Theorem tgpgrp 24043

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 24042	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1146	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6498  (class class class)co 7367  TopOpenctopn 17384  Grpcgrp 18909  inv_gcminusg 18910   Cn ccn 23189  TopMndctmd 24035  TopGrpctgp 24036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-tgp 24038

This theorem is referenced by:  grpinvhmeo  24051  istgp2  24056  oppgtgp  24063  tgplacthmeo  24068  subgtgp  24070  subgntr  24072  opnsubg  24073  clssubg  24074  cldsubg  24076  tgpconncompeqg  24077  tgpconncomp  24078  snclseqg  24081  tgphaus  24082  tgpt1  24083  tgpt0  24084  qustgpopn  24085  qustgplem  24086  qustgphaus  24088  prdstgpd  24090  tsmsinv  24113  tsmssub  24114  tgptsmscls  24115  tsmsxplem1  24118  tsmsxplem2  24119