAFGZH
+ wss 3eqtr3g sseqtrrid ssmd1 ssmd2 jca pm5.1 3syl pm2.61iine ) ABFGZBAFGZH
ZBIJZAIJZKKVEKLVFKLTVFVEMGZVEVFMGZVBVCVEVFVEEUAUBUCZEBDNACNVIUDUEAOPZBOPZ
VBVGHCDABQUFVKVJVCVHHDCBAQUFUGVEKUHZABUMZVBVCTZVDVLRABACUIVLVEIJKIJZBRVEK
IUJBDUKULUNUOVMVBVCVJVKVMVBCDABUPSVJVKVMVCCDABUQSURVBVCUSZUTVFKUHZBAUMZVN
@@ -456180,7 +456194,7 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130,
eqeq1d syl6bb 3adant1 bitrd 3com23 syl3an 3expa eleq1 syl6bir mpbid simpr
imp jca exp31 reximdvai r19.42v syl6ib reximdva ancom bitri anass rexbii2
anbi1i syl6ibr chshii shincl sylancl ad2antrr shsel sylibrd expimpd ssrdv
- syl5bi adantl eqsstr3d chincl mpan2 chslej ad2antrl sstrd ralrimiv dmdbr2
+ syl5bi adantl eqsstrrd chincl mpan2 chslej ad2antrl sstrd ralrimiv dmdbr2
exp32 mp2an sylibr ) ABUAIZABUDIZJZBEKZLZXRXPMZXRAMZBUDIZLZNZEOUBZABUCUEZ
XQYDEOXQXROPZXSYCXQYGXSQZQZXTYABUAIZYBYIXTXRXOMZYJXQYKXTJYHXOXPXRUFUGYHYK
YJLXQYHFYKYJFKZYKPFEUHZYLXOPZQYHYLYJPZYLXRXOUIYHYMYNYOYNYLGKZHKZUJIZJZHBR
@@ -457668,7 +457682,7 @@ Class abstractions (a.k.a. class builders)
( wcel c2o cen wbr csn wceq cuni c1o ccrd cfv cfn 2onn adantl csuc syl6eq
vx wa sylib cdif cv wex com nnfi ax-mp enfi mpbiri diffi syl ensymd simpl
cardidd dif1card syl2anc cardennn mpan2 df-2o eqtr3d suc11reg breqtrd en1
- simpr unieqd vex unisn wss difssd eqsstr3d vsnid sylancl eqeltrd exlimddv
+ simpr unieqd vex unisn wss difssd eqsstrrd vsnid sylancl eqeltrd exlimddv
ssel2 ) BACZADEFZSZABGZUAZRUBZGZHZVSIZACRVQVSJEFWBRUCVQVSVSKLZJEVQWDVSVQV
SMVQAMCZVSMCVPWEVOVPWEDMCZDUDCZWFNDUEUFADUGUHOZAVRUIUJUMUKVQWDPZJPZHWDJHV
QAKLZWIWJVQWEVOWKWIHWHVOVPULABUNUOVPWKWJHVOVPWKDWJVPWGWKDHNADUPUQURQOUSWD
@@ -460327,7 +460341,7 @@ its graph has a given second element (that is, function value).
csdm wo wn csn cv crn wfun w3a wex brdom2 wi c1 chash cfv cfz co wf1o nfv
cfn isfinite2 isfinite4 adantr bren 3adant3 cdif cmpt cin f1of adantl cxp
sylib fconstmpt eqcomi wb simplr fconst2g mpbiri disjdif a1i fun syl21anc
- fz1ssnn undif mpbi feq2i 3adantl3 ssid wfo simpr forn syl5sseqr orcd ssun
+ fz1ssnn undif mpbi feq2i 3adantl3 ssid wfo simpr forn sseqtrrid orcd ssun
f1ofo rnun syl6sseqr dff1o3 simprbi cnvun reseq1i resundir dff1o4 fnresdm
eqtri wfn simpl3 cnveqi cnvxp incom disjsn biimpri syl5eqr syl5eq uneq12d
xpdisjres un0 syl6eq funeqd mpbird nnex difexg ax-mp mptex unex feq1 rneq
@@ -460646,7 +460660,7 @@ its graph has a given second element (that is, function value).
cin syl3anc sseqin2 3sstr4d simpl inundif fneq2i sylibr vex a1i fnsuppres
ffs2 inindif syl121anc mpbid uneq12d eqtrd ad2antrl fconst6g disjdif fun2
jca syl21anc feq12d biimpar cfv fveq1d fconstg ad3antlr syl112anc fvconst
- ffnd fvun2 3eqtrd suppss eqsstr3d reseq1d res0 eqtr4i reseq2i fresaunres1
+ ffnd fvun2 3eqtrd suppss eqsstrrd reseq1d res0 eqtr4i reseq2i fresaunres1
3eqtr4ri jca31 impbida syl5bb f1od ) AFLZBGLZCAUCZUDZIBLZMZDUAHBCUEUFZDUI
ZCNZUAUIZACUGZIUHZUJZUKZEOOKUUHHLZUUIOLUUFUUHCHULUMUUDUUJUUGLZUUNOLZUUEUU
DUUPMUUPUUMOLZUUQUUDUUPUNUUDUURUUPUUDUUKOLZUULOLUURUUAUUBUUSUUCACFUOUPIVA
@@ -463918,7 +463932,7 @@ real number multiplication operation (this has to be defined in the main
xrge00 $p |- 0 = ( 0g ` ( RR*s |`s ( 0 [,] +oo ) ) ) $=
( cxrs cxr cmnf cdif cress co cmnd wcel cc0 cpnf cicc wss cfv wceq ax-mp wn
wbr mpbi cin cvv csn c0g eqid xrs1mnd ccmn xrge0cmn cmnmnd cle wa mnflt0 wb
- clt mnfxr 0xr xrltnle mp2an intnan elxrge0 mtbir iccssxr eqsstr3i 0e0iccpnf
+ clt mnfxr 0xr xrltnle mp2an intnan elxrge0 mtbir iccssxr eqsstrri 0e0iccpnf
difsn ssdif cbs difss df-ss xrex difexg xrsbas ressbas eqtr3i ovex ressress
xrs10 dfss incom eqtr2i oveq2i submnd0 mp4an ) ABCUAZDZEFZGHAIJKFZEFZGHZWEW
CLZIWEHIWFUBMNWDWDUCZUDWFUEHWGUFWFUGOWEWEWBDZWCCWEHZPWJWENWKCBHZICUHQZUIWMW
@@ -465845,7 +465859,7 @@ Formula in property (b) of [Lang] p. 32. (Contributed by Thierry
-> ( W |`s A ) e. Archi ) $=
( vx vy vn wcel wa cfv co cv wbr cn wi wral cmnd wb eqid syl adantl wceq
ctos carchi csubmnd c0g cplt cmg cple wrex submrcl isarchi2 sylan2 biimpa
- cress cbs an32s wss submbas submss eqsstr3d ssralv ralimdv subm0 ad2antrr
+ cress cbs an32s wss submbas submss eqsstrrd ssralv ralimdv subm0 ad2antrr
syld cdif ressle difeq1d pltfval submmnd 3eqtr4d eqidd breq123d ad3antrrr
cid cn0 simplll nnnn0d simpllr eleqtrrd submmulg syl3anc rexbidva imbi12d
simpr ralbidva sylibd mpd resstos syl2anc adantlr mpbird ) BUAFZBUBFZGZAB
@@ -469789,7 +469803,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
crg clinds clvec eqeltrd sralvec lveclmod lbsss clspn islbs4 islinds5
syl21anc rspcdva mpand imp suppss ssfid suppssdm sselda eleq1w anbi2d
fssdm breq1d chvarv iunfi xpfi fveq1d cbvmptv syl6eq suppovss feqmptd
- subrg0 feq1d ressbas2 eqsstr3d iuneq12d xpeq12d 3sstr3d suppssfifsupp
+ subrg0 feq1d ressbas2 eqsstrrd iuneq12d xpeq12d 3sstr3d suppssfifsupp
eqtrd fssd syl32anc breqtrd drgextgsum cmulr csubg subrgsubg subgsubm
csubmnd lmodvscl gsumsubm ressmulr sstrd sseqtr4d ringass syl13anc wb
fmpttd cplusg breq2d rmfsupp2 gsummulc1 3eqtr4rd w3a offval22 ovmpt4g
@@ -469898,7 +469912,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
sseqtrd eqtr3d fveq2d sseldd cplusg 3eqtr3rd 3eqtr3d eleqtrd syl21anc
cvv ovexd w3a fvexd wf wb simpr eqidd oveq1d ad2antrr oveqd mpteq2dva
oveq2d drgextgsum wne crab cfn mptexd fmpttd ovex rgenw mpteqb eqeq1d
- wn ax-mp sylib 3ad2ant1 csn subsubrg simpld clbs lbsss eqsstr3d sstrd
+ wn ax-mp sylib 3ad2ant1 csn subsubrg simpld clbs lbsss eqsstrrd sstrd
csra clss srasubrg cmulr simprd simprl ressabs oveq1i 3eqtr4g 3eqtr4d
syl5eq eleqtrrd simprr subrgmcl eqeltrrd ralrimivva syl12anc sseqtr4d
islss4 biimpar lbslinds sseldi resssca sraaddg eqeltrd lmodvscl 3expb
@@ -470070,7 +470084,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
anasss ad5antr simpllr srasca syl5eq fveq2d simp-4r clspn ad7antr oveq12d
eqtr3d clmod cgsu lveclmod fveq2 cmap wrex cmpt ffvelrnd eleqtrrd cbvmptv
cfsupp oveq1d oveq2d anbi12d ad8antr exlimddv cnzr dimval subsubrg oveq1i
- ressabs 3eqtr4g eqeltrd crn cen wf1 crg drngring ressbas2 eqsstr3d ringcl
+ ressabs 3eqtr4g eqeltrd crn cen wf1 crg drngring ressbas2 eqsstrrd ringcl
cmulr sstrd oveq2 oveq1 cplusg clinds lbslinds sseldi ad3antrrr wn islbs4
0nellinds nelne2 drgext0g neeqtrrd ovexd ovmpt4g drgextvsca simprr simprl
oveqd ovmpod adantllr adantl3r 3eqtr3d linds2eq ex f1opr sylanbrc hasheni
@@ -470408,7 +470422,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
wel oveqd cdr cfield fldextfld1 ccrg isfld fldextfld2 eqeltrrd drgextgsum
simplbi adantlr cvv cmnd crg drngring 3syl ringmnd ad4antr vex a1i elmapi
ad3antrrr wf adantl vsnid syl5eleqr ffvelrnd srasca eqtrd fveq2d eleqtrrd
- sseldd lbsss eqsstr3d snss sylibr srabase ringcl syl3anc oveq12d ressmulr
+ sseldd lbsss eqsstrrd snss sylibr srabase ringcl syl3anc oveq12d ressmulr
gsumsnd eqtr4d 3eqtr3d cur cinvr cui syl2an2r eqeltrd wrex lbslsp r19.29a
wn exlimddv simp-5l simprbi crngcom 3eqtr2d simpld unitinvinv cdif simprd
invrvald sralmod0 clinds lbslinds sseldi eqneltrd nelne2 eldifsn sylanbrc
@@ -471655,7 +471669,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
symgfv cn nnuz syl6eleq eluzfz2 eqtr3d 3eqtr3d ex con3d iffalsed ifeqda
cuz imp cvv simp2 adantr ovexd oveqi mpoeq3ia 3eqtr4d cmat cmdat oveq2d
cbs ovmpt4g mpoeq3dva matecld matbas2d ringidcl marrepval cmin csn cdif
- csubma submaval fzdif2 mpoeq12 difssd eqsstr3d submabas mdetcl ringlidm
+ csubma submaval fzdif2 mpoeq12 difssd eqsstrrd submabas mdetcl ringlidm
wss eqeltrd cmulr smadiadetr fveq1i eqeq12i sylibr submat1n submatminr1
fveq1d eqtr4d 3eqtrd ) APORQVCVDZJDVCZEHVCFHVCVEVDUAVCZTDVCZIVDUVMOPRVF
VCVDZMVCZIVDAUVKOPRVGSVNVDZGVHVDZVCVDZDVCZUVLARCVIZPUVQVIZOUVQVIZUVKUVT
@@ -472012,7 +472026,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
txomap $p |- ( ph -> ( H " A ) e. ( L tX M ) ) $=
( vc va vb vz cima ctx co wcel cxp wss wrex wral cfv wceq simp-6l simpllr
cv wa syl2anc simplr wfn cop opex fnmpoi a1i ctopon toponss xpss12 simprl
- syl fnfvima syl3anc simp-4r wf ffn 3syl fimaproj imass2 ad2antll eqsstr3d
+ syl fnfvima syl3anc simp-4r wf ffn 3syl fimaproj imass2 ad2antll eqsstrrd
3eltr3d xpeq1 eleq2d sseq1d anbi12d xpeq2 rspc2ev syl112anc wb eltx mpbid
r19.21bi adantlr adantr r19.29vva wfun mpofun fvelima mpan adantl r19.29a
ralrimiva mpbird ) AHDUJZKLUKULUMZUFVBZUGVBZUHVBZUNZUMZXNXIUOZVCZUHLUPUGK
@@ -473221,7 +473235,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
( vz cle cordt cfv crest co wceq wtru cxr wcel ax-mp wss cv wa wbr cnvtsr
ccnv cxp cin crn cdm lern df-rn eqtri ctsr letsr a1i crab wb brcnvg simpr
adantlr simplr syl2anc anbi12d ancom syl6bb rabbidva sseldi iccval ancoms
- simpl eqsstr3d eqsstrd adantl ordtrest2 mptru tsrps ordtcnv oveq1i eqtr2i
+ simpl eqsstrrd eqsstrd adantl ordtrest2 mptru tsrps ordtcnv oveq1i eqtr2i
cicc cps ) GUBZCCUCUDHIZVSHIZCJKZGHIZCJKVTWBLMBAFCVSNNGUEVSUFUGGUHUIVSUJO
ZMGUJOZWDUKGUAPULCNQMDULBRZCOZARZCOZSZWFFRZVSTZWKWHVSTZSZFNUMZCQMWJWOWHWK
GTZWKWFGTZSZFNUMZCWJWNWRFNWJWKNOZSZWNWQWPSWRXAWLWQWMWPWGWTWLWQUNWIWFWKCNG
@@ -473461,7 +473475,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
simplr bitr3i syl5bb posrasymb rexbii rexcom rexnal 3bitr3i sselda nelne2
necon3bbid r19.41v syl3anc mpbird pm5.17 rexbidva ex inex1 eqeltri fveq2d
cxp ordttopon prsdm syl5eleq syl5eqel ctg snex cbs fvexi ssfii ordtprsval
- bastg sstri syl5eq syl5sseqr syl unssad simplrl simplrr oran 3expa nrexdv
+ bastg sstri syl5eq sseqtrrid syl unssad simplrl simplrr oran 3expa nrexdv
anbi12i anandi exbii n0f df-rex necon1bbii ineq1d 0in syl6eq eldif ssconb
nfin vex sylanb anass1rs mpbi2and nfun ianor equcom syl6bb syl5bbr 3expia
pm5.32d orbi12i elun andi 3bitr4ri eldifsn bicomi eqrd sseqtr4d nconnsubb
@@ -473945,7 +473959,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a
cicc cmpt wa simpl oveq2d simpr sseldi xmul01 syl eqtrd mpteq2dva 3eqtr4g
fconstmpt c0ex fconst2 sylibr cnconst syl22anc cres eqid oveq1 cbvmptv id
adantl xrmulc1cn cnrest sylancl resmpt ax-mp eqtr4i eqcomi oveq1i 3eltr3g
- letopuni crn ioorp ioossicc eqsstr3i ge0xmulcl syl2anc fmptd frnd cnrest2
+ letopuni crn ioorp ioossicc eqsstrri ge0xmulcl syl2anc fmptd frnd cnrest2
wb syl3anc mpbid oveq2i syl6eleqr eleq2s cico wo clt wbr 0xr pnfxr 0ltpnf
elicoelioo mp3an sylib mpjaodan ) ACJKZDEELMZNZCJOUAMZNZXQXSAXQEJOUQMZUBP
ZNZYDJYBNZYBJUCZDUEZXSYDXQEUFUDPZYBUGMZYCFYHQUBPNZYBQUHZYIYCNUIJOUJZYBYHQ
@@ -476982,7 +476996,7 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be
6-Jul-2017.) $)
esumdivc $p |- ( ph -> ( sum* k e. A B /e C ) = sum* k e. A ( B /e C ) ) $=
( cesum c1 cxdiv co cc0 cpnf wcel wceq syl3anc sseldi cxr cxmu cdiv rpred
- cico wne 1red rpne0d rexdiv cioo ioorp ioossico eqsstr3i rpreccld eqeltrd
+ cico wne 1red rpne0d rexdiv cioo ioorp ioossico eqsstrri rpreccld eqeltrd
cr crp esummulc1 cicc iccssxr wral ralrimiva esumcl syl2anc xdivrec cv wa
nfcv adantr esumeq2dv 3eqtr4d ) ABCEJZKDLMZUAMZBCVLUAMZEJVKDLMZBCDLMZEJAB
CVLEFGHAVLKDUBMZNOUDMZAKUOPDUOPZDNUEZVLVQQAUFADIUCZADIUGZKDUHRAUPVRVQUPNO
@@ -477854,7 +477868,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
( vx vo csiga wcel cuni wceq wa elex cpw wss cdif wral w3a elpwuni biimpa
cv ancom eqcom cfv crn fvssunirn sseli cvv com cdom wbr wi issiga 3imtr4i
3ad2antr1 syl6bi mpcom jca wex isrnsiga simprbi pweq sseq2d difeq1 eleq1d
- wb eleq1 ralbidv 3anbi12d anbi12d syl ibi exlimiv biimprd pwuni syl5sseqr
+ wb eleq1 ralbidv 3anbi12d anbi12d syl ibi exlimiv biimprd pwuni sseqtrrid
simprd jctild anim2d biimpar syl56 impcom impbii ) ABEUAZFZAEUBGZFZBAGZHZ
IWBWDWFWAWCAEBUCUDAUEFZWBWFAWAJWGWBABKZLZBAFZBCRZMZAFZCANZWKUFUGUHWKGAFUI
CAKNZOZIZWFCABUJZWIWNWJWFWOWJWIIWEBHZWIWJIWFWJWIWSABPQWIWJSBWETUKULUMUNUO
@@ -478137,7 +478151,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
$( A set is a subset of the sigma-algebra it generates. (Contributed by
Thierry Arnoux, 24-Jan-2017.) $)
sssigagen $p |- ( A e. V -> A C_ ( sigaGen ` A ) ) $=
- ( vs wcel cv wss cuni cfv crab cint csigagen ssintub sigagenval syl5sseqr
+ ( vs wcel cv wss cuni cfv crab cint csigagen ssintub sigagenval sseqtrrid
csiga ) ABDACEFCAGOHZIJAAKHCAPLABCMN $.
