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Theorem tgpgrp 19624
Description: A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
tgpgrp  |-  ( G  e.  TopGrp  ->  G  e.  Grp )

Proof of Theorem tgpgrp
StepHypRef Expression
1 eqid 2438 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
2 eqid 2438 . . 3  |-  ( invg `  G )  =  ( invg `  G )
31, 2istgp 19623 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  ( invg `  G )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen
`  G ) ) ) )
43simp1bi 1003 1  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756   ` cfv 5413  (class class class)co 6086   TopOpenctopn 14352   Grpcgrp 15402   invgcminusg 15403    Cn ccn 18803  TopMndctmd 19616   TopGrpctgp 19617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-nul 4416
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-iota 5376  df-fv 5421  df-ov 6089  df-tgp 19619
This theorem is referenced by:  grpinvhmeo  19632  istgp2  19637  oppgtgp  19644  tgplacthmeo  19649  subgtgp  19651  subgntr  19652  opnsubg  19653  clssubg  19654  cldsubg  19656  tgpconcompeqg  19657  tgpconcomp  19658  snclseqg  19661  tgphaus  19662  tgpt1  19663  tgpt0  19664  divstgpopn  19665  divstgplem  19666  divstgphaus  19668  prdstgpd  19670  tsmsinv  19697  tsmssub  19698  tgptsmscls  19699  tsmsxplem1  19702  tsmsxplem2  19703
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