Theorem tgpgrp 24022

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 24021	. 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 2113  ‘cfv 6492  (class class class)co 7358  TopOpenctopn 17341  Grpcgrp 18863  inv_gcminusg 18864   Cn ccn 23168  TopMndctmd 24014  TopGrpctgp 24015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-tgp 24017

This theorem is referenced by:  grpinvhmeo  24030  istgp2  24035  oppgtgp  24042  tgplacthmeo  24047  subgtgp  24049  subgntr  24051  opnsubg  24052  clssubg  24053  cldsubg  24055  tgpconncompeqg  24056  tgpconncomp  24057  snclseqg  24060  tgphaus  24061  tgpt1  24062  tgpt0  24063  qustgpopn  24064  qustgplem  24065  qustgphaus  24067  prdstgpd  24069  tsmsinv  24092  tsmssub  24093  tgptsmscls  24094  tsmsxplem1  24097  tsmsxplem2  24098