$}
@@ -479233,7 +479247,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
= sum* n e. N ( M ` ( A \ U_ k e. ( 1 ..^ n ) B ) ) ) $=
( cfv c1 wcel wa cn wss ciun cv cfzo co cdif cesum iundisjcnt fveq2d wral
cmeas com cdom wbr wdisj wceq crn cuni measbase syl adantr simpll fzossnn
- csiga simpr syl5sseqr cuz simplr eleqtrd elfzouz2 fzoss2 3syl sseqtr4d wo
+ csiga simpr sseqtrrid cuz simplr eleqtrd elfzouz2 fzoss2 3syl sseqtr4d wo
mpjaodan sselda wsb sbimi sban sbv clelsb3 anbi12i bitri csb sbsbc wb cvv
wsbc sbcel1g elv nfcv cbvcsb csbid eqtri eleq1i 3imtr3i syl2anc ralrimiva
3bitri sigaclfu2 difelsiga eqimss sseq1 mpbiri jaoi nnct ssct iundisj2cnt
@@ -480609,7 +480623,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
c1 2re a1i cc0 wne 2ne0 reexpclzd 2cnd expne0d redivcld 1red readdcld
simplr rexrd icossre syl2anc eqsstrd ex rexlimivv sylbi mprgbir sseli
elpwid xpss12 syl2an adantr ctx ctop cioo ctg eqeltri txtopi uniretop
- cfv retop unieqi eqtr4i txunii topopn dya2iocuni mp2b unissd eqsstr3d
+ cfv retop unieqi eqtr4i txunii topopn dya2iocuni mp2b unissd eqsstrrd
elpwi rexlimiva ax-mp eqssi ) DUBZUCZLLUDZXIXJMKNZXJMZKXHKXHXJUEXKXHO
XKCNZBNZUDZUFZBFUBZPCXQPXLCBXQXQXOXKDJXMXNCUHBUHUGUIXPXLCBXQXQXMXQOZX
NXQOZQZXPXLXTXPQXKXOXJXTXPUJXTXOXJMZXPXRXMLMXNLMYAXSXRXMLXQLUKZXMXQYB
@@ -481411,7 +481425,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
xrletri3 fveq2 nnex simpr sseldd ad2antrr cima cdif wdisj cnvimass 3syl
esumrnmpt2 c1 cin 3adant1r wfn cfn mpan2 ax-mp disjss1 syl6sseq sseqin2
sylib eqbrtrrd fssdm ffun difpreima fimacnv difeq1d uncom difun2 eqtr3i
- difeq1i eqsstri sspreima eqsstr3d fvimacnvi wf1o simpr3 fresf1o disjrdx
+ difeq1i eqsstri sspreima eqsstrrd fvimacnvi wf1o simpr3 fresf1o disjrdx
difss fvres disjeq2dv bitr3d mpbid biimpa syl21anc 3eqtr3rd ciun uniiun
disjss3 elpwiuncl syl5eqel cfz cbvesum syl6breq fz1ssnn fzfi fnfi resss
ffn fnssres rnfi rnss cbvdisj disjun0 sylbi sylc carsgclctunlem1 unissd
@@ -482322,7 +482336,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
wi cr crrh wf cioo ctg ctopn czlm cds cxp cres xpeq12i reseq12i eqtri
fveq2i eqid cdr crrext syl5eqel rrextdrg syl5eqelr cnrg rrextnrg cnlm
syl rrextnlm cchr wceq rrextchr syl5eqr ccusp rrextcusp cuss rrextust
- cmetu rrhf feq1i ffund rge0ssre fdmd syl5sseqr funfvima2 syl2anc sylc
+ cmetu rrhf feq1i ffund rge0ssre fdmd sseqtrrid funfvima2 syl2anc sylc
sylibr cuni cmeas csiga dmmeas csigagen fvexi a1i sgsiga sibfmbl frnd
mbfmf unieqi w3a cfn sylancl sstri eqeltrd unisg syl5eq tpsuni eqtr4d
mp1i sseqtrd ssdifd sselda eldifad simp2 eleq1 3anbi2d eleq1d imbi12d
@@ -488693,7 +488707,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
rpsqrtcn $p |- ( sqrt |` RR+ ) e. ( RR+ -cn-> RR+ ) $=
( vx csqrt crp cres ccncf co wcel wf cv cdm cc cr wceq sqrtf ax-mp wb mpbir
wss cc0 cpnf cfv wa wral rpssre ax-resscn sstri fdm sseqtr4i sseli rpsqrtcl
- rgen wfun ffun ffvresb cico cioo ioossico eqsstr3i resabs1 resqrtcn rescncf
+ rgen wfun ffun ffvresb cico cioo ioossico eqsstrri resabs1 resqrtcn rescncf
jca ioorp mp2 eqeltrri cncffvrn mp2an ) BCDZCCEFGZCCVHHZVJAIZBJZGZVKBUACGZU
BZACUCZVOACVKCGVMVNCVLVKCKVLCLKUDUEUFZKKBHZVLKMNKKBUGOUHUIVKUJVBUKBULZVJVPP
VRVSNKKBUMOACCBUNOQCKRVHCLEFZGVIVJPVQBSTUOFZDZCDZVHVTCWARZWCVHMCSTUPFWAVCST
@@ -493425,7 +493439,7 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a
$( First-order logic and set theory. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.) (New usage is discouraged.) $)
bnj1241 $p |- ( ( ph /\ ps ) -> C C_ B ) $=
- ( wa wceq eqcomd adantl wss adantr eqsstr3d ) ABHECDBCEIABECGJKACDLBFMN
+ ( wa wceq eqcomd adantl wss adantr eqsstrrd ) ABHECDBCEIABECGJKACDLBFMN
$.
$}
@@ -495586,7 +495600,7 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a
w-bnj15 wfn csuc wral w3a csn cdif cab c-bnj18 wss weu wne 1onn 1n0 ne0ii
eldifsn mpbir2an biid eqid bnj852 r19.2z sylancr euex bnj31 rexcom4 sylib
bnj1198 simp2 reximi sylbi df-rex 19.41v simprbi cdm bnj900 fveq2 ssiun2s
- abid ssiun2 bnj882 syl6sseqr sstrd eqsstr3d exlimiv ) ABUBCAHIZDJZWGEJZUC
+ abid ssiun2 bnj882 syl6sseqr sstrd eqsstrrd exlimiv ) ABUBCAHIZDJZWGEJZUC
ZKWGLZABCMZNZFJZUDZWHHWNWGLGWMWGLZABGJMONUAFPUEZUFZEPKUGUHZQZDUIZHZDRWKAB
CUJZUKZWFWSDXAWFWQDRZEWRQWSDRWFWQDULZXDEWRWFWRKUMXEEWRUEXEEWRQSWRSWRHSPHS
KUMUNUOSPKUQURUPWLWPGAWRBDFECWLUSZWPUSZWRUTZVAXEEWRVBVCWQDVDVEWQEDWRVFVGW
@@ -500943,7 +500957,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by
f1osn sneqd fzsplit cz 1p1e2 oveq1i uneq12i syl6req cin snssd incom
fzsn wn cle clt 1lt2 1re 2re ltnlei mpbi elfzle1 disjsn mpbir eqtri
mto uneqdifeq mpbid reseq2 f1oeq2 f1oeq3 3bitrd fzp1ss subfacp1lem4
- eqsstr3i fveq1d r19.21bi sylan2 eleq2i eldifsn subfacp1lem2b fvresi
+ eqsstrri fveq1d r19.21bi sylan2 eleq2i eldifsn subfacp1lem2b fvresi
sseli bitri adantl sylan2br adantlr expr pm2.61dne syl2an ralrimiva
ffvelrn cpr difexg fex prex unex coex resex fvres sylan9eq ralbidva
f1oun mpanr12 sylancr bitrd biimpa syldan wo eleqtrrd nelne2 simp2d
@@ -503135,7 +503149,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by
cvmscld $p |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) ->
A e. ( Clsd ` ( C |`t ( `' F " U ) ) ) ) $=
( vx wcel cuni wss wceq syl2anc cin c0 ccvm co cfv w3a ccnv cima csn cdif
- crest ccld ctop cvmtop1 3ad2ant1 cvmsuni 3ad2ant2 cvmsss eqsstr3d restuni
+ crest ccld ctop cvmtop1 3ad2ant1 cvmsuni 3ad2ant2 cvmsss eqsstrrd restuni
unissd eqid difeq1d cun unisng 3ad2ant3 uneq2d uniun undif1 simp3 ssequn2
snssd sylib syl5eq unieqd eqtrd syl5eqr eqtr3d difss unissi syl5sseq ciun
wb cv uniiun ineq2i incom iunin2 3eqtr4i wa wne eldifsn wn nesym cvmsdisj
@@ -503816,7 +503830,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by
syl6eleq fzsn ax-mp 1ex syl6eqr eluzfz1 1m1e0 oveq1i nnne0d div0d a1d
syl5eq elnnuz biimpi peano2fzr ex imim1d simplbi elfznn adantr fveq2i
cioo crn ctg ssun1 nnnn0d nn0ge0d nngt0d divge0 syl22anc ltp1d ltdiv1
- syl112anc ltled w3a elicc2 mpbir3and iccsplit syl5sseqr unieqi eqtr4i
+ syl112anc ltled w3a elicc2 mpbir3and iccsplit sseqtrrid unieqi eqtr4i
uniretop restcldi ssun2 eqeltri restuni cin simprbi cnf mpbird ax-1cn
retop pncan sylancl cop wfun cdm ffund ssiun2s peano2rem ltm1d ubicc2
fdmd eleqtrrd funssfv fveq2d fveq12d cvmliftlem9 3eqtr2d opeq2d sneqd
@@ -505886,7 +505900,7 @@ codes is an increasing chain (with respect to inclusion). (Contributed
( vb com wcel wa wss cfv wi cv wceq fveq2 sseq2d imbi2d va vx vu vv vy vi
vk vz csuc weq ssidd a1i pm2.27 adantl simpr c1st cgna cmap c2nd cin cdif
co wrex cgol cop csn cres wral crab wo copab ssun1 simpl simplll satfvsuc
- cun syl2an23an syl5sseqr adantr sstrd ex syld com23 findsg impcom ) AJKBJ
+ cun syl2an23an sseqtrrid adantr sstrd ex syld com23 findsg impcom ) AJKBJ
KZLZEFKZDGKZLZBAMZBCNZACNZMZOWGWKWJWNWGWKWJWNOZWJWLIPZCNZMZOWJWLWLMZOZWJW
LUAPZCNZMZOZWJWLXAUIZCNZMZOWOIUAABWPBQZWRWSWJXHWQWLWLWPBCRSTIUAUJZWRXCWJX
IWQXBWLWPXACRSTWPXEQZWRXGWJXJWQXFWLWPXECRSTWPAQZWRWNWJXKWQWMWLWPACRSTWTWF
@@ -506633,7 +506647,7 @@ codes is an increasing chain (with respect to inclusion). (Contributed
height ` N + 1 ` . (Contributed by AV, 20-Oct-2023.) $)
fmlasssuc $p |- ( N e. _om -> ( Fmla ` N ) C_ ( Fmla ` suc N ) ) $=
( vx vu vv vi com wcel cfmla cfv cv cgna co wceq wrex cgol cab csuc ssun1
- wo cun fmlasuc syl5sseqr ) AFGAHIZBJZCJZDJKLMDUCNUDUEEJOMEFNSCUCNBPZTUCAQ
+ wo cun fmlasuc sseqtrrid ) AFGAHIZBJZCJZDJKLMDUCNUDUEEJOMEFNSCUCNBPZTUCAQ
HIUCUFRBDCEAUAUB $.
$}
@@ -507340,7 +507354,7 @@ codes is an increasing chain (with respect to inclusion). (Contributed
-> ( ( M Sat E ) ` _om ) : ( Fmla ` _om ) --> ~P ( M ^m _om ) ) $=
( vx vy wcel wa com cfmla cfv co wf cv ciun wss wral wbr adantl wb cpw wo
cmap csat satff 3expa csdm cen w3o entric wpss nnsdomo pm3.22 anim2i eqid
- pssss satfsschain imp syl2an orcd ex sylbid weq ssid fveq2 syl5sseqr olcd
+ pssss satfsschain imp syl2an orcd ex sylbid weq ssid fveq2 sseqtrrid olcd
nneneq syl6bi ancoms impel 3jaod mpd expr ralrimiv jca ralrimiva fvex syl
fiun satom wceq fmla a1i feq12d mpbird ) BCGZADGZHZIJKZBIUCLUAZIBAUDLZKZM
EIENZJKZOZWKEIWNWLKZOZMZWIWOWKWQMZWQFNZWLKZPZXBWQPZUBZFIQZHZEIQWSWIXGEIWI
@@ -516343,7 +516357,7 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $=
ndmfv 3orrot incom word nodmord ordirr disjsn xpdisj1 eqtr3d resundir un0
uneq2d eqcomi 3eqtr4g wrel resdm sssucid resabs1 mp1i 3brtr4d elexi prid2
funrel noextend sucelon sltres soasym df-suc reseq2i resundi eqtri rabbii
- dmres necom inteqi necomd syl5eqss eqsstr3d df-ss biimpar ralrimiv funres
+ dmres necom inteqi necomd syl5eqss eqsstrrd df-ss biimpar ralrimiv funres
nesym eqfunfv mpbir2and wfn funfn 1oex prid1 neeq1d 3brtr3d brtp ecase23d
3bitri simprd neeq1 con4i fnressn opeq2d sneqd uneq12d eqnbrtrd jaodan
mpdan ) CBHZCAUSIJKABUBZLZBMUCZBUDHZDMHZUEZEUSZDIJZEBUBZUEZCDIJZDCUFZUGZU
@@ -516603,7 +516617,7 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $=
noetalem4 $p |- ( ( ( A C_ No /\ A e. _V ) /\ ( B C_ No /\ B e. _V ) ) ->
( bday ` Z ) C_ suc U. ( bday " ( A u. B ) ) ) $=
( csur wss wcel cbday cun wceq ax-mp eqtri con0 cvv wa cdm cima nosupno
- cuni csuc cfv bdayval syl nosupbday eqsstr3d adantr unss1 simpll simplr
+ cuni csuc cfv bdayval syl nosupbday eqsstrrd adantr unss1 simpll simplr
simprr noetalem1 syl3anc cdif c1o csn cxp dmeqi dmun wne 1oex snnz dmxp
c0 uneq2i undif2 syl6eq imaundi unieqi uniun suceq word crn imassrn wfo
bdayfo forn sseqtri ssorduni ordsucun mp2an a1i 3sstr4d ) ELMZEUANZUBZF
@@ -522229,7 +522243,7 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and
( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) ) ->
( A Line B ) = { x e. ( EE ` N ) | x Colinear <. A , B >. } ) $=
( cn wcel cee cfv wne w3a wa cline2 co cv cop ccolin wbr cab cin crab wss
- fvline wceq liness eqsstr3d df-ss sylib eqtr4d dfrab2 syl6eqr ) DEFBDGHZF
+ fvline wceq liness eqsstrrd df-ss sylib eqtr4d dfrab2 syl6eqr ) DEFBDGHZF
CUKFBCIJKZBCLMZANBCOPQZARZUKSZUNAUKTULUMUOUPABCDUBZULUOUKUAUPUOUCULUOUMUK
UQBCDUDUEUOUKUFUGUHUNAUKUIUJ $.
$}
@@ -523546,7 +523560,7 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and
` A ) ) = A <-> E. c e. ( Clsd ` J ) A = ( ( int ` J ) ` c ) ) ) $=
( ctop wcel wss wa ccl cfv wceq cv ccld wrex clscld syl2anc ntrss syl3anc
cnt eqcom biimpi fveq2 rspceeqv syl2an cldrcl ntrss2 clsss2 ntridm ntrss3
- ex cldss mpdan clsss3 sscls eqsstr3d eqssd adantl id syl5ibrcom rexlimdva
+ ex cldss mpdan clsss3 sscls eqsstrrd eqssd adantl id syl5ibrcom rexlimdva
2fveq3 eqeq12d impbid ) BFGZACHIZABJKZKZBTKZKZALZADMZVIKZLZDBNKZOZVFVKVPV
FVHVOGAVJLZVPVKABCEPVKVQVJAUAUBDVHVOVMVJAVLVHVIUCUDUEUKVFVNVKDVOVFVLVOGZI
VKVNVMVGKZVIKZVMLZVRWAVFVRVTVMVRVEVLCHZVSVLHZVTVMHVLBUFZVLBCEULZVRVMVLHZW
@@ -524476,7 +524490,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $=
filnetlem3 $p |- ( H = U. U. D /\
( F e. ( Fil ` X ) -> ( H C_ ( F X. X ) /\ D e. DirRel ) ) ) $=
( vv vw vz cfv wcel wss wa cv c1st wbr cvv vu cuni wceq cfil cxp wi cdm
- cdir crn cun cid cres dmresi filnetlem2 dmss ax-mp eqsstr3i ssun1 sstri
+ cdir crn cun cid cres dmresi filnetlem2 dmss ax-mp eqsstrri ssun1 sstri
simpli dmrnssfld simpri uniss mp2b unixpss unidm sseqtri eqssi csn ciun
wral filelss xpss2 syl ralrimiva iunxpconst syl6sseq syl5eqss wrel ccom
ss2iun ccnv a1i relopabi jctil wex cin c0 wne simpl adantr simprl xp1st
@@ -524528,7 +524542,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $=
dfss3 rexiunxp 3bitri fileln0 adantlr r19.9rzv rspcv adantl sstr2 com12
ssid mpii ralrimivw impbid1 bitr3d syl5bb 3bitrd pm5.32da filn0 wo snnz
jctil neanior sylnibr ralrimiva r19.2z rexnal sseq1i 3bitr3i necon3abii
- xpeq0 iunss sylibr cid dmresi filnetlem2 eqsstr3i dmxpid sseqtri tailfb
+ xpeq0 iunss sylibr cid dmresi filnetlem2 eqsstrri dmxpid sseqtri tailfb
eqssi elfm filfbas elfg eqrdv fgfil eqtr2d jca feq1 oveq2 fveq1d spcegv
3bitr4d sylc dmeq fveq2 rneqd fveq2d exbidv rspcev ) FHUDMZNZCUENZCUFZH
DUGZVBZFCUHMZUIZHUYHUJVCZMZOZPZDUKZIUGZUFZHUYHVBZFUYQUHMZUIZUYLMZOZPZDU
@@ -527942,9 +527956,9 @@ Utility lemmas or strengthenings of theorems in the main part (biconditional
bj-mpgs.min $e |- ph $.
bj-mpgs.maj $e |- ( ( ph /\ A. x ph ) -> ps ) $.
$( From a closed form theorem (the major premise) with an antecedent in the
- "strong necessity" modality form (in the language of modal logic),
- deduce the inference ` |- ph => |- ps ` . Strong necessity is stronger
- than necessity, and equivalent to it when ~ sp (modal T) is available.
+ "strong necessity" modality (in the language of modal logic), deduce the
+ inference ` |- ph => |- ps ` . Strong necessity is stronger than
+ necessity, and equivalent to it when ~ sp (modal T) is available.
Therefore, this theorem is stronger than ~ mpg when ~ sp is not
available. (Contributed by BJ, 1-Nov-2023.) $)
bj-mpgs $p |- ps $=
@@ -528378,12 +528392,6 @@ Utility lemmas or strengthenings of theorems in the main part (biconditional
) BCDZAEZBFZQAGZBHZSUABFZUAQAUAGGZBHSUBGABCIUCRUABQAUAJKLTBMNABCOP $.
$}
- $( See ~ sbft . This proof is from Tarski's FOL together with ~ sp (and its
- dual). (Contributed by BJ, 22-Dec-2020.) $)
- bj-ssbft $p |- ( F/ x ph -> ( [ t / x ] ph <-> ph ) ) $=
- ( wnf wsb wex wal spsbe wi df-nf biimpi sp syl56 19.8a stdpc4 impbid ) ABDZ
- ABCEZARABFZQABGZAABCHQSTIABJKZABLMASQTRABNUAABCOMP $.
-
${
$d y x $. $d y ph $.
