Theorem tgpgrp 24034

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 2737	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2737	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24033	. 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 6500  (class class class)co 7368  TopOpenctopn 17353  Grpcgrp 18875  inv_gcminusg 18876   Cn ccn 23180  TopMndctmd 24026  TopGrpctgp 24027

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 2709  ax-nul 5253

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-tgp 24029

This theorem is referenced by:  grpinvhmeo  24042  istgp2  24047  oppgtgp  24054  tgplacthmeo  24059  subgtgp  24061  subgntr  24063  opnsubg  24064  clssubg  24065  cldsubg  24067  tgpconncompeqg  24068  tgpconncomp  24069  snclseqg  24072  tgphaus  24073  tgpt1  24074  tgpt0  24075  qustgpopn  24076  qustgplem  24077  qustgphaus  24079  prdstgpd  24081  tsmsinv  24104  tsmssub  24105  tgptsmscls  24106  tsmsxplem1  24109  tsmsxplem2  24110