MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgpgrp Structured version   Unicode version

Theorem tgpgrp 20404
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 2467 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
2 eqid 2467 . . 3  |-  ( invg `  G )  =  ( invg `  G )
31, 2istgp 20403 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  ( invg `  G )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen
`  G ) ) ) )
43simp1bi 1011 1  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   ` cfv 5588  (class class class)co 6285   TopOpenctopn 14680   Grpcgrp 15730   invgcminusg 15731    Cn ccn 19531  TopMndctmd 20396   TopGrpctgp 20397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6288  df-tgp 20399
This theorem is referenced by:  grpinvhmeo  20412  istgp2  20417  oppgtgp  20424  tgplacthmeo  20429  subgtgp  20431  subgntr  20432  opnsubg  20433  clssubg  20434  cldsubg  20436  tgpconcompeqg  20437  tgpconcomp  20438  snclseqg  20441  tgphaus  20442  tgpt1  20443  tgpt0  20444  qustgpopn  20445  qustgplem  20446  qustgphaus  20448  prdstgpd  20450  tsmsinv  20477  tsmssub  20478  tgptsmscls  20479  tsmsxplem1  20482  tsmsxplem2  20483
  Copyright terms: Public domain W3C validator