$( A special case of ~ sbequ2 . (Contributed by BJ, 22-Dec-2020.) $)
@@ -531275,10 +531283,10 @@ FOL part ( ~ bj-ru0 ) and then two versions ( ~ bj-ru1 and ~ bj-ru ).
from which the theorem follows. QED
2. Suppose that S also contains (the FOL version of) modal logic KB and
- commutation of quantifiers ~ alcom and ~ excom (possibly weakend by a DV
+ commutation of quantifiers ~ alcom and ~ excom (possibly weakened by a DV
condition on the quantifying variables), and that S can be axiomatized
such that the only axioms with a DV condition involving a formula variable
- are among ~ ax-5 , ~ ax5e , ax5ea . If the scheme
+ are among ~ ax-5 , ~ ax5e , ~ ax5ea . If the scheme
(PHI_1 ` & ... & ` PHI_n ` => ` PHI_0, DV)
@@ -531320,9 +531328,9 @@ variables all in OC(PHI) ` \ { ph } ` , so ` ( F// x ` PHI,
` \ { ph } } ) ` by ~ a1i . For the induction step, PHI is either an
implication, a negation, a conjunction, a disjunction, a biconditional, a
universal or an existential quantification of formulas where ` x ` does
- not occur. We use respectively ~ bj-nnfim , ~ bj-nnfnt , nnfant, nnfort,
- nnfbit, ~ bj-nnfalt , ~ bj-nnfext . For instance, in the implication
- case, if we have by induction hypothesis
+ not occur. We use respectively ~ bj-nnfim , ~ bj-nnfnt , ~ bj-nnfant ,
+ ~ bj-nnfort , nnfbit, ~ bj-nnfalt , ~ bj-nnfext . For instance, in the
+ implication case, if we have by induction hypothesis
` ( ( A. x `_1 ` ... A. x `_m ` F// x ph -> F// x ` PHI),
` { { x , a } | a e. ` OC(PHI) ` \ { ph } } ) ` and ` ( ( A. y `_1
@@ -531442,16 +531450,18 @@ quantifiers appear (since they typically require ~ ax-10 to work with),
( wnnf wex wal wi wnf bj-nnfea df-nf sylibr ) ABCABDABEFABGABHABIJ $.
$( A variable is nonfree in a formula if and only if it is nonfree in its
- negation. The foward implication is intuitionistically valid.
+ negation. The foward implication is intuitionistically valid (and that
+ direction is sufficient for the purpose of recursively proving that some
+ formulas have a given variable not free in them, like ~ bj-nnfim ).
Intuitionistically, ` |- ( F// x -. ph <-> F// x -. -. ph ) ` . See
~ nfnt . (Contributed by BJ, 28-Jul-2023.) $)
bj-nnfnt $p |- ( F// x ph <-> F// x -. ph ) $=
( wex wi wal wa wn wnnf eximal alimex anbi12ci df-bj-nnf 3bitr4i ) ABCADZAA
BEDZFAGZBCPDZPPBEDZFABHPBHNROQAABIAABJKABLPBLM $.
- $( A variable is nonfree in a theorem. Note that the antecedent is in the
- form of what is called "strong necessity" in modal logic. (Contributed by
- BJ, 28-Jul-2023.) $)
+ $( A variable is nonfree in a theorem. The antecedent is in the "strong
+ necessity" modality of modal logic in order not to require ~ sp (modal T).
+ (Contributed by BJ, 28-Jul-2023.) $)
bj-nnftht $p |- ( ( ph /\ A. x ph ) -> F// x ph ) $=
( wal wa wex wi wnnf ax-1 anim12i df-bj-nnf sylibr ) AABCZDABEZAFZALFZDABGA
NLOAMHLAHIABJK $.
@@ -531493,20 +531503,72 @@ quantifiers appear (since they typically require ~ ax-10 to work with),
wi syl6 ) ACDBCDEZABOZCFZRORRCGZORCDSACGBCFOQRABCHABCIJQRACFBCGOTABCKABCLPR
CMN $.
+ $( Nonfreeness in both conjuncts implies nonfreeness in the conjunction.
+ (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof
+ using the definition of conjunction in terms of implication and negation,
+ so using ~ bj-nnfim , ~ bj-nnfnt and ~ bj-nnfbi , but we want a proof
+ valid in intuitionistic logic. (Proof modification is discouraged.) $)
+ bj-nnfant $p |- ( ( F// x ph /\ F// x ps ) -> F// x ( ph /\ ps ) ) $=
+ ( wnnf wa wex wi wal df-bj-nnf 19.40 prth syl5 id alanimi syl6 anim12i an4s
+ syl2anb sylibr ) ACDZBCDZEABEZCFZUBGZUBUBCHZGZEZUBCDTACFZAGZAACHZGZEBCFZBGZ
+ BBCHZGZEUGUAACIBCIUIUMUKUOUGUIUMEZUDUKUOEZUFUCUHULEUPUBABCJUHAULBKLUQUBUJUN
+ EUEAUJBUNKABUBCUBMNOPQRUBCIS $.
+
+ ${
+ bj-nnfand.1 $e |- ( ph -> F// x ps ) $.
+ bj-nnfand.2 $e |- ( ph -> F// x ch ) $.
+ $( Nonfreeness in both conjuncts implies nonfreeness in the conjunction.
+ Deduction form. Note: compared with the proof of ~ bj-nnfant , it has a
+ few more essential steps (this difference would be reduced by adding
+ deduction forms of ~ bj-nnfe and ~ bj-nnfa ) and has fewer total steps
+ (since there are fewer intermediate formulas to build) and is slightly
+ easier to understand. This statement is of intermediate complexity: for
+ simpler statements, closed-style proofs will generally be shorter than
+ deduction-style proofs, while for more complex statements, the opposite
+ will be true (and deduction-style proofs will generally be easier to
+ understand). (Contributed by BJ, 19-Nov-2023.)
+ (Proof modification is discouraged.) $)
+ bj-nnfand $p |- ( ph -> F// x ( ps /\ ch ) ) $=
+ ( wa wex wi wal wnnf 19.40 bj-nnfe syl anim12d syl5 bj-nnfa alanimi syl6
+ id df-bj-nnf sylanbrc ) ABCGZDHZUCIUCUCDJZIUCDKUDBDHZCDHZGAUCBCDLAUFBUGCA
+ BDKZUFBIEBDMNACDKZUGCIFCDMNOPAUCBDJZCDJZGUEABUJCUKAUHBUJIEBDQNAUICUKIFCDQ
+ NOBCUCDUCTRSUCDUAUB $.
+ $}
+
+ $( Nonfreeness in both disjuncts implies nonfreeness in the disjunction.
+ (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof
+ using the definition of disjunction in terms of implication and negation,
+ so using ~ bj-nnfim , ~ bj-nnfnt and ~ bj-nnfbi , but we want a proof
+ valid in intuitionistic logic. (Proof modification is discouraged.) $)
+ bj-nnfort $p |- ( ( F// x ph /\ F// x ps ) -> F// x ( ph \/ ps ) ) $=
+ ( wnnf wa wo wex wi wal df-bj-nnf 19.43 pm3.48 syl5bi 19.33 anim12i syl2anb
+ syl6 an4s sylibr ) ACDZBCDZEABFZCGZUBHZUBUBCIZHZEZUBCDTACGZAHZAACIZHZEBCGZB
+ HZBBCIZHZEUGUAACJBCJUIUMUKUOUGUIUMEZUDUKUOEZUFUCUHULFUPUBABCKUHAULBLMUQUBUJ
+ UNFUEAUJBUNLABCNQORPUBCJS $.
+
$( If two formulas are equivalent for all ` x ` , then nonfreeness of ` x `
in one of them is equivalent to nonfreeness in the other. Compare
~ nfbiit . From this and ~ bj-nnfim and ~ bj-nnfnt , one can prove
analogous nonfreeness conservation results for other propositional
- operators.
-
- Note: The antecedent is in the "strong necessity" modality of modal logic
- (see also ~ bj-nnftht ) in order not to require ~ sp (modal T).
+ operators. The antecedent is in the "strong necessity" modality of modal
+ logic (see also ~ bj-nnftht ) in order not to require ~ sp (modal T).
(Contributed by BJ, 27-Aug-2023.) $)
bj-nnfbi $p |- ( ( ( ph <-> ps ) /\ A. x ( ph <-> ps ) ) ->
( F// x ph <-> F// x ps ) ) $=
( wb wal wa wex wi wnnf bj-hbyfrbi bj-hbxfrbi anbi12d df-bj-nnf 3bitr4g ) A
BDZOCEFZACGAHZAACEHZFBCGBHZBBCEHZFACIBCIPQSRTABCJABCKLACMBCMN $.
+ ${
+ bj-nnfbii.1 $e |- ( ph <-> ps ) $.
+ $( If two formulas are equivalent for all ` x ` , then nonfreeness of ` x `
+ in one of them is equivalent to nonfreeness in the other. Compare
+ ~ nfbii . From this and ~ bj-nnfim and ~ bj-nnfnt , one can prove
+ analogous nonfreeness conservation results for other propositional
+ operators. (Contributed by BJ, 18-Nov-2023.) $)
+ bj-nnfbii $p |- ( F// x ph <-> F// x ps ) $=
+ ( wb wnnf bj-nnfbi bj-mpgs ) ABEACFBCFECDABCGH $.
+ $}
+
${ $d x ph $.
$( A non-occurring variable is nonfree in a formula. (Contributed by BJ,
28-Jul-2023.) $)
@@ -531612,6 +531674,12 @@ quantifiers appear (since they typically require ~ ax-10 to work with),
$.
$}
+ $( Version of ~ sbft using ` F// ` , proved from core axioms. (Contributed
+ by BJ, 19-Nov-2023.) $)
+ bj-sbft $p |- ( F// x ph -> ( [ t / x ] ph <-> ph ) ) $=
+ ( wnnf wsb wex spsbe bj-nnfe syl5 wal bj-nnfa stdpc4 syl6 impbid ) ABDZABCE
+ ZAPABFOAABCGABHIOAABJPABKABCLMN $.
+
$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
@@ -532266,6 +532334,15 @@ sethood hypotheses (compare ~ opth ). (Contributed by BJ, 6-Oct-2018.) $)
ZBCKZLCMDNABOPZCQZACQZLDERBUCUDFGSUFUGUECHTACITSABCDEUAUB $.
$}
+ $( Join of ~ elsng and ~ elsn2g . (Contributed by BJ, 18-Nov-2023.) $)
+ bj-elsn12g $p |- ( ( A e. V \/ B e. W ) -> ( A e. { B } <-> A = B ) ) $=
+ ( wcel csn wceq wb elsng elsn2g jaoi ) ACEABFEABGHBDEABCIABDJK $.
+
+ $( Biconditional version of ~ elsng . (Contributed by BJ, 18-Nov-2023.) $)
+ bj-elsnb $p |- ( A e. { B } <-> ( A e. _V /\ A = B ) ) $=
+ ( cvv wcel csn wceq wb wi wa elsng elelb mpbi ) ACDZABEZDZABFZGHOMPIGABCJPA
+ NKL $.
+
${
$d y f A $.
$( Remove hypothesis from ~ pwcfsdom . Illustration of how to remove a
@@ -535458,7 +535535,7 @@ Complex numbers (supplements)
( vx vy vz con0 wcel cfv wss wi wceq c0 cvv fveq2 crdg cv wa wral nfsbc1v
wsbc 0ex rzal sbceq1a mpbid vtoclef csuc wo vex elsuc csb cun ssun1 csbex
fvex unex nfcv cmpt nfmpt1 nfcxfr nfrdg nffv nfcsb1 nfun ax-mp id csbeq1a
- rdgeq1 uneq12d rdgsucmptf mpan2 syl5sseqr sstr2 syl5com imim2d imp sseq1d
+ rdgeq1 uneq12d rdgsucmptf mpan2 sseqtrrid sstr2 syl5com imim2d imp sseq1d
syl5ibrcom adantr jaod syl5bi ralimdv2 cab df-sbc sucex sseq2d raleqbi1dv
cbvabv elab2 bitri syl6ibr wlim ciun ssiun2 adantl rdglim2a mpan sseqtr4d
ex ralrimiva eleq2i abid sylibr a1d tfindes rsp syl eleq1 sseq12d imbi12d
@@ -536214,7 +536291,7 @@ Complex numbers (supplements)
syldan sylan2 adantlrr isfinite adantrr ad2antrr mpd eqeq2d sylibr df-rex
ex unieq a1i eximdv exlimdv sylan2b adantlr sseld wfn f1fn fniunfv cldopn
sseqtrd ancomd eqcomd eqimss ssun4 uniun ssun3 uncom undif1 eqtri ssequn2
- biimpi syl5eq eqsstr3d sylanr1 sylanr2 f1f frn topopn snssd uniss ancom1s
+ biimpi syl5eq eqsstrrd sylanr1 sylanr2 f1f frn topopn snssd uniss ancom1s
eqssd mpand impr wf1o f1f1orn f1oen3g sylancr enen1 endom snfi mpbi unctb
difopn sdomdom sylancl syl6bi impcom adantll elpw2g biimprd imbi2i ralbii
ad2ant2lr simprbi cbvrexv breq1 anbi12d pweq ineq1d rexeqdv rspccv mp2and
@@ -536449,13 +536526,13 @@ the following slightly extended version (related to ~ pm2.65 ):
$}
${
- wl-luk-syl5.1 $e |- ( ph -> ps ) $.
- wl-luk-syl5.2 $e |- ( ch -> ( ps -> th ) ) $.
+ wl-luk-imtrid.1 $e |- ( ph -> ps ) $.
+ wl-luk-imtrid.2 $e |- ( ch -> ( ps -> th ) ) $.
$( A syllogism rule of inference. The first premise is used to replace the
second antecedent of the second premise. Copy of ~ syl5 with a
different proof. (Contributed by Wolf Lammen, 17-Dec-2018.)
(New usage is discouraged.) (Proof modification is discouraged.) $)
- wl-luk-syl5 $p |- ( ch -> ( ph -> th ) ) $=
+ wl-luk-imtrid $p |- ( ch -> ( ph -> th ) ) $=
( wi wl-luk-imim1i wl-luk-syl ) CBDGADGFABDEHI $.
$}
@@ -536474,7 +536551,7 @@ the following slightly extended version (related to ~ pm2.65 ):
by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-con4i $p |- ( ps -> ph ) $=
- ( wn ax-luk3 wl-luk-syl5 wl-luk-pm2.18d ) BAADBDBACBAEFG $.
+ ( wn ax-luk3 wl-luk-imtrid wl-luk-pm2.18d ) BAADBDBACBAEFG $.
$}
${
@@ -536502,7 +536579,7 @@ the following slightly extended version (related to ~ pm2.65 ):
(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-mpi $p |- ( ph -> ch ) $=
- ( wn wl-luk-a1i wl-luk-syl5 wl-luk-pm2.18d ) ACCFZBACBJDGEHI $.
+ ( wn wl-luk-a1i wl-luk-imtrid wl-luk-pm2.18d ) ACCFZBACBJDGEHI $.
$}
${
@@ -536515,13 +536592,13 @@ the following slightly extended version (related to ~ pm2.65 ):
$}
${
- wl-luk-syl6.1 $e |- ( ph -> ( ps -> ch ) ) $.
- wl-luk-syl6.2 $e |- ( ch -> th ) $.
+ wl-luk-imtrdi.1 $e |- ( ph -> ( ps -> ch ) ) $.
+ wl-luk-imtrdi.2 $e |- ( ch -> th ) $.
$( A syllogism rule of inference. The second premise is used to replace
the consequent of the first premise. Copy of ~ syl6 with a different
proof. (Contributed by Wolf Lammen, 17-Dec-2018.)
(New usage is discouraged.) (Proof modification is discouraged.) $)
- wl-luk-syl6 $p |- ( ph -> ( ps -> th ) ) $=
+ wl-luk-imtrdi $p |- ( ph -> ( ps -> th ) ) $=
( wi wl-luk-imim2i wl-luk-syl ) ABCGBDGECDBFHI $.
$}
@@ -536529,8 +536606,8 @@ the following slightly extended version (related to ~ pm2.65 ):
17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-ax3 $p |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) $=
- ( wn wi ax-luk3 ax-luk1 wl-luk-syl5 ax-luk2 wl-luk-syl6 ) ACZBCZDZBJADZABKA
- DLMBAEJKAFGAHI $.
+ ( wn wi ax-luk3 ax-luk1 wl-luk-imtrid ax-luk2 wl-luk-imtrdi ) ACZBCZDZBJADZ
+ ABKADLMBAEJKAFGAHI $.
$( ~ ax-1 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen,
17-Dec-2018.) (New usage is discouraged.)
@@ -536544,8 +536621,8 @@ the following slightly extended version (related to ~ pm2.65 ):
17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-pm2.27 $p |- ( ph -> ( ( ph -> ps ) -> ps ) ) $=
- ( wi wn wl-luk-ax1 ax-luk1 wl-luk-syl ax-luk2 wl-luk-syl6 ) AABCZBDZBCZBAKA
- CJLCAKEKABFGBHI $.
+ ( wi wn wl-luk-ax1 ax-luk1 wl-luk-syl ax-luk2 wl-luk-imtrdi ) AABCZBDZBCZBA
+ KACJLCAKEKABFGBHI $.
${
wl-luk-com12.1 $e |- ( ph -> ( ps -> ch ) ) $.
@@ -536554,7 +536631,7 @@ the following slightly extended version (related to ~ pm2.65 ):
17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-com12 $p |- ( ps -> ( ph -> ch ) ) $=
- ( wi wl-luk-pm2.27 wl-luk-syl5 ) ABCEBCDBCFG $.
+ ( wi wl-luk-pm2.27 wl-luk-imtrid ) ABCEBCDBCFG $.
$}
$( From a wff and its negation, anything follows. Theorem *2.21 of
@@ -536571,7 +536648,7 @@ the following slightly extended version (related to ~ pm2.65 ):
(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-con1i $p |- ( -. ps -> ph ) $=
- ( wn wl-luk-pm2.21 wl-luk-syl5 wl-luk-pm2.18d ) BDZAADBHACBAEFG $.
+ ( wn wl-luk-pm2.21 wl-luk-imtrid wl-luk-pm2.18d ) BDZAADBHACBAEFG $.
$}
${
@@ -536581,8 +536658,8 @@ the following slightly extended version (related to ~ pm2.65 ):
different proof. (Contributed by Wolf Lammen, 17-Dec-2018.)
(New usage is discouraged.) (Proof modification is discouraged.) $)
wl-luk-ja $p |- ( ( ph -> ps ) -> ch ) $=
- ( wi wn wl-luk-con1i wl-luk-imim2i wl-luk-syl5 wl-luk-pm2.18d ) ABFZCCGAL
- CACDHBCAEIJK $.
+ ( wi wn wl-luk-con1i wl-luk-imim2i wl-luk-imtrid wl-luk-pm2.18d ) ABFZCCG
+ ALCACDHBCAEIJK $.
$}
$( A closed form of syllogism (see ~ syl ). Theorem *2.05 of
@@ -536631,7 +536708,7 @@ In classical logic (our logic) this is always true. In intuitionistic
by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.)
(Proof modification is discouraged.) $)
wl-luk-pm2.04 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) $=
- ( wi wl-luk-ax1 wl-luk-ax2 wl-luk-syl5 ) BABDABCDDACDBAEABCFG $.
+ ( wi wl-luk-ax1 wl-luk-ax2 wl-luk-imtrid ) BABDABCDDACDBAEABCFG $.
$(
@@ -536715,7 +536792,7 @@ indicating the (minimal) length of the longest chain involved.
wl-impchain-mp-2.a $e |- ( th -> ( ch -> ps ) ) $.
$( Use to replace the consequent. $)
wl-impchain-mp-2.b $e |- ( ps -> ph ) $.
- $( This theorem is in fact a copy of ~ wl-luk-syl6 , and repeated here to
+ $( This theorem is in fact a copy of ~ wl-luk-imtrdi , and repeated here to
demonstrate a recursive proof scheme. The number '2' in the theorem
name indicates that a chain of length 2 is modified. (Contributed by
Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.)
