Theorem tgpgrp 24065

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 2741	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2741	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24064	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1152	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2121  ‘cfv 6489  (class class class)co 7360  TopOpenctopn 17379  Grpcgrp 18904  inv_gcminusg 18905   Cn ccn 23211  TopMndctmd 24057  TopGrpctgp 24058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-nul 5231

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-ov 7363  df-tgp 24060

This theorem is referenced by:  grpinvhmeo  24073  istgp2  24078  oppgtgp  24085  tgplacthmeo  24090  subgtgp  24092  subgntr  24094  opnsubg  24095  clssubg  24096  cldsubg  24098  tgpconncompeqg  24099  tgpconncomp  24100  snclseqg  24103  tgphaus  24104  tgpt1  24105  tgpt0  24106  qustgpopn  24107  qustgplem  24108  qustgphaus  24110  prdstgpd  24112  tsmsinv  24135  tsmssub  24136  tgptsmscls  24137  tsmsxplem1  24140  tsmsxplem2  24141