Theorem tgpgrp 23998

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 2729	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2729	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 23997	. 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 6499  (class class class)co 7369  TopOpenctopn 17360  Grpcgrp 18847  inv_gcminusg 18848   Cn ccn 23144  TopMndctmd 23990  TopGrpctgp 23991

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 2701  ax-nul 5256

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 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-ov 7372  df-tgp 23993

This theorem is referenced by:  grpinvhmeo  24006  istgp2  24011  oppgtgp  24018  tgplacthmeo  24023  subgtgp  24025  subgntr  24027  opnsubg  24028  clssubg  24029  cldsubg  24031  tgpconncompeqg  24032  tgpconncomp  24033  snclseqg  24036  tgphaus  24037  tgpt1  24038  tgpt0  24039  qustgpopn  24040  qustgplem  24041  qustgphaus  24043  prdstgpd  24045  tsmsinv  24068  tsmssub  24069  tgptsmscls  24070  tsmsxplem1  24073  tsmsxplem2  24074