@@ -540760,7 +540837,7 @@ curry M LIndF ( R freeLMod I ) ) ) $=
unex coex fz1ssfz0 syl6ss fveq2 breq2d oveq12d simprbi elfznn0 nn0red
cc f1ofn df-pr indi 3eqtr3g elfzuz3 reldisj crn imassrn fnima elfzelz
ltled cn0 nn0p1nn elfz syl3anc 1red ltle imp leadd1dd peano2zd imass2
- eqsstr3d eqssd wfo f1ofo forn fnresi eqtri mp2an nnuz syl6eleq ssdifd
+ eqsstrrd eqssd wfo f1ofo forn fnresi eqtri mp2an nnuz syl6eleq ssdifd
fzss1 imaundi eqtr3d iftrue vex weq oveq2 oveq1 3eqtr4d lenlt biimpar
oveq1d lep1d letrd eluzp1p1 ltadd1dd elfzle1 syldan iffalse pm2.61dan
mpteq2dva eqtr4d op1std op2ndd id imaeq1 oveqan12d sylan9eqr sylanbrc
@@ -544069,7 +544146,7 @@ curry M LIndF ( R freeLMod I ) ) ) $=
sstr mtbird syl5bi imp cxp cmap cabs mp3an3 cico icossxr peano2re c2 cdiv
1rp mpani resubcl posdif elrpd rphalfcld rehalfcld jctl ltsubrpd retopbas
biimpd ctop bastg mblfinlem2 indif2 df-ss syl5eq opncld incld simpr simpl
- difeq1d ssdif2d dfin4 inss2 eqsstr3i anbi1d eqcoms biantrud bitr4d rspcev
+ difeq1d ssdif2d dfin4 inss2 eqsstrri anbi1d eqcoms biantrud bitr4d rspcev
sseq1 fveq2 an12s adantrrr adantlr simp-6r difdif2 fveq2i unssi ad3antrrr
cun ssdifss ssinss1 ovolun syl22anc sylancl eleq1w vtoclga subaddd biimpi
ssdif oveq2d ad2antll eqtr3d breqtrd ltsub23d lelttrd ad4antr ssun2 incom
@@ -545426,7 +545503,7 @@ curry M LIndF ( R freeLMod I ) ) ) $=
0le0 iftrue 3brtr4d ex iffalse ifbi ifan eqtri breq2d reex eqidd ad2antrr
cvv cima csup crn imassrn cico frnd icossxr syl6ss syl5ss supxrcl wne cxp
c0 ltrelxr ssbri brxp ioon0 r19.2z wfn ffnd ioossre qbtwnxr anim1i anasss
- crab qre rpxr syl2an cicc ioorp ioossicc eqsstr3i sseli cmul covol ioombl
+ crab qre rpxr syl2an cicc ioorp ioossicc eqsstrri sseli cmul covol ioombl
cin cdm elioore xrltnle biimpar xrre2 syl32anc mnfxr mnfle xrlelttrd xrre
mblvol ltsubposd eliooord ltaddpos2d lttrd ovolioo syl5eq eqeltrd mulgt0d
iooin ifbid mpteq2dv rpge0 sylanbrc itg2const eqtrd fvoveq1 imbrov2fvoveq
@@ -546697,7 +546774,7 @@ curry M LIndF ( R freeLMod I ) ) ) $=
cico reex off abstrid ralrimiva ovexd eqidd fvexd absmul oveq1i mulid2d
absi eqtr2d offval2 ofrfval2 itg2le syl3anc ccom cofmpt resubcl crn csn
absf iunin2 imaiun iunid imaeq2i eqtr3i ineq2i cnvimass cnvimarndm fdmd
- dmmpti sseqtri syl5sseqr df-ss sylib frnd sselda ad2antrr 2thd ltaddsub
+ dmmpti sseqtri sseqtrrid df-ss sylib frnd sselda ad2antrr 2thd ltaddsub
sylan syl3an3 3comr 3expa readdcl adantll elioopnf rexr 3bitr4rd bibi1d
oveq2 syl5ibrcom pm5.32rd adantllr syldan rabbidv imaeq1d vex mptiniseg
cnveqd 3eqtr4d eqtr3d cfn i1frn mbfss mbfima i1fima ralrimivw finiunmbl
@@ -549475,7 +549552,7 @@ curry M LIndF ( R freeLMod I ) ) ) $=
cv cbl ciun cpw cfn wrex cxp cres metres2 syl5eqel istotbnd3 baib simpllr
sspwb sylib ssrind simprl sseldd simprr wel cxmet metxmet ad4antr simplbi
wb elfpw adantl sselda simp-4r sseqin2 eleqtrrd rpxrd blres syl3anc inss1
- syl6eqss ralrimiva ss2iun adantrr eqsstr3d ex reximdv2 ralimdva sylbid c2
+ syl6eqss ralrimiva ss2iun adantrr eqsstrrd ex reximdv2 ralimdva sylbid c2
cxr jca cdiv wi simpr rphalfcld oveq2 iuneq2d sseq2d rexbidv rspcv wne wf
crab simprbi ad2antrl ssrab2 ssfi sylancl oveq1 ineq1d incom syl6eq dfin5
wex weq neeq1d rabn0 syl6bb rgen cima mpbird wfn elpreima imaeq2d eleq12d
@@ -549844,7 +549921,7 @@ counterexample is the discrete extended metric (assigning distinct
necon3i wor w3a ltso fisupcl mpan sseldd cxmet cmin metxmet ad2antrr 1red
fimaxre2 syl2anc cvv elrnmpt1 mpan2 adantl suprub syl31anc leaddsub mpbid
wb blss2 syl33anc ralrimiva nfcv nfmpt1 nfrn nfsup nfov nfss oveq1 sseq1d
- nfv cbvral sylibr iunss eqsstr3d cxr rpxrd blssm eqssd rspceeqv ralrimdva
+ nfv cbvral sylibr iunss eqsstrrd cxr rpxrd blssm eqssd rspceeqv ralrimdva
rexlimdvaa isbnd baib sylibrd sylc ) ABUBGHZABUCGHZCDIZCIZJAUDGZKZUEZBLZD
BUFUGUHZUIZABUJGHZABUKJMHUULCUUNUUOEIZUUPKZUEZBLZDUUTUIZEMNZUVAULUULUUMUV
HCDABEUMUOUVGUVAEJMUVCJLZUVFUUSDUUTUVIUVEUURBUVICUUNUVDUUQUVCJUUOUUPUPUQU
@@ -550516,7 +550593,7 @@ counterexample is the discrete extended metric (assigning distinct
selpw anbi1i csn cun ffn 0z ssv int0 sseqtr4i ssun1 anim2i imim1i ssun2
ralsn sscls sseq2 syl5ibrcom anim12i intunsn syl6sseqr rexlimdvv reeanv
ssin ss2in cbvrexv 3imtr3g expd findcard2 com12 impr ffnd eqeltri mpan2
- sylcom 1z uzn0 wfun fnfun fndm syl5sseqr funfvima2 necomd neneqd nrexdv
+ sylcom 1z uzn0 wfun fnfun fndm sseqtrrid funfvima2 necomd neneqd nrexdv
ne0i 0ex zex pwex frn ssexi abrexex elfi sylnibr cmptop cmpfi ibi fveq2
notbid neeq1d syl6bb rspccv syl3c wrel lmrel r19.23v albii eleq2 ralbii
ceqsalv ralcom4 bitr3i elintab rspceeqv elab mpbir intss1 clsss3 sselda
@@ -550619,7 +550696,7 @@ counterexample is the discrete extended metric (assigning distinct
cdm expr c1 nnuz 1zzd simpl iscmet3 mpbird cbl co cab cpw cin wss metxmet
cxmet cxr id rpxr blopn syl3an 3com23 eleq1a syl rexlimdva adantlr abssdv
3expa ad2antrr mopnuni blcntr syl3an1 elabrex adantl elunii syl2anc nfre1
- ovex nfcv nfab nfuni dfss3f sylibr eqsstr3d unissd eqssd eqid cmpcov elin
+ ovex nfcv nfab nfuni dfss3f sylibr eqsstrrd unissd eqssd eqid cmpcov elin
syl3anc ancom bitri anbi1i anass rexbii2 sylib eqcom eqeq1d anbi1d syl5bb
syl5rbb elpwi ssabral anim2i syl6bi reximdv mpd istotbnd sylanbrc jca ) A
CUBIJZBUCJZKZACUDIJZACUEIJZYFYGUFCELZUGZYIBUAIUNJZMZEAUHIZNYFYLEYMYFYIYMJ
@@ -550963,7 +551040,7 @@ the next (since the empty set has a finite subcover, the
cc0 wb elrp 1re lediv1 mp3an12 sylbi mpbii fveq2 opeq12d opex fvmpt
3re op2nd breqtrrd ssbl syl221anc oveq1 ovmpo oveq1d eqtr3d 1st2nd2
op1st df-ov syl6reqr ctop mopntop blssm mopnuni eqid simprr blsscls
- ccl sscls syl23anc eqsstr3d rpre ccom clm heiborlem6 caublcls mpdan
+ ccl sscls syl23anc eqsstrrd rpre ccom clm heiborlem6 caublcls mpdan
3expia imp blhalf syl22anc sstr2 cpw cin csn unisng snssd snex elpw
biimpar sylibr snfi a1i elind unieq rspcev sylan syldan sseq1 elab2
notbid con2bii sylib ex syld mt2d rexlimddv nrexdv pm2.21dd ) AUBCV
@@ -551094,7 +551171,7 @@ the next (since the empty set has a finite subcover, the
iuneq2 syl5eq syl simp2r 3eqtr2rd iuneq1 rspceeqv 3expia adantrrr exlimdv
fofi mpd rexlimdvaa syld ralrimdva pwex inex1 com eqid simpl weq iuneq2dv
oveq2d nn0ennn nnenom entri axcc4 syl6 elpwi cab copab pweqd ineq1d feq3d
- cmpo wn biimpar adantrr cbviunv id syl5sseqr fss syl2anr ffvelrnda elpwid
+ cmpo wn biimpar adantrr cbviunv id sseqtrrid fss syl2anr ffvelrnda elpwid
inss1 sselda ovex ovmpo biimprd ralimdva impr fveq2 iuneq1d eqtrd cbvralv
simplr heiborlem10 exp32 syl5 ralrimiv ex imp iscmp sylanbrc jca impbii )
ACUBUCKZBUFKZLACUGUCKZACUHUCKZLZABCDUIUWIUWEUWFUWGUWEUWHACUJZUKUWIBULKZBU
@@ -553339,7 +553416,7 @@ the next (since the empty set has a finite subcover, the
/\ B e. ( X \ { Z } ) ) -> ( A H B ) e. ( X \ { Z } ) ) $=
( wcel cxp wa co wceq adantl cdm wss eleq2d cdrng crngo csn cdif cres cgr
isdrngo1 ovres crn wi eqid grpocl 3expib grporndm difss xpss12 mp2an fdmd
- rngosm syl5sseqr ssdmres sylib adantr dmeqd dmxpid syl6eq anbi12d 3imtr3d
+ rngosm sseqtrrid ssdmres sylib adantr dmeqd dmxpid syl6eq anbi12d 3imtr3d
eqtrd imp eqeltrrd 3impb syl3an1b ) CUALCUBLZEFGUCZUDZVPMZUEZUFLZNZAVPLZB
VPLZABEOZVPLZCDEFGHIJKUGVTWAWBWDVTWAWBNZNABVROZWCVPWEWFWCPVTABVPVPEUHQVTW
EWFVPLZVTAVRUIZLZBWHLZNZWFWHLZWEWGVSWKWLUJVNVSWIWJWLABVRWHWHUKULUMQVTWIWA
@@ -553356,7 +553433,7 @@ the next (since the empty set has a finite subcover, the
A. x e. ( X \ { Z } ) E. y e. ( X \ { Z } ) ( y H x ) = U ) ) ) $=
( vz wcel wa co wceq wrex adantr vu vv vw cdrng csn cdif cxp cres cgr wne
crngo cv wral isdrngo1 dvrunz sylbir crn cdm grporndm adantl difss xpss12
- wss mp2an rngosm syl5sseqr ssdmres sylib dmeqd dmxpid syl6eq eqtrd eleq2d
+ wss mp2an rngosm sseqtrrid ssdmres sylib dmeqd dmxpid syl6eq eqtrd eleq2d
fdmd biimpar cgn cfv grpoinvcl adantll cgi grpolinv cmagm cexid cin cmndo
eqid rngomndo mndomgmid syl sseqtri rngorn1eq syl5sseq c1st rneqi rngo1cl
eqtri eldifsn sylanbrc grpomndo mndoismgmOLD syl31anc oveq1 eqeq1d rspcev
@@ -554856,7 +554933,7 @@ the next (since the empty set has a finite subcover, the
Madsen, 10-Jun-2010.) $)
igenss $p |- ( ( R e. RingOps /\ S C_ X ) -> S C_ ( R IdlGen S ) ) $=
( vj crngo wcel wss wa cv cidl cfv crab cint cigen co ssintub igenval
- syl5sseqr ) AHIBDJKBGLJGAMNZOPBABQRGBUBSABGCDEFTUA $.
+ sseqtrrid ) AHIBDJKBGLJGAMNZOPBABQRGBUBSABGCDEFTUA $.
$( The ideal generated by a set is an ideal. (Contributed by Jeff Madsen,
10-Jun-2010.) $)
@@ -563630,7 +563707,7 @@ inference form (e.g.,
(Contributed by NM, 15-Jan-2015.) $)
lsmsat $p |- ( ph -> E. p e. A ( p C_ T /\ Q C_ ( p .(+) U ) ) ) $=
( vr vy vz vq cv csn clspn cfv wceq cbs cdif wrex wss co wa wcel clmod wb
- eqid islsat syl mpbid cplusg simp3 3ad2ant1 eqsstr3d lsmcl syl3anc eldifi
+ eqid islsat syl mpbid cplusg simp3 3ad2ant1 eqsstrrd lsmcl syl3anc eldifi
w3a 3ad2ant2 lspsnel5 mpbird lsssssubg sseldd lsmelval syl2anc wne lssne0
csubg wi adantr simpr2 lssel simpr3 lspsnel5a simpl3 simpr1 oveq1d simp2r
lsatlspsn2 lmod0vlid 3eqtrd sneqd fveq2d eqsstrd lsmub2 sstrd sseq1 oveq1
@@ -563763,7 +563840,7 @@ inference form (e.g.,
( vy wcel wa cv wss crab cuni cfv wceq syl2anc syl3anc clmod eleq1 simplr
c0g wne csn simplll cbs simpllr lssel lspsncl lspid lsatlss adantr rabss2
eqid uniss 3syl unimax lssss eqsstrd adantl ad2antrr lsatlspsn2 lspsnel5a
- sstrd simpr sseq1 elrab sylanbrc elssuni syl lspss eqsstr3d sseldd simpll
+ sstrd simpr sseq1 elrab sylanbrc elssuni syl lspss eqsstrrd sseldd simpll
lspsnid lspcl syldan lss0cl pm2.61ne ex ssrdv simpl fveq2d eqtrd sseqtrd
eqssd ) FUAKZDCKZLZDAMZDNZABOZPZEQZWKJDWPWKJMZDKZWQWPKZWKWRLZWSFUDQZWPKZW
QXAWQXAWPUBWTWQXAUEZLZWQUFEQZWPWQXDXEXEEQZWPXDWIXECKZXFXERWIWJWRXCUGZXDWI
@@ -566826,7 +566903,7 @@ Functionals and kernels of a left vector space (or module)
<-> ( G =/= .0. /\ H = .0. ) ) ) $=
( cfv wne wceq wa simpr wpss cbs wss df-pss eqid clvec clmod lveclmod syl
wcel lkrssv adantr psssstrd pssned sylan2br clsh simplr ad3antrrr simpllr
- wn ad2antrr eqsstr3d eqssd wb lkrshp4 necon1bbid mpbird pm2.21dd lkrshpor
+ wn ad2antrr eqsstrrd eqssd wb lkrshp4 necon1bbid mpbird pm2.21dd lkrshpor
wo mpjaodan lshpcmp mpbid necon3ad impr jca simprr eqcomd sseqtrd neeqtrd
ex simprl impbida syl5bb lkr0f2 necon3bid anbi12d bitrd ) ADFPZEFPZUAZWIG
UBPZQZWJWLRZSZDHQZEHRZSWKWIWJUCZWIWJQZSZAWOWIWJUDZAWTWOAWTSZWMWNWTAWKWMXA
@@ -576865,7 +576942,7 @@ Cambridge University Press (1988). Unlike them, we do not assume that
( vk wcel cv wral cfv catm cvv wss co wbr wi wa cab wceq elex cpsubsp cjn
fveq2 syl6eqr sseq2d oveqd breq2d breqd imbi1d raleqbidv 2ralbidv anbi12d
cple bitrd abbidv df-psubsp cpw fvexi pwex selpw anbi1i abbii ssab2 ssexi
- eqsstr3i fvmpt syl5eq syl ) EBPEUAPZCGQZAUBZHQZJQZIQZDUCZFUDZWAVSPZUEZHAR
+ eqsstrri fvmpt syl5eq syl ) EBPEUAPZCGQZAUBZHQZJQZIQZDUCZFUDZWAVSPZUEZHAR
ZIVSRJVSRZUFZGUGZUHEBUIVRCEUJSWKNOEVSOQZTSZUBZWAWBWCWLUKSZUCZWLVBSZUDZWFU
EZHWMRZIVSRJVSRZUFZGUGWKUAUJWLEUHZXBWJGXCWNVTXAWIXCWMAVSXCWMETSAWLETULMUM
ZUNXCWTWHJIVSVSXCWSWGHWMAXDXCWRWEWFXCWRWAWDWQUDWEXCWPWDWAWQXCWODWBWCXCWOE
@@ -577964,7 +578041,7 @@ their common point (atom). (Contributed by NM, 2-May-2013.)
paddunssN $p |- ( ( K e. B /\ X C_ A /\ Y C_ A )
-> ( X u. Y ) C_ ( X .+ Y ) ) $=
( vp vq vr wcel wss w3a cun cv cfv co wrex eqid cjn cple wbr crab paddval
- ssun1 syl5sseqr ) DBLEAMFAMNEFOZIPJPKPDUAQZRDUBQZUCKFSJESIAUDZOUHEFCRUHUK
+ ssun1 sseqtrrid ) DBLEAMFAMNEFOZIPJPKPDUAQZRDUBQZUCKFSJESIAUDZOUHEFCRUHUK
UFABCUIDUJEFKJIUJTUITGHUEUG $.
$( Member of projective subspace sum with at least one empty set.
@@ -578028,14 +578105,14 @@ their common point (atom). (Contributed by NM, 2-May-2013.)
analog.) (Contributed by NM, 3-Jan-2012.) $)
sspadd1 $p |- ( ( K e. B /\ X C_ A /\ Y C_ A ) -> X C_ ( X .+ Y ) ) $=
( vp vq vr wcel wss cun cv cfv co wrex ssun1 eqid w3a cjn cple crab sstri
- wbr paddval syl5sseqr ) DBLEAMFAMUAEFNZIOJOKODUBPZQDUCPZUFKFRJERIAUDZNZEE
+ wbr paddval sseqtrrid ) DBLEAMFAMUAEFNZIOJOKODUBPZQDUCPZUFKFRJERIAUDZNZEE
FCQEUIUMEFSUIULSUEABCUJDUKEFKJIUKTUJTGHUGUH $.
$( A projective subspace sum is a superset of its second summand.
( ~ ssun2 analog.) (Contributed by NM, 3-Jan-2012.) $)
sspadd2 $p |- ( ( K e. B /\ X C_ A /\ Y C_ A ) -> X C_ ( Y .+ X ) ) $=
( vp vq vr wcel wss w3a cun cv cfv co wrex eqid cjn cple crab ssun2 ssun1
- wbr sstri wceq paddval 3com23 syl5sseqr ) DBLZEAMZFAMZNFEOZIPJPKPDUAQZRDU
+ wbr sstri wceq paddval 3com23 sseqtrrid ) DBLZEAMZFAMZNFEOZIPJPKPDUAQZRDU
BQZUFKESJFSIAUCZOZEFECRZEUOUSEFUDUOURUEUGULUNUMUTUSUHABCUPDUQFEKJIUQTUPTG
HUIUJUK $.
@@ -579662,7 +579739,7 @@ one element is a lattice line (expressed as the join ` P .\/ Q ` ).
$( A set of atoms is included in its projective subspace closure.
(Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) $)
pclssidN $p |- ( ( K e. V /\ X C_ A ) -> X C_ ( U ` X ) ) $=
- ( vy wcel wss wa cv cpsubsp cfv crab cint ssintub eqid pclvalN syl5sseqr
+ ( vy wcel wss wa cv cpsubsp cfv crab cint ssintub eqid pclvalN sseqtrrid
) CDIEAJKEHLJHCMNZOPEEBNHEUAQHAUABCDEFUARGST $.
