Theorem tgpgrp 23993

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 2731	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2731	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 23992	. 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 2111  ‘cfv 6481  (class class class)co 7346  TopOpenctopn 17325  Grpcgrp 18846  inv_gcminusg 18847   Cn ccn 23139  TopMndctmd 23985  TopGrpctgp 23986

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 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-tgp 23988

This theorem is referenced by:  grpinvhmeo  24001  istgp2  24006  oppgtgp  24013  tgplacthmeo  24018  subgtgp  24020  subgntr  24022  opnsubg  24023  clssubg  24024  cldsubg  24026  tgpconncompeqg  24027  tgpconncomp  24028  snclseqg  24031  tgphaus  24032  tgpt1  24033  tgpt0  24034  qustgpopn  24035  qustgplem  24036  qustgphaus  24038  prdstgpd  24040  tsmsinv  24063  tsmssub  24064  tgptsmscls  24065  tsmsxplem1  24068  tsmsxplem2  24069