Theorem tgpgrp 24021

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 24020	. 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 2109  ‘cfv 6536  (class class class)co 7410  TopOpenctopn 17440  Grpcgrp 18921  inv_gcminusg 18922   Cn ccn 23167  TopMndctmd 24013  TopGrpctgp 24014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-tgp 24016

This theorem is referenced by:  grpinvhmeo  24029  istgp2  24034  oppgtgp  24041  tgplacthmeo  24046  subgtgp  24048  subgntr  24050  opnsubg  24051  clssubg  24052  cldsubg  24054  tgpconncompeqg  24055  tgpconncomp  24056  snclseqg  24059  tgphaus  24060  tgpt1  24061  tgpt0  24062  qustgpopn  24063  qustgplem  24064  qustgphaus  24066  prdstgpd  24068  tsmsinv  24091  tsmssub  24092  tgptsmscls  24093  tsmsxplem1  24096  tsmsxplem2  24097