$}
@@ -580271,7 +580348,7 @@ one element is a lattice line (expressed as the join ` P .\/ Q ` ).
( ._|_ ` ( ._|_ ` s ) ) = s ) } ) $=
( vk wcel cvv cv wss cfv wceq wa cab catm cpolN elex cpscN syl6eqr sseq2d
fveq2 fveq1d fveq12d eqeq1d abbidv df-psubclN cpw fvexi pwex selpw anbi1i
- anbi12d abbii ssab2 eqsstr3i ssexi fvmpt syl5eq syl ) DBKDLKZCFMZANZVEEOZ
+ anbi12d abbii ssab2 eqsstrri ssexi fvmpt syl5eq syl ) DBKDLKZCFMZANZVEEOZ
EOZVEPZQZFRZPDBUAVDCDUBOVKIJDVEJMZSOZNZVEVLTOZOZVOOZVEPZQZFRVKLUBVLDPZVSV
JFVTVNVFVRVIVTVMAVEVTVMDSOAVLDSUEGUCUDVTVQVHVEVTVPVGVOEVTVODTOEVLDTUEHUCZ
VTVEVOEWAUFUGUHUPUIJFUJVKAUKZAADSGULUMVKVEWBKZVIQZFRWBWDVJFWCVFVIFAUNUOUQ
@@ -603098,7 +603175,7 @@ are all translations (for a fiducial co-atom ` W ` ). (Contributed by
diassdvaN $p |- ( ( ( K e. Y /\ W e. H ) /\ ( X e. B /\ X .<_ W ) )
-> ( I ` X ) C_ V ) $=
( vf wcel wa cfv wbr cv ctrl crab eqid diaval ssrab2 wceq dvavbase adantr
- cltrn syl5sseqr eqsstrd ) EJRHCRSZIARIHFUASZSZIDTQUBHEUCTTZTIFUAZQHEUKTTZ
+ cltrn sseqtrrid eqsstrd ) EJRHCRSZIARIHFUASZSZIDTQUBHEUCTTZTIFUAZQHEUKTTZ
UDZGAUQUSQCDEFJHIKLMUSUEZUQUENUFUPUSUTGURQUSUGUNGUSUHUOUSBCEGHJMVAOPUIUJU
LUM $.
$}
@@ -603406,7 +603483,7 @@ are all translations (for a fiducial co-atom ` W ` ). (Contributed by
( cfv co cplusg ccom chlt wcel wceq eqid dvavadd syl12anc wne dia2dimlem4
wa simpld eqtr2d csubg clmod wss clvec dvalvec lveclmod lsssssubg syl cbs
3syl wbr atbase simprd dialss sseldd csn dia1dim2 dia2dimlem3 fveq2d eqss
- eqsstr3d dvavbase eleqtrrd lspsnel5 mpbird dia2dimlem2 lsmelvali syl22anc
+ eqsstrrd dvavbase eleqtrrd lspsnel5 mpbird dia2dimlem2 lsmelvali syl22anc
syl2anc sylib eqeltrd cabl lmodabl lsmcom syl3anc eleqtrd ) AKTNVIZJNVIZE
VJZYAXTEVJZAKCLUBVKVIZVJZYBAYECLVLZKAPVMVNUAMVNWAZCIVNZLIVNZYEYFVOUPVGVEY
DIUBCLMPVMUAUGUHUJYDVPZVQVRABCDFIKLMPQUAUCUFUGUHUPUSAKIVNDKVIDVSUTWBVEVFV
@@ -606069,7 +606146,7 @@ all translations (for a fiducial co-atom ` W ` ). (Contributed by NM,
/\ S .<_ ( Q .\/ X ) ) -> ( N ` { <. F , O >. } ) C_ ( I ` X ) ) $=
( chlt wcel wa wbr wn w3a cop csn cfv wceq simp1 simp21 simp22 ltrniotacl
co syl3anc dib1dim2 syl2anc wss cdlemn2 wb simp23 dibord syl121anc mpbird
- trlcl trlle eqsstr3d ) NULUMRKUMUNZCAUMCROUOUPUNZEAUMEROUOUPUNZSBUMSROUOU
+ trlcl trlle eqsstrrd ) NULUMRKUMUNZCAUMCROUOUPUNZEAUMEROUOUPUNZSBUMSROUOU
NZUQZECSMVFOUOZUQZJQURUSPUTZJDUTZLUTZSLUTZWFVTJFUMZWIWGVAVTWDWEVBZWFVTWAW
BWKWLVTWAWBWCWEVCVTWAWBWCWEVDACEFIJKNORUAUCUDUEUKVEVGZBDFGHJKLNPQRTUDUEUF
UGUIUHUJVHVIWFWIWJVJZWHSOUOZABCDEFIJKMNORSTUAUBUCUDUEUFUKVKWFVTWHBUMZWHRO
@@ -609581,7 +609658,7 @@ of phi(x) is independent of the atom q." (Contributed by NM,
wn wi syl clatglbcl syl2anc dihlss dihglblem5 adantr simpr lpssat ex ccnv
w3a crn simp1l dih1dimat adantlr 3adant3 dihcnvid2 wbr simp3l ssiin sylib
wral wb wf1o wf1 dihf11 f1f1orn 3syl f1ocnvdm sselda dihord sseq1d bitr3d
- syl3anc ralbidva mpbird simp1ll simp1rl clatleglb eqsstr3d pm2.21fal syld
+ syl3anc ralbidva mpbird simp1ll simp1rl clatleglb eqsstrrd pm2.21fal syld
simp3r rexlimdv3a mtoi dfpss3 notbii iman anclb 3bitr2i mpd eqss sylibr )
KUHUIZNIUIZUJZFCUKZFULUMZUJZUJZFHUNZJUNZAFAVBZJUNZUOZUKZYTYQUKZUJZYQYTUPY
OUUAUUCAUEUFCFUFVBUEVBNKUQUNZVCUPUEFURUFCUSZHIJNKUTUNUNZKLUUDNOPUUDVASTUU
@@ -609891,7 +609968,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $=
-> ( N ` X ) = ( I ` ( ._|_ ` ( `' I ` X ) ) ) ) $=
( vy vz wcel wa cfv wss wceq eqid wbr chlt crn cv cbs crab cglb ccnv cdvh
dihrnss dochval syldan cple clat ad2antrr hlclat ssrab2 clatglbcl sylancl
- hllat ccla dihcnvcl ssid dihcnvid2 syl5sseqr fveq2 sseq2d elrab clatglble
+ hllat ccla dihcnvcl ssid dihcnvid2 sseqtrrid fveq2 sseq2d elrab clatglble
a1i sylanbrc syl3anc wral adantr sseq1d simpll simpr dihord bitr3d biimpd
wb expimpd syl5bi ralrimiv clatleglb mpbird latasymd fveq2d eqtrd ) CUANZ
FANZOZGBUBNZOZGDPZGLUCZBPZQZLCUDPZUEZCUFPZPZEPZBPZGBUGPZEPZBPWKWLGFCUHPPZ
@@ -610262,7 +610339,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $=
(Contributed by NM, 7-Aug-2014.) $)
dochocsp $p |- ( ph -> ( ._|_ ` ( N ` X ) ) = ( ._|_ ` X ) ) $=
( cfv wcel wss syl2anc chlt wa dvhlmod lspssv lspssid dochss syl3anc cdih
- clmod crn wceq eqid dochcl dochoc dochssv dochspss eqsstr3d eqssd ) AIEQZ
+ clmod crn wceq eqid dochcl dochoc dochssv dochspss eqsstrrd eqssd ) AIEQZ
FQZIFQZADUARHCRUBZUSGSZIUSSZUTVASOABUIRZIGSZVCABCDHJKOUCZPIEGBMNUDTAVEVFV
DVGPIEGBMNUETBCDFGHIUSJKMLUFUGAVAVAFQZFQZUTAVBVAHDUHQQZUJRZVIVAUKOAVBVFVK
OPBCVJDFGHIJVJULZKMLUMTCVJDFHVAJVLLUNTAVBVHGSZUSVHSVIUTSOAVBVAGSZVMOAVBVF
@@ -610479,7 +610556,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $=
( wcel wss cfv a1i dochss syl3anc 3adant3 3adant2 chlt wa w3a simp1 simp2
cun cin simp3 unssd ssun1 ssun2 ssind cdih crn wceq eqid dochcl dihmeetcl
syl12anc dochoc syl2anc dochssv ssinss1 dochocss unss12 inss1 inss2 sstrd
- syl eqsstr3d eqssd ) CUAMFBMUBZGENZHENZUCZGHUFZDOZGDOZHDOZUGZVOVQVRVSVOVL
+ syl eqsstrrd eqssd ) CUAMFBMUBZGENZHENZUCZGHUFZDOZGDOZHDOZUGZVOVQVRVSVOVL
VPENZGVPNZVQVRNVLVMVNUDZVOGHEVLVMVNUEVLVMVNUHUIZWBVOGHUJPABCDEFGVPIJKLQRV
OVLWAHVPNZVQVSNWCWDWEVOHGUKPABCDEFHVPIJKLQRULVOVTVTDOZDOZVQVOVLVTFCUMOOZU
NZMZWGVTUOWCVOVLVRWIMZVSWIMZWJWCVLVMWKVNABWHCDEFGIWHUPZJKLUQSVLVNWLVMABWH
@@ -611824,7 +611901,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $=
( wcel cfv vv chlt adantr simpr dochsatshp c0g wne wrex csn cbs wceq cdih
wa cv crn wss eqid lssss syl dochcl syl2anc dochoc dvhlmod lshpne eqnetrd
clmod wb dochssv dochn0nv mpbird dochlss lssne0 mpbid w3a clspn lspsnel5a
- 3ad2ant1 simp2 lssel dihlsprn dochord eqsstr3d clvec dvhlvec simp1r simp3
+ 3ad2ant1 simp2 lssel dihlsprn dochord eqsstrrd clvec dvhlvec simp1r simp3
lsatlspsn2 syl3anc lshpcmp fveq2d eqeltrd rexlimdv3a mpd dochsat impbida
eqtrd ) ACBSZCHTZJSZAWQUMBCEFGHIJKMLOPAGUBSIFSUMZWQQUCAWQUDUEAWSUMZWRHTZB
SZWQXAUAUNZEUFTZUGZUAXBUHZXCXAXBXEUIUGZXGXAXHXBHTZEUJTZUGZXAXIWRXJAXIWRUK
@@ -613039,7 +613116,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
by NM, 16-Feb-2015.) $)
lcfl9a $p |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) $=
( cfv wceq c0g wa dochoc1 adantr wss dvhlmod lkrssv sneq fveq2d chlt wcel
- csn eqid doch0 syl sylan9eqr eqsstr3d eqssd 3eqtr4d simpr wne cdih dochcl
+ csn eqid doch0 syl sylan9eqr eqsstrrd eqssd 3eqtr4d simpr wne cdih dochcl
crn snssd syl2anc dochoc clsh clvec dvhlvec cdif simprl eldifsn dochsnshp
sylanbrc simprr wb lkrshp4 mpbid lshpcmp 3eqtr3d pm2.61da2ne ) ADGUBZHUBZ
HUBZWFUCKBUDUBZWFIAKWIUCZUEZIHUBZHUBZIWHWFAWMIUCZWJABEFHIJLNMORUFZUGWKWGW
@@ -613494,7 +613571,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
kernel of ` G ` , so the kernels of ` G ` and the sum are comparable.
(Contributed by NM, 18-Jan-2015.) $)
lclkrlem2r $p |- ( ph -> ( L ` G ) C_ ( L ` ( E .+ G ) ) ) $=
- ( cfv cin co wss wceq lclkrlem2p 3sstr4d sseqin2 sylib dvhlmod eqsstr3d
+ ( cfv cin co wss wceq lclkrlem2p 3sstr4d sseqin2 sylib dvhlmod eqsstrrd
csn lkrin ) ALPVKZJPVKZWDVLZJLDVMPVKAWDWEVNWFWDVOAUCWBSVKUBWBSVKWDWEABC
DEFGHIJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBVCVDVEVHVI
VJVPVGVFVQWDWEVRVSACDKJLPIULUTUMUNAIMOUAVAVCVEVTUQURWCWA $.
@@ -613506,7 +613583,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
( cfv clsh wcel co wceq csn chlt cdih crn wss snssd eqid dochcl syl2anc
wa dochoc ad2antrr lclkrlem2r clvec dvhlvec simplr simpr lshpcmp eqtr3d
mpbid fveq2d 3eqtr4d dochoc1 dvhlmod ldualvaddcl lkrshpor adantr lkrssv
- wo mpjaodan eqsstr3d eqssd ) ALPVKZIVLVKZVMZJLDVNZPVKZSVKZSVKZXLVOZXHTV
+ wo mpjaodan eqsstrrd eqssd ) ALPVKZIVLVKZVMZJLDVNZPVKZSVKZSVKZXLVOZXHTV
OZAXJWEZXLXIVMZXOXLTVOZXQXRWEZUCVPZSVKZSVKZSVKZYBXNXLAYDYBVOZXJXRAOVQVM
UAMVMWEZYBUAOVRVKVKZVSVMZYEVEAYFYATVTYHVEAUCTUPWAIMYGOSTUAYAVAYGWBZVCUE
VBWCWDMYGOSUAYBVAYIVBWFWDWGXTXMYCSXTXLYBSXTXHXLYBXTXHXLVTZXHXLVOAYJXJXR
@@ -613561,7 +613638,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
wceq dihprrn wss lssss syl dochoccl mpbid dochexmid dvhlvec cin csn cun
lclkrlem2n wa snssd dochdmj1 syl3anc df-pr fveq2i unssd dochocsp syl5eq
chlt ineq12d 3eqtr4d lkrin eqsstrd csubg clmod lsssssubg sseldd dochlss
- wb syl2anc lkrlss lsmlub mpbi2and eqsstr3d eqssd ) AIKCVIZOVJZSAJXTOSHU
+ wb syl2anc lkrlss lsmlub mpbi2and eqsstrrd eqssd ) AIKCVIZOVJZSAJXTOSHU
DUKUSAHLNTUTVBVDVKZABCJIKHUKULUMYBUPUQVLZVMASUAUBVNZQVJZYERVJZDVIZYAADH
VOVJZHLNRSTYEUTVAVBUDYHVPZVCVDAYHQSHUAUBUDYIURYBUNUOVQZAYETNVRVJVJZVSVT
YFRVJYEWAAHLYKNQSTUAUBUTVBUDURYKVPZVDUNUOWBAHLYKNRSTYEUTYLVBUDVAVDAYEYH
@@ -614058,7 +614135,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
lcfrlem6 $p |- ( ph -> ( X .+ Y ) e. E ) $=
( co cv cfv ciun wcel wrex syl6eleq eliun sylib clmod clss dvhlmod adantr
chlt cbs wss clfn eqid lssel sylan wceq ldualvbase eleqtrd lkrssv dochlss
- syl2anc simpr csn eqsstr3d lcfrlem4 lspsnel5 3imtr4d imp lssvacl syl22anc
+ syl2anc simpr csn eqsstrrd lcfrlem4 lspsnel5 3imtr4d imp lssvacl syl22anc
wa ex reximdva mpd sylibr syl6eleqr ) AOPCUKZFHFULZKUMZMUMZUNZGAWLWOUOZFH
UPZWLWPUOAOWOUOZFHUPZWRAOWPUOWTAOGWPUHUGUQFOHWOURUSAWSWQFHAWMHUOZWFZWSWQX
BWSWFEUTUOZWOEVAUMZUOZWSPWOUOZWQXBXCWSAXCXAAEIJNQSUEVBZVCZVCXBXEWSXBJVDUO
@@ -614448,7 +614525,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
inss2 dvhlvec lcfrlem22 lsatel clss dvhlmod clmod wcel clfn cv crab c0g
lcfrlem10 lkrlss syl2anc lsatssv sseldd ellkr2 mpbird lspsnel5a eqsstrd
csubg wa wb lsssssubg syl eldifad prssi dochlss lspprcl lcfrlem17 snssd
- chlt lssincl syl3anc syl5eqel mpbi2and eqsstr3d clsh dochsnshp wne cdif
+ chlt lssincl syl3anc syl5eqel mpbi2and eqsstrrd clsh dochsnshp wne cdif
lsmlub lcfrlem13 eldifsni lduallkr3 lshpcmp mpbid ) AUDUEHVKZVLZUAVMZUE
QVMZSVMZVNYMYOVOAYMUDUEVPZUAVMZFMVQVMZVKZYOAEFHYRMORTUAUBUCUDUEUFUGUHUI
UJUKULUMUNUOUPUQURUSYRVRZVSAYQYOVNZFYOVNZYSYOVNZAYQUDQVMSVMZYOVTYOABCDE
@@ -614626,7 +614703,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
cmulr dvhlvec cv crab c0g lcfrlem10 clss wcel dvhlmod lcfrlem22 lsatlssel
lssel syl2anc lcfrlem2 eqsstrd lcfrlem28 lsatel lcfrlem30 lkrlss lcfrlem3
clmod lspsnel5a csubg wa lsssssubg syl chlt eldifad dochlss sseldd lsmlub
- wb prssi syl3anc mpbi2and eqsstr3d clsh lcfrlem17 dochsnshp wne lcfrlem34
+ wb prssi syl3anc mpbi2and eqsstrrd clsh lcfrlem17 dochsnshp wne lcfrlem34
lduallkr3 mpbird lshpcmp mpbid ) AUGUHIVPZVQUDVRZGUAVRZVSYOYPVTAYOUGUHWAZ
UDVRZFNWBVRZVPZYPAEFIYSNQTUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBYSWCZWD
AYRYPVSZFYPVSZYTYPVSZAYRUGSVRZUAVRUHSVRZUAVRWEYPABCDEFIJKLMNOQRSTUAUCUDUE
@@ -614964,7 +615041,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
kernels. (Contributed by NM, 13-Mar-2015.) $)
lcdvbase $p |- ( ph -> V = B ) $=
( cld cfv cress co cbs chlt eqid lcdval2 fveq2d syl5eq wss wceq cv ssrab2
- crab eqsstri clmod dvhlmod ldualvbase syl5sseqr ressbas2 syl eqtr4d ) AKD
+ crab eqsstri clmod dvhlmod ldualvbase sseqtrrid ressbas2 syl eqtr4d ) AKD
UBUCZBUDUEZUFUCZBAKCUFUCVGPACVFUFABCVEDEFGHIJLUGMNOQRSVEUHZUATUIUJUKABVEU
FUCZULBVGUMAFBVIBEUNIUCZJUCJUCVJUMZEFUPFTVKEFUOUQAVEFVIDURRVHVIUHZADGHLMQ
UAUSUTVABVIVFVEVFUHVLVBVCVD $.
