Theorem tgpgrp 24087

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 ∈ TopGrp → 𝐺 ∈ Grp)

Proof of Theorem tgpgrp

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2736	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24086	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1145	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2107  ‘cfv 6560  (class class class)co 7432  TopOpenctopn 17467  Grpcgrp 18952  inv_gcminusg 18953   Cn ccn 23233  TopMndctmd 24079  TopGrpctgp 24080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-tgp 24082

This theorem is referenced by:  grpinvhmeo  24095  istgp2  24100  oppgtgp  24107  tgplacthmeo  24112  subgtgp  24114  subgntr  24116  opnsubg  24117  clssubg  24118  cldsubg  24120  tgpconncompeqg  24121  tgpconncomp  24122  snclseqg  24125  tgphaus  24126  tgpt1  24127  tgpt0  24128  qustgpopn  24129  qustgplem  24130  qustgphaus  24132  prdstgpd  24134  tsmsinv  24157  tsmssub  24158  tgptsmscls  24159  tsmsxplem1  24162  tsmsxplem2  24163