Theorem tgpgrp 23965

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 23964	. 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 6511  (class class class)co 7387  TopOpenctopn 17384  Grpcgrp 18865  inv_gcminusg 18866   Cn ccn 23111  TopMndctmd 23957  TopGrpctgp 23958

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 5261

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 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-tgp 23960

This theorem is referenced by:  grpinvhmeo  23973  istgp2  23978  oppgtgp  23985  tgplacthmeo  23990  subgtgp  23992  subgntr  23994  opnsubg  23995  clssubg  23996  cldsubg  23998  tgpconncompeqg  23999  tgpconncomp  24000  snclseqg  24003  tgphaus  24004  tgpt1  24005  tgpt0  24006  qustgpopn  24007  qustgplem  24008  qustgphaus  24010  prdstgpd  24012  tsmsinv  24035  tsmssub  24036  tgptsmscls  24037  tsmsxplem1  24040  tsmsxplem2  24041