@@ -615659,7 +615736,7 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane).
mapdval2N $p |- ( ph
-> ( M ` T ) = { f e. C | E. v e. T ( O ` ( L ` f ) ) = ( N ` { v } ) } ) $=
( cfv cv wss crab csn wceq wrex chlt mapdvalc wcel wa clsa wi c0g dvhlmod
- clmod ad3antrrr simplr eqid islsati syl2anc simprr eqsstr3d adantr simprl
+ clmod ad3antrrr simplr eqid islsati syl2anc simprr eqsstrrd adantr simprl
cbs lspsnel5 mpbird reximssdv ex lss0cl simpr lspsn0 eqtr4d sneq rspceeqv
syl fveq2d adantlr a1d lcfl1lem simplbi adantl dochsat0 mpjaodan 3ad2ant1
w3a simp3 simp2 lspsnel5a eqsstrd rexlimdv3a impbid rabbidva eqtrd ) AEMU
@@ -616336,7 +616413,7 @@ Part of property (e) in [Baer] p. 40. (Contributed by NM,
mapdin $p |- ( ph -> ( M ` ( X i^i Y ) ) = ( ( M ` X ) i^i ( M ` Y ) ) ) $=
( cfv wss wcel mapdord cin inss1 clmod dvhlmod lssincl mpbiri inss2 ssind
syl3anc ccnv clcd eqid clss mapdcl2 eleqtrrd mapdincl mapdcnvid2 syl6eqss
- crn mapdrn2 lcdlmod mapdcnvcl mpbid mpbird eqsstr3d eqssd ) AHIUAZFQZHFQZ
+ crn mapdrn2 lcdlmod mapdcnvcl mpbid mpbird eqsstrrd eqssd ) AHIUAZFQZHFQZ
IFQZUAZAVHVIVJAVHVIRVGHRHIUBABCDEFGVGHJLMKNACUCSHBSIBSVGBSACDEGJLNUDOPBHI
CMUEUIZOTUFAVHVJRVGIRHIUGABCDEFGVGIJLMKNVLPTUFUHAVKVKFUJQZFQZVHADEFGVKJKN
AGEUKQQZCDEFGVIVJJKLVOULZNAVIVOUMQZFUSZAVOHBVQCDEFGJKLMVPVQULZNOUNZAVOVQD
@@ -617311,7 +617388,7 @@ be in (Fy)*..." (Contributed by NM, 18-Mar-2015.) $)
= ( ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { Z } ) )
i^i ( ( N ` { ( X .- Z ) } ) .(+) ( N ` { Y } ) ) ) ) $=
( vj va vb vd ve csn cfv cin clmod wcel cabl clvec lveclmod syl lmodabl
- co eldifad ablsubsub4 sneqd fveq2d lmodvsubcl syl3anc eqsstr3d ablsub32
+ co eldifad ablsubsub4 sneqd fveq2d lmodvsubcl syl3anc eqsstrrd ablsub32
wss lspsntrim eqtrd ssind cv wceq wrex wa lsmspsn anbi12d syl5bb wi w3a
elin simp11 cpr wn wne cdif simp12l simp12r simp2l simp2r baerlem5alem1
simp13 simp3 lmodvacl lspsneli eqeltrd 3exp rexlimdvv impd sylbid ssrdv
@@ -617339,7 +617416,7 @@ be in (Fy)*..." (Contributed by NM, 18-Mar-2015.) $)
i^i ( ( N ` { ( X .- ( Y .+ Z ) ) } ) .(+) ( N ` { X } ) ) ) ) $=
( vj va vb vd ve csn cfv cin clmod wcel wss clvec lveclmod syl lspsntri
co eldifad syl3anc lmodvacl lmodvsubcl lspsnsub lmodabl ablnncan fveq2d
- cabl sneqd eqtrd lspsntrim eqsstr3d ssind wceq wrex elin lsmspsn syl5bb
+ cabl sneqd eqtrd lspsntrim eqsstrrd ssind wceq wrex elin lsmspsn syl5bb
cv wa anbi12d wi w3a simp11 cpr wn simp12l simp12r simp2l simp2r simp13
wne cdif simp3 baerlem5blem1 cgrp crg lmodring ringgrp grpinvcl syl2anc
4syl lspsneli eqeltrd 3exp rexlimdvv impd sylbid ssrdv eqssd ) APRCVMZV
@@ -626455,7 +626532,7 @@ family of sets (implicit). (Contributed by Stefan O'Rear,
A. x e. NN0 ( F ` x ) C_ ( F ` ( x + 1 ) ) ) ->
E. y e. NN0 A. z e. ( ZZ>= ` y ) ( F ` z ) = ( F ` y ) ) $=
( vw va vb vc cfv wcel cn0 cv wss wral wceq wa wrex wb cnacs wf caddc w3a
- c1 co crn cuni fvssunirn simplrr syl5sseqr simpll3 simplrl simpr incssnn0
+ c1 co crn cuni fvssunirn simplrr sseqtrrid simpll3 simplrl simpr incssnn0
cuz syl3anc eqssd ralrimiva cipo cdrs cvv wne cun cpw frn 3ad2ant2 elpw2g
c0 3ad2ant1 mpbird elex syl cc0 wfn ffn 0nn0 fnfvelrn sylancl cr ad2antrl
ne0d nn0re ad2antll cle cz nn0z eluz syl2an biimpar adantll ssequn1 sylib
@@ -633614,7 +633691,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by
function. (Contributed by Stefan O'Rear, 18-Jan-2015.) $)
dnnumch2 $p |- ( ph -> A C_ ran F ) $=
( vx cv cres wf1o con0 wrex crn wss dnnumch1 wi wfo wceq f1ofo forn resss
- syl rnss mp1i eqsstr3d a1i rexlimdvw mpd ) AKLZDEUMMZNZKOPDEQZRZAKBCDEFGH
+ syl rnss mp1i eqsstrrd a1i rexlimdvw mpd ) AKLZDEUMMZNZKOPDEQZRZAKBCDEFGH
IJSAUOUQKOUOUQTAUODUNQZUPUOUMDUNUAURDUBUMDUNUCUMDUNUDUFUNERURUPRUOEUMUEUN
EUGUHUIUJUKUL $.
@@ -634964,7 +635041,7 @@ is the base set of some (relabeled) Abelian group. (Contributed by
S e. dom card ) $=
( vx va vc vd vb cfv cbs cgrp wcel cv wceq cvv wss wb wbr sseldd ad2antrr
wa co char cun cima wrex ccrd cdm wfn basfn ssv fvelimab mp2an cdom harcl
- cxp con0 onenon ax-mp xpnum cwdom ssun1 simpr syl5sseqr fvex ssex syl a1i
+ cxp con0 onenon ax-mp xpnum cwdom ssun1 simpr sseqtrrid fvex ssex syl a1i
cplusg w3a simp1l 3ad2ant1 simp2 ssun2 simp3 grpcl syl3anc simp1r eleqtrd
simplll simprl simprr simplr grplcan syl13anc grprcan wn harndom wdomnumr
eqid unxpwdom3 sylib numdom sylancr rexlimiva sylbi ) AAUAGZUBZHIUCJZBKZH
@@ -636053,7 +636130,7 @@ of ideals (the usual "pure ring theory" definition). (Contributed by
$( The ring-span of a set contains the set. (Contributed by Stefan O'Rear,
30-Nov-2014.) $)
rgspnssid $p |- ( ph -> A C_ U ) $=
- ( vt cv wss csubrg cfv crab cint ssintub rgspnval syl5sseqr ) ABLMNLDOPZQ
+ ( vt cv wss csubrg cfv crab cint ssintub rgspnval sseqtrrid ) ABLMNLDOPZQ
RBELBUBSALBCDEFGHIJKTUA $.
rgspnmin.sr $e |- ( ph -> S e. ( SubRing ` R ) ) $.
@@ -637451,7 +637528,7 @@ is in the span of P(i)(X), so there is an R-linear combination of
itgcl nnne0d cmul itgmulc2 cioo eqidd oveq1 oveq2d adantl ioossicc syldan
simpr a1i mulcld fvmptd itgeq2dv cdv crn ctg ccnfld ctopn reelprrecn 1nn0
cpr nn0addcld nn0cnd 1cnd addcld wf cdm fmpttd ssidd dvexp mpteq2dv eqtrd
- pncand feq1d mpbird fdmd syl5sseqr dvres3 syl22anc reseq1d resmpt 3eqtr3d
+ pncand feq1d mpbird fdmd sseqtrrid dvres3 syl22anc reseq1d resmpt 3eqtr3d
mp1i eqid tgioo2 cnt iccntr dvmptres2 ioossre sstri cncfmptc mulcncf cvol
eqeltrd ioombl iblss ftc2 wral fveq1d ralrimivw itgeq2 oveq1d cxr cle wbr
rexrd ubicc2 recnd lbicc2 oveq12d itgioo 3eqtr3rd mvllmuld ) AEUAUBKZBCDU
@@ -638575,6 +638652,47 @@ is in the span of P(i)(X), so there is an R-linear combination of
DFPACEGPHBCQR $.
$}
+$( May move to main section after ~ alephsuc . $)
+
+ $( ` ( aleph `` 1o ) ` is the least uncountable ordinal. (Contributed by RP,
+ 18-Nov-2023.) $)
+ aleph1min $p |- ( aleph ` 1o ) = |^| { x e. On | _om ~< x } $=
+ ( c1o cale cfv c0 csuc com cv csdm wbr con0 crab cint fveq2i char wcel wceq
+ df-1o ax-mp eqtri 0elon alephsuc aleph0 ccrd cdm omelon onenon harval2 ) BC
+ DEFZCDZGAHIJAKLMZBUICRNUJGODZUKUJECDZODZULEKPUJUNQUAEUBSUMGOUCNTGUDUEPZULUK
+ QGKPUOUFGUGSAGUHSTT $.
+
+$( May move to main section after ~ alephiso . $)
+
+ ${
+ $d x y z $.
+ $( ` aleph ` is a strictly order-preserving mapping of ` On ` onto the
+ class of all infinite cardinal numbers. (Contributed by RP,
+ 18-Nov-2023.) $)
+ alephiso2 $p |- aleph Isom _E , ~< ( On , { x e. ran card | _om C_ x } ) $=
+ ( vy vz con0 cv ccrd cfv wceq wa cab cep cale wiso csdm wf1o wb wral wcel
+ wbr df-isom com wss crn crab alephiso iscard4 anbi1ci abbii df-rab eqtr4i
+ f1oeq3 ax-mp wel alephon epelg alephord2 alephord 3bitr2d bibi2d ralbidva
+ mp1i ralbiia anbi12i 3bitr4i mpbi ) DUAAEZUBZVFFGVFHZIZAJZKKLMZDVGAFUCZUD
+ ZKNLMZAUEDVJLOZBEZCEZKSZVPLGZVQLGZKSZPZCDQZBDQZIDVMLOZVRVSVTNSZPZCDQZBDQZ
+ IVKVNVOWEWDWIVJVMHVOWEPVJVFVLRZVGIZAJVMVIWKAVHWJVGVFUFUGUHVGAVLUIUJVJVMDL
+ UKULWCWHBDVPDRZWBWGCDWLVQDRIZWAWFVRWMWAVSVTRZBCUMWFVTDRWAWNPWMVQUNVSVTDUO
+ VAVPVQUPVPVQUQURUSUTVBVCBCDVJKKLTBCDVMKNLTVDVE $.
+ $}
+
+ ${
+ $d x y $.
+ $( ` aleph ` is a strictly order-preserving mapping of ` On ` onto the
+ class of all infinite cardinal numbers. (Contributed by RP,
+ 18-Nov-2023.) $)
+ alephiso3 $p |- aleph Isom _E , ~< ( On , ( ran card \ _om ) ) $=
+ ( vx vy con0 com wss ccrd crn crab cep csdm cale wiso cdif alephiso2 wceq
+ cv wb wcel wn omelon cen wbr wral elrncard simplbi ontri1 sylancr rabbiia
+ dfdif2 eqtr4i isoeq5 ax-mp mpbi ) CDAPZEZAFGZHZIJKLZCUPDMZIJKLZANUQUSOURU
+ TQUQUNDRSZAUPHUSUOVAAUPUNUPRZDCRUNCRZUOVAQTVBVCBPUNUAUBSBUNUCBUNUDUEDUNUF
+ UGUHAUPDUIUJCUQUSIJKUKULUM $.
+ $}
+
$( May move to main section after ~ infdjuabs . $)
@@ -639504,7 +639622,7 @@ of all sets ( ~ inex1g ).
relation. (Contributed by RP, 30-Jul-2020.) $)
clrellem $p |- ( ph -> Rel |^| { x | ( X C_ x /\ ps ) } ) $=
( vy cv wss wa wrel ccnv cvv wcel 3syl wrex cint cnvexg wceq dfrel2 sylib
- wex cab cnvss eqsstr3d relcnv jca31 cleq2lem releq anbi12d rexab2 biimpri
+ wex cab cnvss eqsstrrd relcnv jca31 cleq2lem releq anbi12d rexab2 biimpri
a1i elabd relint ) AEDMZNBOZVAPZOZDUGZLMZPZLVBDUHZUAZVHUBPAVDEFQZQZNZCOZV
KPZODVKAFRSVJRSVKRSGFRUCVJRUCTAVLCVNAEEQZQZVKAEPVPEUDHEUEUFAEFNVOVJNVPVKN
JEFUIVOVJUITUJKVNAVJUKURULVAVKUDVBVMVCVNBCVAVKEIUMVAVKUNUOUSVIVEVBVGVCLDV
@@ -640017,7 +640135,7 @@ Transitive relations (not to be confused with transitive classes).
$( Concrete construction of a superclass of relation ` R ` which is a
transitive relation. (Contributed by RP, 20-Jul-2020.) $)
trrelsuperrel2dg $p |- ( ph -> ( R C_ S /\ ( S o. S ) C_ S ) ) $=
- ( wss ccom cdm crn cxp cun ssun1 syl5sseqr ccnv syl syl5eqssr sylib ax-mp
+ ( wss ccom cdm crn cxp cun ssun1 sseqtrrid ccnv syl syl5eqssr sylib ax-mp
wceq ssequn1 eqtri xptrrel ssun2 sstri a1i coeq12d coundir wrel cocnvcnv1
relcnv relssdmrn dmcnvcnv rncnvcnv xpeq12i syl6sseq coss1 cocnvcnv2 coss2
coundi syl6eq 3sstr4d jca ) ABCECCFZCEABBGZBHZIZJZBCBVEKDLAVEVEFZVFVBCVGV
@@ -640291,7 +640409,7 @@ natural numbers (including zero) is equivalent to the existence of at
$( A set is a subset of its image under the identity relation.
(Contributed by RP, 22-Jul-2020.) $)
fvilbd $p |- ( ph -> R C_ ( _I ` R ) ) $=
- ( cid cfv ssid cvv wcel wceq fvi syl syl5sseqr ) ABBBDEZBFABGHMBICBGJKL
+ ( cid cfv ssid cvv wcel wceq fvi syl sseqtrrid ) ABBBDEZBFABGHMBICBGJKL
$.
$}
@@ -640669,7 +640787,7 @@ over the natural numbers (including zero) is equivalent to the
( wcel cn wceq wa cv wss wi crelexp co c1 sseq1d imbi2d vx vy ccom ciun
cfv cvv simplr simpr oveq1d iuneq12d elex adantr nnex iunex a1i fvmptd2
ovex relexp1g anbi1d wral caddc oveq2 weq simprl w3a simp2l relexpaddnn
- simp1 1nn syl3anc simp2rr simp3 simp2rl trrelssd eqsstr3d 3exp ralrimiv
+ simp1 1nn syl3anc simp2rr simp3 simp2rl trrelssd eqsstrrd 3exp ralrimiv
a2d nnind com12 iunss sylibr ex sylbird sseq1 syl5ibr mpcom alrimiv ) B
EIZDJKZLZBFMZNZWLWLUCWLNZLZBAUEZWLNZOZFWPCJBCMZPQZUDZKZWKWRWKGBCDGMZWSP
QZUDXAUFAUFHWKXCBKZLZCDJXDWTWIWJXEUGXFXCBWSPWKXEUHUIUJWIBUFIWJBEUKULXAU
@@ -640773,7 +640891,7 @@ over the natural numbers (including zero) is equivalent to the
vx vy wa cun cres ccom w3a cfv ciun simplr simpr oveq1d iuneq12d adantr
cvv elex nn0ex ovex iunex a1i fvmptd2 cc0 relexp0g relexp1g 3anbi12d cn
wral wo elnn0 caddc weq simpr2 simp1 simp2l relexpaddnn syl3anc simp2r3
- 1nn simp3 simp2r2 trrelssd eqsstr3d 3exp a2d nnind simpr1 syl5ibr sylbi
+ 1nn simp3 simp2r2 trrelssd eqsstrrd 3exp a2d nnind simpr1 syl5ibr sylbi
jaoi com12 ralrimiv iunss sylibr ex sylbird sseq1 mpcom alrimiv ) BEIZD
JKZUFZUABUBBUCUGUHZFLZMZBXFMZXFXFUIXFMZUJZBAUKZXFMZNZFXKCJBCLZOPZULZKZX
DXMXDGBCDGLZXNOPZULXPURAURHXDXRBKZUFZCDJXSXOXBXCXTUMYAXRBXNOXDXTUNUOUPX
@@ -640838,7 +640956,7 @@ over the natural numbers (including zero) is equivalent to the
( wcel crelexp co cc0 wss wceq wi oveq2 oveq1d sseq1d imbi2d cid cun cres
c1 cvv ax-mp vx vy cn0 cn wo elnn0 cv weq relexp1g ssid syl6eqss w3a ccom
caddc simp2 simp1 wa relexpsucnnr syl2anc cdm crn ovex coexg relexp0g syl
- dmcoss rncoss unss12 mp2an ssres2 resundi ssun1 syl5sseqr adantr sseqtr4i
+ dmcoss rncoss unss12 mp2an ssres2 resundi ssun1 sseqtrrid adantr sseqtr4i
mpan ssun2 simpr unssd syl5eqss 3adant1 sstrd eqsstrd 3exp a2d relexp0idm
syl5ss nnind sylan9eq eqimss ex jaoi sylbi impcom ) BUCDZACDZABEFZGEFZAGE
FZHZWOBUDDZBGIZUEWPWTJZBUFXAXCXBWPAUAUGZEFZGEFZWSHZJWPAREFZGEFZWSHZJWPAUB
@@ -640896,7 +641014,7 @@ over the natural numbers (including zero) is equivalent to the
( wcel crelexp co ccom caddc wss cc0 wceq w3a syl cres ccnv oveq2d 3ad2ant3
c1 eqtrd cn0 cn wo wi elnn0 biimpi relexpaddnn eqimss 3exp cuz cfv elnn1uz2
c2 cid cdm crn wrel relco dfrel2 ax-mp cnvco cnvresid coeq2i coires1 3eqtri
- cun cnvss resss sstri eqsstr3i cnvcnvss a1i simp1 relexp0g relexp1g coeq12d
+ cun cnvss resss sstri eqsstrri cnvcnvss a1i simp1 relexp0g relexp1g coeq12d
simp2 oveq12d addid2d 3sstr4d cnvexg relexpuzrel syl2anc eluz2nn relexpnndm
1cnd cvv df-rn ssun2 syl6ss relssres syl5eq eluzge2nn0 relexpcnv 3eqtr4d wb
simp3 3adant1 cnveqb sylancr mpbird coeq1d oveq1d eluzelcn jaod syl5bi jaoi
@@ -640993,7 +641111,7 @@ over the natural numbers (including zero) is equivalent to the
dftrcl3 $p |- t+ = ( r e. _V |-> U_ n e. NN ( r ^r n ) ) $=
( vz va vk vt vs cvv cv wss ccom wa cmpt cn crelexp co ciun wcel c1 wceq
ctcl cab cint df-trcl cfv relexp1g 1nn oveq1 iuneq2d oveq2 cbviunv syl6eq
- nnex weq cbvmptv ov2ssiunov2 mp3an23 eqsstr3d cuz nnuz 1nn0 iunrelexpuztr
+ nnex weq cbvmptv ov2ssiunov2 mp3an23 eqsstrrd cuz nnuz 1nn0 iunrelexpuztr
cn0 wb wi wal fvex trcleq2lem a1i alrimiv elabgt sylancr mpbir2and intss1
syl wral vex elab eqid iunrelexpmin1 mpan2 19.21bi syl5bi ralrimiv sylibr
ssint eqssd ovex iunex fvmpt eqtrd mpteq2ia eqtri ) UABHBIZCIZJWOWOKWOJLZ
@@ -641116,7 +641234,7 @@ over the natural numbers (including zero) is equivalent to the
ctcl ccom cint csn cun cop df-br trclfv 3ad2ant3 elimasng 3adant3 3bitr4d
a1i breqd wrex intimasn 3ad2ant1 wal wex cxp simpl3 snex vex xpex sylancl
wi unexg trclfvlb unssad trclfvcotrg cin wne simpl1 inelcm syl2anc xpima2
- c0 snidg unssbd imass1 eqsstr3d imaundir simpr imassrn rnxpss sstri unssd
+ c0 snidg unssbd imass1 eqsstrrd imaundir simpr imassrn rnxpss sstri unssd
crn syl5eqss trclimalb2 eqssd sbcan csb sbcssg csbconstg csbvargi sseq12i
fvex bitri csbcog coeq12i anbi12i sbceq2g csbima12 imaeq1i imaeq2i 3eqtri
eqtri eqeq2i sylbbr syl21anc spesbcd ex eqeq1 imaeq1 eqeq2d rexab2 syl6bb
@@ -641217,7 +641335,7 @@ over the natural numbers (including zero) is equivalent to the
dfrtrcl3 $p |- t* = ( r e. _V |-> U_ n e. NN0 ( r ^r n ) ) $=
( vz va vk vt vs cvv cv wss ccom w3a cmpt cn0 crelexp ciun wcel cc0 sseq2
co crtcl cid cdm crn cun cres cab cint df-rtrcl relexp0g nn0ex 0nn0 oveq1
- cfv weq iuneq2d oveq2 cbviunv syl6eq cbvmptv ov2ssiunov2 mp3an23 eqsstr3d
+ cfv weq iuneq2d oveq2 cbviunv syl6eq cbvmptv ov2ssiunov2 mp3an23 eqsstrrd
c1 relexp1g 1nn0 cuz wceq nn0uz iunrelexpuztr wb wal fvex coeq12d sseq12d
wi id 3anbi123d a1i alrimiv elabgt sylancr mpbir3and intss1 syl wral elab
eqid iunrelexpmin2 mpan2 19.21bi syl5bi ralrimiv ssint sylibr eqssd iunex
@@ -641303,7 +641421,7 @@ over the natural numbers (including zero) is equivalent to the
( vi vj vk vd crelexp cc0 c1 cn cn0 cun wss wceq wcel ax-mp cv ciun oveq2
1nn co cvv va vb vc cpr crcl ctcl crtcl dfrcl4 dftrcl3 dfrtrcl3 prex nnex
df-n0 uncom df-pr uneq1i unass ssequn1 mpbi uneq2i 3eqtrri 3eqtri cbviunv
- csn snssi ss2iun relexp1g elv ssiun2s eqsstr3i ovex iunex 0nn0 1nn0 prssi
+ csn snssi ss2iun relexp1g elv ssiun2s eqsstrri ovex iunex 0nn0 1nn0 prssi
a1i mp2an sseli relexpss1d mprg eqsstri eqtr4i 1ex prid2 c0ex ssid unss12
prid1 iuneq1 iunxun iunxsn cin c0 wne vex nnssnn0 inelcm iunrelexp0 mp3an
eqtri uneq12i 3sstr4i comptiunov2i ) ABCEFGUDZHIUEUFUGUAUBUCDAUAUHBUBUICU
@@ -647915,7 +648033,7 @@ base set if and only if the neighborhoods (convergents) of every point
$( The image of a mapping from A is nonempty if A is nonempty.
