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Theorem tgpgrp 19782
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 2454 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
2 eqid 2454 . . 3  |-  ( invg `  G )  =  ( invg `  G )
31, 2istgp 19781 . 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 1758   ` cfv 5527  (class class class)co 6201   TopOpenctopn 14480   Grpcgrp 15530   invgcminusg 15531    Cn ccn 18961  TopMndctmd 19774   TopGrpctgp 19775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-iota 5490  df-fv 5535  df-ov 6204  df-tgp 19777
This theorem is referenced by:  grpinvhmeo  19790  istgp2  19795  oppgtgp  19802  tgplacthmeo  19807  subgtgp  19809  subgntr  19810  opnsubg  19811  clssubg  19812  cldsubg  19814  tgpconcompeqg  19815  tgpconcomp  19816  snclseqg  19819  tgphaus  19820  tgpt1  19821  tgpt0  19822  divstgpopn  19823  divstgplem  19824  divstgphaus  19826  prdstgpd  19828  tsmsinv  19855  tsmssub  19856  tgptsmscls  19857  tsmsxplem1  19860  tsmsxplem2  19861
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