Theorem tgpgrp 24204

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 2769	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2769	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24203	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1161	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2149  ‘cfv 6537  (class class class)co 7411  TopOpenctopn 17474  Grpcgrp 19000  inv_gcminusg 19001   Cn ccn 23350  TopMndctmd 24196  TopGrpctgp 24197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414  df-tgp 24199

This theorem is referenced by:  grpinvhmeo  24212  istgp2  24217  oppgtgp  24224  tgplacthmeo  24229  subgtgp  24231  subgntr  24233  opnsubg  24234  clssubg  24235  cldsubg  24237  tgpconncompeqg  24238  tgpconncomp  24239  snclseqg  24242  tgphaus  24243  tgpt1  24244  tgpt0  24245  qustgpopn  24246  qustgplem  24247  qustgphaus  24249  prdstgpd  24251  tsmsinv  24274  tsmssub  24275  tgptsmscls  24276  tsmsxplem1  24279  tsmsxplem2  24280