(Contributed by Stanislas Polu, 9-Mar-2020.) $)
wnefimgd $p |- ( ph -> ( F " A ) =/= (/) ) $=
- ( cdm cin wss wceq ssid fdmd syl5sseqr sseqin2 sylib eqnetrd imadisjlnd
+ ( cdm cin wss wceq ssid fdmd sseqtrrid sseqin2 sylib eqnetrd imadisjlnd
c0 ) ADBADGZBHZBRABSITBJABBSBKABCDFLMBSNOEPQ $.
$}
@@ -649434,7 +649552,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing
( va cv wcel c0 cuni wa cpr wceq wss wrex csn eleq1 rexbidv 0ex prid1 a1i
anbi1d rspcdva adantr wn simprl simpr 0nep0 snnzg necomd nelprd elnelneqd
wne syl adantrr adantrl wo elpri eleq2s orcomd ord sylc unieqd snex unipr
- cun uncom un0 3eqtri syl6eq simprrr eqsstr3d snssg biimprd eleq2w anbi12d
+ cun uncom un0 3eqtri syl6eq simprrr eqsstrrd snssg biimprd eleq2w anbi12d
unieq sseq1d rexlimddvcbv ) ADBNZOZPMNZOZWIQZWGUAZRZPCNZOZWNQZWGUAZRZCMHA
GNZWNOZWQRZCHUBWRCHUBGPPUCZSZPWSPTZXAWRCHXDWTWOWQWSPWNUDUIUELPXCOAPXBUFUG
UHUJAWIHOZWMRZRZDFOZDUCZWGUAZWHAXHXFJUKXGXIWKWGXGWKPXISZQZXIXGWIXKXGXEWIX
@@ -649457,7 +649575,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing
( va cv wcel c0 csn wa cpr wceq cuni wrex eleq1 anbi1d rexbidv p0ex prid2
wss a1i rspcdva simpl simprl simpr wne 0nep0 necomi 0ex sneqr eqcomd nsyl
wn neqned nelprd adantr elnelneqd adantrr adantrl elpri eleq2s ord unieqd
- wo sylc cun snex unipr df-pr eqtr4i syl6eq simprrr eqsstr3d prssg sylancr
+ wo sylc cun snex unipr df-pr eqtr4i syl6eq simprrr eqsstrrd prssg sylancr
cvv wb biimprd simprd eleq2w unieq sseq1d anbi12d rexlimddvcbv ) AEBNZOZP
QZMNZOZWPUAZWMUHZRZWOCNZOZXAUAZWMUHZRZCMHAGNZXAOZXDRZCHUBXECHUBGPWOSZWOXF
WOTZXHXECHXJXGXBXDXFWOXAUCUDUELWOXIOAPWOUFUGUIUJAWPHOZWTRZRZPWMOZWNXMAPES
@@ -649702,7 +649820,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing
$( Lemma for ~ grumnud . (Contributed by Rohan Ridenour, 13-Aug-2023.) $)
grumnudlem $p |- ( ph -> G e. M ) $=
( vw vv va wcel cv cpw wss wel wa wrex cuni wi wral wal cgru gruss 3expia
- syl3an1 alrimiv pwss ccoll cun wceq w3a ssun1 simp3 syl5sseqr simp1l3 wex
+ syl3an1 alrimiv pwss ccoll cun wceq w3a ssun1 simp3 sseqtrrid simp1l3 wex
sylibr wbr simp1r simpr unieqd simpl eqtr4d adantll simpll3 simprd simpld
weq eqeltrd eleqtrrd 3jca simpl2 rr-spce simp1l1 syl simp2 gruuni syl2anc
rr-rspce simpl1 gruel syl3anc 3ad2ant1 sylan rexbidva mpbird rexex adantr
@@ -660848,7 +660966,7 @@ reference definition (axiom) in set.mm. Also, a theorem or
( vu vv vz copab cvv cv wcel wa vex cop wceq wex cab exbii eximi biimpi
wi cxp wss wrel pm3.2i a1i ssopab2i biantru abbii ax6ev equcom mpbi opeq2
idn1 e1a idn2 opeq1 eqeq1 biimprd e12 eqeq2 biimpd in2 in1 19.37iv 19.37v
- e2 ax-mp 19.9v ss2abi eqsstr3i sseqtri df-opab 3sstr4i df-xp eqcomi sstri
+ e2 ax-mp 19.9v ss2abi eqsstrri sseqtri df-opab 3sstr4i df-xp eqcomi sstri
syl df-rel biimpri e0a ) ABCGZHHUAZUBZWAUCZWABIZHJZCIZHJZKZBCGZWBAWIBCWIA
WFWHBLCLUDZUEUFWJDIZHJZEIZHJZKZDEGZWBFIZWEWGMZNZWIKZCOZBOZFPZWRWLWNMZNZWP
KZEOZDOZFPZWJWQXDXFEOZDOZFPZXJXDWTCOZBOZFPXMXOXCFXNXBBWTXACWIWTWKUGQQUHXO
@@ -671971,7 +672089,7 @@ closed under the multiplication ( ' X ' ) of a finite number of
mullimc $p |- ( ph -> ( X x. Y ) e. ( H limCC D ) ) $=
( cc crp vz vy vw vc va vd vb ve vf cmul climc wcel wne cmin cabs cfv clt
co cv wbr wa wi wral limccl sseldi mulcld simpr adantr mulcn2 syl3anc w3a
- wrex fmptd cdm dmmptd wf wss limcrcl simp2d eqsstr3d simp3d ellimc3 mpbid
+ wrex fmptd cdm dmmptd wf wss limcrcl simp2d eqsstrrd simp3d ellimc3 mpbid
syl simprd r19.21bi adantrr adantrl reeanv sylanbrc cle cif ifcl 3ad2ant2
nfv nfra1 nfan nf3an simp11l simp1rl 3ad2ant1 simp12 simp13l jca31 simp1r
simp2 simp3l simplll simp1lr simp3r simp1l sselda syl2anc cr rpre ltletrd
@@ -672089,7 +672207,7 @@ closed under the multiplication ( ' X ' ) of a finite number of
sseq1 anbi12d rspcev syl12anc ioossre jctil elioore snssd sylancr 3adant1
wceq isnei mpbird neeq1d rspccva syl2anc ralrimivv simplbda eleq2i biimpi
ineq1 3ad2ant2 simp1 simp3l simpr adantr snssg syl jca tg2 cxp cpw wf wfn
- 3exp ioof ovelrn mp2b simpll eleqtrd simplr eqsstr3d ex reximdv rexlimiva
+ 3exp ioof ovelrn mp2b simpll eleqtrd simplr eqsstrrd ex reximdv rexlimiva
ffn 3syl simpl3r sstr expcom anim2d reximdva rexlimdv adantlr nfra1 nfra2
nfan inss1 simp3r syl5ss inss2 ssind simpllr 3ad2ant1 simp2 rsp2 rexlimd
mpd syl3c ssn0 ralrimiva impbida bitrd ) ACBDUBLLMZJUHZBCUCZUDZUEZUIUFZJU
@@ -672230,7 +672348,7 @@ closed under the multiplication ( ' X ' ) of a finite number of
mullimcf $p |- ( ph -> ( B x. C ) e. ( H limCC D ) ) $=
( vz wcel cc clt wa crp vy vw vc va vd vb ve vf cmul co climc cv wne cmin
cabs cfv wbr wi wral limccl sseldi mulcld simpr adantr mulcn2 syl3anc w3a
- wrex cdm fdmd wss limcrcl syl simp2d eqsstr3d simp3d ellimc3 mpbid simprd
+ wrex cdm fdmd wss limcrcl syl simp2d eqsstrrd simp3d ellimc3 mpbid simprd
r19.21bi adantrr adantrl reeanv sylanbrc cle cif ifcl 3ad2ant2 nfra1 nfan
nfv nf3an simp11l simp1rl 3ad2ant1 jca simp12 simp13l jca31 simp1r simp3l
simp2 simplll simp1lr simp3r simp1l sselda syl2anc subcld abscld ad2antrl
@@ -672661,7 +672779,7 @@ closed under the multiplication ( ' X ' ) of a finite number of
$( The domain of a maps-to function with a limit. (Contributed by Glauco
Siliprandi, 11-Dec-2019.) $)
limcmptdm $p |- ( ph -> A C_ CC ) $=
- ( cdm cc dmmptd wf wss wcel climc co w3a limcrcl syl simp2d eqsstr3d ) AC
+ ( cdm cc dmmptd wf wss wcel climc co w3a limcrcl syl simp2d eqsstrrd ) AC
GKZLABGCDLHIMAUDLGNZUDLOZFLPZAEGFQRPUEUFUGSJFEGTUAUBUC $.
$}
@@ -678597,7 +678715,7 @@ distinct definitions for the same symbol (limit of a sequence).
mpdan leloe ltpnfd ssinss1 ioossre unima 3eqtrd 3adant2 elinel2 mpbir3and
xrnltled anim2i adantll simplbda 3adant3 3adant1r eqsstrd eqeq1d pm2.65da
simp1 ltled lenlt ltpnf ancom ancli vtoclg sylan9eqr neqne necomd leneltd
- sylc wne condan pm5.6 3adantll2 olci inss ax-mp sseli ralrimiva syl5sseqr
+ sylc wne condan pm5.6 3adantll2 olci inss ax-mp sseli ralrimiva sseqtrrid
funimass4 eqeltrrd nelneq necon3bi ad4antlr simp2d leltne simp3d ioossicc
necom inss2 sstri sslin rexlimdv dfss3 syld3an1 iscnp cncnp fnssres fvres
3exp eqfnfvd ) AFGHEUUAZUUBUGZUUCUGSZFCDUUDUGZUUEZEUHAWUDUIWUBFUUFZFUAUJZ
@@ -679038,7 +679156,7 @@ distinct definitions for the same symbol (limit of a sequence).
a1i cfv cif cxr rexrd iccid syl oveq2 sylan9req eqcomd eleqtrd mpteq12dva
elsni iftrued syl5eq recnd cncfdmsn eqeltrd sseldd oveq1d simpll wo eqcom
adantlr biimpi con3i adantl simplr pm4.56 lttrid mpbird ccnfld ctopn eqid
- ccn 0ss crest ctop cnfldtop rest0 ax-mp eqcomi cncfcn mp2an eqsstr3i icc0
+ ccn 0ss crest ctop cnfldtop rest0 ax-mp eqcomi cncfcn mp2an eqsstrri icc0
wb mpteq1d mpt0 3eqtrd 0cnf syl6eqel pm2.61dan ) ACDUAUBZGCDUCPZQUDPZRZAX
QUEZBCDEFGHYABUJJACUFRZXQKSADUFRZXQLSAXQUGAFCDUHPQUDPRXQMSAHFDUKPRXQNSAEF
CUKPZRXQOSUIAXQULZUEZCDUMZXTAYGXTYEAYGUEZGCUNZQUDPZXSYHYIEUNZUDPZYJGAYLYJ
@@ -679561,7 +679679,7 @@ distinct definitions for the same symbol (limit of a sequence).
( cres cdv co cfv cc wss wceq cnt wf syl22anc wfn ffn fnresdm 3syl oveq2d
dvres ctop wcel cuni crest cnfldtopon resttopon sylancr syl5eqel topontop
ctopon toponuni sseqtrd eqid ntridm syl2anc fveq2d 3eqtr3d reseq2d ntrss2
- syl eqsstr3d sstrd 3eqtr4rd ) ABCFNZOPZBCOPZFDUAQZQZNZVOBCGNOPZABRSZFRCUB
+ syl eqsstrrd sstrd 3eqtr4rd ) ABCFNZOPZBCOPZFDUAQZQZNZVOBCGNOPZABRSZFRCUB
ZFBSZWBVNVRTHJIIFFBDCELKUIUCAVMCBOAWACFUDVMCTJFRCUEFCUFUGUHAVOGVPQZNZVOGN
VSVRAWCGVOAVQVPQZVQWCGADUJUKZFDULZSZWEVQTADBUSQZUKZWFADEBUMPZWIKAERUSQUKV
TWKWIUKELUNHBERUOUPUQZBDURVIZAFBWGIAWJBWGTWLBDUTVIVAZFDWGWGVBZVCVDAVQGVPM
@@ -682142,7 +682260,7 @@ distinct definitions for the same symbol (limit of a sequence).
Glauco Siliprandi, 11-Dec-2019.) $)
iblsplit $p |- ( ph -> ( x e. U |-> C ) e. L^1 ) $=
( vk cmpt wcel cr cc0 cfv wa cc eqidd vy cibl cmbf cv ci cexp co cdiv cre
- cle wbr cif citg2 c3 cfz wral fmpttd cres cun syl5sseqr resmptd imdistani
+ cle wbr cif citg2 c3 cfz wral fmpttd cres cun sseqtrrid resmptd imdistani
ssun1 sseld isibl2 mpbid simpld eqeltrd ssun2 eqcomd mbfres2cn caddc cvol
syl cdm mbfdm2 adantr cin covol wceq cpnf cicc cxr adantlr ax-icn elfznn0
a1i expcld ad2antlr wne cz elfzelz expne0d divcld recld rexrd simpr pnfge
@@ -682919,7 +683037,7 @@ distinct definitions for the same symbol (limit of a sequence).
wral fdmd cncfperiod elrab simprr nfre1 nfan 3jca 3ad2ant1 mpbird rexlimd
nfv 3exp mpd sylan2b cmin resubcld pncand eqcomd lesub1dd eqbrtrd breqtrd
reseq2d oveq1d cibl a1i 1cnd ssid cvol cdv resmptd eqtrd cc0 3eqtrd eqrdv
- rspceeqv sylanbrc impbida eqsstr3d feqresmpt iccshift 3eltr4d ioosscn csn
+ rspceeqv sylanbrc impbida eqsstrrd feqresmpt iccshift 3eltr4d ioosscn csn
npcand constcncfg cxp fconstmpt ioovolcl iblconst syl5eqelr elind crn ctg
ioombl cnt oveq2d addcld fmpttd ccnfld ctopn tgioo2 dvres syl22anc iccntr
cpr reelprrecn dvmptid 0cnd dvmptc dvmptadd reseq1d ioossre 1p0e1 mpteq2i
@@ -689006,7 +689124,7 @@ approximated is nonnegative (this assumption is removed in a later
wss ax-resscn resmpt 3eqtrd ccncf constcncfg cncfmpt1f cncfmptssg addcncf
cdm fsumcncf difssd eldifsn mpbir2an divcncf eqeltrd cibl halfcld clt wbr
recn zcnd 0red elfzle1 ltletrd gt0ne0d fsumcl addcld dvmptid tgioo2 reopn
- 0lt1 2cnd simpr coscld mp1i eqcomd fmpttd wral ralrimiva dmmptg syl5sseqr
+ 0lt1 2cnd simpr coscld mp1i eqcomd fmpttd wral ralrimiva dmmptg sseqtrrid
wf dvsinax dmeqd sseqtr4d dvcnre reseq1d ax-mp syl6eq mpteq2dva dvmptfsum
divcan3d eqtrd dvmptadd iccssred iccntr dvmptres2 syl5eq fvmpt2d itgeq2dv
cnt 3eqtr4d ioosscn ssid coscn mulc1cncf csn cdif cncfmptc mp3an ioossicc
@@ -689169,7 +689287,7 @@ approximated is nonnegative (this assumption is removed in a later
ioossre cn nnred readdcld sselda remulcld resincld 2re remulcli rehalfcld
pire crn ctg iooretop cnt nncnd pm3.2i ccnfld ctopn tgioo2 dvres syl21anc
retop ccld cha rehaus elioored uniretop difopn isopn3i reseq2d reelprrecn
- sncld dvsinax syl5sseqr dvres3 syl22anc syl6eq eqimss oveq2i mulcli sseli
+ sncld dvsinax sseqtrrid dvres3 syl22anc syl6eq eqimss oveq2i mulcli sseli
cpr 2cn 1cnd div13d halfcl mulid1d dvasinbx mul32d recidd mulid2d resmptd
ccom wfn ralrimiva fnmpt eldifi eleq1w anbi2d neeq1d imbi12d elioore 3syl
oveq1 0re pipos gtneii sineq0 mtbid eqneltrd crp 2rp rpmulcl mp2an neqned
@@ -695699,7 +695817,7 @@ approximated is nonnegative (this assumption is removed in a later
crab rexlimdv ralrimiva sylibr simpll fourierdlem15 ioossicc fourierdlem1
3exp rabss sseli cncfperiod elrab simprbi nfre1 nfan elioo2 simp2d simp3d
nfv 3jca rexlimd mpd pncand ltsub1dd eqbrtrd npcand rspcev sylanbrc eqrdv
- breqtrd impbida reseq2d feqresmpt 3eltr3d syl5sseqr sseldi anbi2d eqeq12d
+ breqtrd impbida reseq2d feqresmpt 3eltr3d sseqtrrid sseldi anbi2d eqeq12d
eleq1 imbi12d limcperiod eleqtrd eliccre adantll ad2antlr iccgelb iccleub
chvarv necomd fvres 3eqtrrd mpteq2dva syl5eq syl6eq iftrued lbicc2 limccl
ltled gtned neneqd ubicc2 3eqtr4rd ad4ant14 stoic1a eliccd fssresd neqned
@@ -696650,7 +696768,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u
cioc simp3d breq12d rspccva ancli eleq1 anbi2d eqeq12d eleq1i rgenw rabbi
rexbii mpbi fourierdlem65 breqtrrd id sseq12d fourierdlem79 fourierdlem10
anabsi7 wne neqne adantl leneltd ltletrd eliood fvres ifeq2da cdm ioosscn
- fdm feq2d ioossre syl5sseqr syl5eqel iooshift eqsstr3d syl6breqr divcan1d
+ fdm feq2d ioossre sseqtrrid syl5eqel iooshift eqsstrrd syl6breqr divcan1d
c2 gt0ne0d negsubdi2d cmpt oveq2 redivcld remulcld readdcld fvmptd mulcld
flcld zred pncan2d divnegd 3eqtr4d divcan4d znegcld adantlr sylan2 ccnfld
simpr fperiodmul ctopn cico crest reseq2d eleq12d nfmpt1 nfcxfr nfcv nffv
@@ -696931,7 +697049,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u
simp3d breq12d rspccva eleq1 anbi2d eleq1i rgenw rabbi mpbi fourierdlem65
eqeq12d rexbii anabsi7 mpdan breqtrrd sseq12d fourierdlem79 fourierdlem10
c2 lelttrd wne neqne necomd adantl leneltd eliood fvres ifeq2da cdm feq2d
- fdm ioosscn ioossre syl5sseqr syl5eqel iooshift eqsstr3d gt0ne0d divcan1d
+ fdm ioosscn ioossre sseqtrrid syl5eqel iooshift eqsstrrd gt0ne0d divcan1d
syl6breqr negsubdi2d cmpt id oveq2 redivcld zred remulcld readdcld fvmptd
flcld zcnd mulcld pncan2d divnegd 3eqtr4d divcan4d znegcld adantlr sylan2
simpr fperiodmul ccnfld ctopn csn crest ancli reseq2d eleq12d sylc nfmpt1
@@ -697067,7 +697185,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u
vtoclg npcand ioossre feqresmpt iooshift mpteq1d biimpar elioore resubcld
eqtr3d eleq2d 3adant2 simp3 ioogtlb iooltub fourierdlem15 lelttrd ltletrd
eliood fvres ltled recn 3eqtr2rd mpteq2dva 3eqtrd ioosscn rexbidv cbvrexv
- eqeq2d syl6bb cbvrabv cncfshift eqeltrd cdm ffdm syl5sseqr eqcomi 3sstr4d
+ eqeq2d syl6bb cbvrabv cncfshift eqeltrd cdm ffdm sseqtrrid eqcomi 3sstr4d
syl5eqssr ioossicc sseli fourierdlem1 limcperiod syl5eq reseq2d pm2.61dan
fdm ) AUNIUOUPZBCIUQURZDIUQURZUSURZBUTZLVAZVBZBCDUSURZVUEVBZVCZAUYTVDZBCD
EFGUJUNOVEURZUJUTZFVAZIUQURZVFZIJKLUKJUTZFVAZVUPUUAUQURZFVAZUSURZUKUTZVUQ
@@ -699787,7 +699905,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u
jca32 vtoclg sylc dirkerper adantll resmpt ax-mp ccom rexrd breq2d npcand
reseq1i addcomd ioogtlb breq1d iooltub eliood fmptco constcncfg cncfmptid
ltadd2dd ssid addcncf ioosscn cncfmptssg cncfco feqmptd cncfss dirkercncf
- mp2an sseldi cncff mulcncf csn cdif crn wne syl5sseqr ssdmres sylib gtned
+ mp2an sseldi cncff mulcncf csn cdif crn wne sseqtrrid ssdmres sylib gtned
fdm eldifsn sylanbrc rnmptss eqidd leidd iccssred resmptd rescncf cnlimci
syl6ss eqcomi addcld limciccioolb limccog mullimcf adantllr ltned 3eltr3d
fcompt limcicciooub resabs1 fourierdlem110 subnegd itgeq1d 3eqtr3a ltleii
@@ -701593,7 +701711,7 @@ those for the more general case of a piecewise smooth function (see
( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) ) = ( F
` X ) ) $=
( co cr cc cioo crn ctg cfv cpi cneg cdm cdif cfn cdv cres dmeqi wss wceq
- c0 ioossre ccncf wcel wf cncff fdm syl5sseqr ssdmres sylib syl5eq difeq2d
+ c0 ioossre ccncf wcel wf cncff fdm sseqtrrid ssdmres sylib syl5eq difeq2d
3syl difid syl6eq 0fin syl6eqel rescncf mpsyl oveq1d 3eltr4d cv cico cpnf
a1i climc wne pire renegcli rexri icossre mp2an eldifi sseldi wa limcresi
cxr reseq1i resres eqtr2i oveq1i sseqtri adantr simpr cnlimci ne0d sylan2
@@ -713253,7 +713371,7 @@ its dimensional volume (the product of its length in each dimension,
( ( i ( H ` X ) ( B ` i ) ) \ ( i ( H ` X ) ( A ` i ) ) ) ) $=
( vk wcel wceq wa cr adantlr vf vh cv cfv cico co cixp cdif ciin wal wral
wb nfv nfcv nfixp1 nfel nfan cmnf cif wfn ixpfn ad2antlr wss fveq2 oveq2d
- cioo iftrue eqtr4d eqimss syl ioossre iffalse syl5sseqr pm2.61i cxr mnfxr
+ cioo iftrue eqtr4d eqimss syl ioossre iffalse sseqtrrid pm2.61i cxr mnfxr
wn a1i ffvelrnda rexrd icossre syl2anc simpl simpr oveq12d adantll sseldd
fvixp mnfltd clt wbr icoltub syl3anc eliood sseldi ralrimiva elixp sylibr
jca vex cfn cmpo cmpt equequ1 ifbid cbvixpv mpoeq3ia mpteq2i eqtri adantr
@@ -725765,7 +725883,7 @@ The modulo (remainder) operation - extension
elfzle1 1red elfzel1 zred letr syl3anc mpan2d syl5com syl3anbrc elfzel2
imp ad4antr nnred elfzle2 elfzolt2 lelttrd elfzo2 iccpartipre cmin cmap
cr cxr ad3antrrr fzoval wss elfzo0le 0le1 0red mpani sylbid mpd 0zd jca
- elfzoel2 ssfzo12bi mpbird eqsstr3d sselda iccpartimp simprd exp31 com23
+ elfzoel2 ssfzo12bi mpbird eqsstrrd sselda iccpartimp simprd exp31 com23
smonoord elfzuz iccpartiltu breq2 3imtr4d exp4c com13 com3r sylbi imp32
anbi2d ralrimivva ) ACUFZDUFZIJZUVEBKZUVFBKZIJZLZCDMEUANZUVLAUVEUVLOZUV
FUVLOZUVKAUVMUVEMUBZPEUANZUCZOZUVNUVKLZAUVLUVQUVEAUVLUVOMPUDNZEUANZUCZU
@@ -726266,6 +726384,26 @@ valid cases ( ` (/) ` is the last symbol) and invalid cases ( ` (/) `
$.
$}
+ ${
+ $d a x ph $. $d a y ph $.
+ $( Setvar variables are interchangeable in a wff they do not appear in.
+ (Contributed by SN, 23-Nov-2023.) $)
+ ichv $p |- [ x <> y ] ph $=
+ ( va wich wsb wb wal sbv sbbii bitri gen2 df-ich mpbir ) ABCEACDFZBCFZDBF
+ ZAGZCHBHRBCQADBFAPADBPABCFAOABCACDIJABCIKJADBIKLABCDMN $.
+ $}
+
+ ${
+ $d a x $. $d a y $. $d a ph $.
+ ichf.1 $e |- F/ x ph $.
+ ichf.2 $e |- F/ y ph $.
+ $( Setvar variables are interchangeable in a wff they are not free in.
+ (Contributed by SN, 23-Nov-2023.) $)
+ ichf $p |- [ x <> y ] ph $=
+ ( va wich wsb wb wal sbf sbbii bitri sbv gen2 df-ich mpbir ) ABCGACFHZBCH
+ ZFBHZAIZCJBJUABCTAFBHASAFBSABCHARABCACFEKLABCDKMLAFBNMOABCFPQ $.
+ $}
+
${
$d a ph $. $d a x $.
$( A setvar variable is always interchangeable with itself. (Contributed
@@ -726451,6 +726589,16 @@ valid cases ( ` (/) ` is the last symbol) and invalid cases ( ` (/) `
DETUAUJCDETUB $.
$}
+ $( Two setvar variables are always interchangeable when there are two
+ universal quantifiers. (Contributed by SN, 23-Nov-2023.) $)
+ ich2al $p |- [ x <> y ] A. x A. y ph $=
+ ( wal nfa1 nfa2 ichf ) ACDZBDBCHBEACBFG $.
+
+ $( Two setvar variables are always interchangeable when there are two
+ existential quantifiers. (Contributed by SN, 23-Nov-2023.) $)
+ ich2ex $p |- [ x <> y ] E. x E. y ph $=
+ ( wex nfe1 excom nfxfr ichf ) ACDZBDZBCIBEJABDZCDCABCFKCEGH $.
+
${
$d ph x y $. $d ps x y $. $d a b x y $.
$( If two setvar variables are interchangeable in two wffs, then they are
@@ -734110,7 +734258,7 @@ their group (addition) operations are equal for all pairs of elements of
resmgmhm2 $p |- ( ( F e. ( S MgmHom U ) /\ X e. ( SubMgm ` T ) )
-> F e. ( S MgmHom T ) ) $=
( vx vy cmgmhm co wcel cfv wa cmgm cbs wf cv cplusg wceq eqid wral simpld
- csubmgm mgmhmrcl submgmrcl anim12i wss mgmhmf submgmbas submgmss eqsstr3d
+ csubmgm mgmhmrcl submgmrcl anim12i wss mgmhmf submgmbas submgmss eqsstrrd
fss syl2an mgmhmlin 3expb adantlr ressplusg ad2antlr oveqd ralrimivva jca
eqtr4d ismgmhm sylanbrc ) DACIJKZEBUCLZKZMZANKZBNKZMAOLZBOLZDPZGQZHQZARLZ
JDLZVNDLZVODLZBRLZJZSZHVKUAGVKUAZMDABIJKVEVIVGVJVEVICNKACDUDUBEBUEUFVHVMW
@@ -738374,7 +738522,7 @@ and the morphisms (ring homomorphisms) to mappings of the corresponding
srhmsubc $p |- ( U e. V -> J e. ( Subcat ` ( RingCat ` U ) ) ) $=
( vx vy wcel cfv co wa crg crh wceq adantr vg vf vz cringc csubc cssc wbr
chomf cv ccid cop cco wral cin wss vtoclri ssriv sslin mp1i syl5eqss chom
- eleq1w ssid cbs simpl srhmsubclem2 adantrr adantrl ringchom syl5sseqr cvv
+ eleq1w ssid cbs simpl srhmsubclem2 adantrr adantrl ringchom sseqtrrid cvv
eqid cmpo a1i oveq12 adantl simprl simprr ovexd ovmpod homfval ralrimivva
3sstr4d cxp wfn ovex fnmpoi homffn id eqcomd sqxpeqd fneq2d mpbiri inex1g
ringcbas isssc mpbir2and cid cres elin2 sylbi idrhm ringcid simpr 3eltr4d
@@ -738621,7 +738769,7 @@ homomorphisms between unital rings (in the same universe). (Contributed
srhmsubcALTV $p |- ( U e. V -> J e. ( Subcat ` ( RingCatALTV ` U ) ) ) $=
( vx vy wcel cfv co wa crg crh wceq adantr vg vf vz cringcALTV csubc cssc
chomf wbr cv ccid cop cco wral cin wss eleq1w vtoclri ssriv mp1i syl5eqss
- sslin chom ssid cbs eqid simpl srhmsubcALTVlem1 adantrr adantrl syl5sseqr
+ sslin chom ssid cbs eqid simpl srhmsubcALTVlem1 adantrr adantrl sseqtrrid
ringchomALTV cvv oveq12 adantl simprl simprr ovexd ovmpod homfval 3sstr4d
cmpo a1i ralrimivva cxp ovex fnmpoi homffn id ringcbasALTV eqcomd sqxpeqd
wfn fneq2d mpbiri inex1g isssc mpbir2and cid cres elin2 sylbi ringcidALTV
@@ -739444,7 +739592,7 @@ Ordered group sum operation (extension)
$( Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.) $)
gsumsplit2f $p |- ( ph -> ( G gsum ( k e. A |-> X ) ) =
( ( G gsum ( k e. C |-> X ) ) .+ ( G gsum ( k e. D |-> X ) ) ) ) $=
- ( cmpt cgsu cres eqid fmptdf gsumsplit cun ssun1 syl5sseqr resmptd oveq2d
+ ( cmpt cgsu cres eqid fmptdf gsumsplit cun ssun1 sseqtrrid resmptd oveq2d
co ssun2 oveq12d eqtrd ) AHGBJUBZUCUMHUQDUDZUCUMZHUQEUDZUCUMZFUMHGDJUBZUC
UMZHGEJUBZUCUMZFUMABCDEFUQHIKMNOPQAGBJCUQLRUQUEUFSTUAUGAUSVCVAVEFAURVBHUC
AGBDJADEUHZDBDEUIUAUJUKULAUTVDHUCAGBEJAVFEBEDUNUAUJUKULUOUP $.
@@ -743261,7 +743409,7 @@ The natural logarithm on complex numbers (extension)
|-> ( ( F ` x ) / ( G ` x ) ) ) ) $=
( cc wf wcel co cc0 csupp cres cdiv cfv cvv wceq fex eqtrd syl2anc wfn cv
w3a cfdiv cof cmpt 3adant2 3adant1 wa fdivval offres wss ffn 3ad2ant1 cdm
- suppssdm fdm eqcomd 3ad2ant2 syl5sseqr fnssres ovexd inidm adantl offval
+ suppssdm fdm eqcomd 3ad2ant2 sseqtrrid fnssres ovexd inidm adantl offval
fvres ) BFCGZBFDGZBEHZUBZCDUCIZCDJKIZLZDVKLZMUDZIZAVKAUAZCNZVPDNZMIUEVICO
HZDOHZVJVOPVFVHVSVGBFECQUFVGVHVTVFBFEDQUGVSVTUHVJCDVNIVKLVOCDOOUIVKMCDOOU
JRSVIAVKVKVQVRMVKVLVMOOVICBTZVKBUKZVLVKTVFVGWAVHBFCULUMVIDUNZVKBDJUOVGVFB
@@ -743299,7 +743447,7 @@ The natural logarithm on complex numbers (extension)
fdivpm $p |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V )
-> ( F /_f G ) e. ( CC ^pm A ) ) $=
( cc wf wcel w3a cvv cc0 csupp cfdiv wss cpm cnex a1i simp3 fdivmptf cdm
- co suppssdm wceq fdm eqcomd 3ad2ant2 syl5sseqr elpm2r syl22anc ) AEBFZAEC
+ co suppssdm wceq fdm eqcomd 3ad2ant2 sseqtrrid elpm2r syl22anc ) AEBFZAEC
FZADGZHZEIGZUKCJKTZEBCLTZFUNAMUOEANTGUMULOPUIUJUKQABCDRULCSZUNACJUAUJUIAU
PUBUKUJUPAAECUCUDUEUFEAUNUOIDUGUH $.
@@ -743308,7 +743456,7 @@ The natural logarithm on complex numbers (extension)
refdivpm $p |- ( ( F : A --> RR /\ G : A --> RR /\ A e. V )
-> ( F /_f G ) e. ( RR ^pm A ) ) $=
( cr wf wcel w3a cvv cc0 csupp co cfdiv wss cpm reex a1i simp3 refdivmptf
- cdm suppssdm wceq fdm eqcomd 3ad2ant2 syl5sseqr elpm2r syl22anc ) AEBFZAE
+ cdm suppssdm wceq fdm eqcomd 3ad2ant2 sseqtrrid elpm2r syl22anc ) AEBFZAE
CFZADGZHZEIGZUKCJKLZEBCMLZFUNANUOEAOLGUMULPQUIUJUKRABCDSULCTZUNACJUAUJUIA
UPUBUKUJUPAAECUCUDUEUFEAUNUOIDUGUH $.
@@ -748182,7 +748330,7 @@ the relevant induction theorem (such as ~ tfi ) to the other class.
14-Mar-2022.) $)
setrecsres $p |- ( ph -> B = setrecs ( ( F |` ~P B ) ) ) $=
( vx cpw cres csetrecs cv wss cfv wi wa wceq id resss a1i setrecsss wcel
- syl6sseqr sylan9ssr selpw sylbir syl eqid cvv vex setrec1 adantl eqsstr3d
+ syl6sseqr sylan9ssr selpw sylbir syl eqid cvv vex setrec1 adantl eqsstrrd
fvres ex alrimiv setrec2v eqssd ) ABCBGZHZIZABUSCFDAFJZUSKZUTCLZUSKZMFAVA
VCAVANZVBUTURLZUSVDUTBKZVEVBOZVAAUTUSBVAPZAUSCIBAURCEURCKACUQQRSDUAZUBVFU
TUQTVGFBUCUTUQCULUDUEVAVEUSKAVAUTUSURUSUFUTUGTVAFUHRVHUIUJUKUMUNUOVIUP $.
@@ -748320,7 +748468,7 @@ the relevant induction theorem (such as ~ tfi ) to the other class.
elpglem2 $p |- ( ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg )
-> E. x ( x C_ Pg
/\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) $=
- ( c1st cfv cpg wss c2nd wa cv cpw wcel wceq wi fvex unex isseti syl5sseqr
+ ( c1st cfv cpg wss c2nd wa cv cpw wcel wceq wi fvex unex isseti sseqtrrid
cun elpw2 sylibr sseq1 unss syl6bbr biimprd ssun1 id vex ssun2 jca jctird
eximii 19.37iv ) BCDZEFBGDZEFHZAIZEFZUMUPJZKZUNURKZHZHZAUPUMUNRZLZUOVBMAA
VCUMUNBCNBGNOPVDUOUQVAVDUQUOVDUQVCEFUOUPVCEUAUMUNEUBUCUDVDUSUTVDUMUPFUSVD
@@ -750013,7 +750161,7 @@ to Davis and Putnam (1960). (Contributed by David A. Wheeler,
ccrg csubrg cdr resubdrg subrgring cghm cmhm cgim reloggim gimghm gsummhm
ghmmhm subrgsubg simprl mulcomd caofcom fsumrpcl eqeltrd eqeltrrd ringcmn
simprr remulcl cicc simpl iccssioo2 syl6sseq cico ioossico 0lt1 breqtrrid
- eqsstr3i logccv 3adant1 simp21 remulcld resubcld simp22 readdcld eliooord
+ eqsstrri logccv 3adant1 simp21 remulcld resubcld simp22 readdcld eliooord
elioore simpld elrpd simprd 1m0e1 syl6breqr ltsub13d rpaddcl ltnegd mpbid
cmin fvmptd oveq12d negdid eqtr4d 3brtr4d scvxcvx jensen div1d mulcl rpcn
submid ssriv fssd fmptco 3brtr3d lenegcon1d eqbrtrrd efle reeflogd ) ADCE
@@ -750189,7 +750337,6 @@ to Davis and Putnam (1960). (Contributed by David A. Wheeler,
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
-
$( (End of Larry Lesyna's mathbox.) $)
$(
@@ -750198,7 +750345,6 @@ to Davis and Putnam (1960). (Contributed by David A. Wheeler,
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
-
$( Replacement of a nested antecedent with an outer antecedent. Commuted
simplificated form of elimination of a nested antecedent. Also holds
intuitionistically. Polish prefix notation: CCCpqrCsCqr . (Contributed
@@ -750510,7 +750656,6 @@ with simplest antecedents (i.e., in the corresponding ordering of binary
( wi adh-minimp-ax1 adh-minimp-ax2 ax-mp ) AABCZCZAACZCZHGCZAGACCJAGDAGAEFH
IGCCJKCAABEHIGEFF $.
-
$( (End of Adhemar's mathbox.) $)
$( End $[ set-mbox.mm $] $)