Theorem tgpgrp 24235

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 2763	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2763	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24234	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1163	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2143  ‘cfv 6536  (class class class)co 7410  TopOpenctopn 17469  Grpcgrp 18995  inv_gcminusg 18996   Cn ccn 23381  TopMndctmd 24227  TopGrpctgp 24228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-tgp 24230

This theorem is referenced by:  grpinvhmeo  24243  istgp2  24248  oppgtgp  24255  tgplacthmeo  24260  subgtgp  24262  subgntr  24264  opnsubg  24265  clssubg  24266  cldsubg  24268  tgpconncompeqg  24269  tgpconncomp  24270  snclseqg  24273  tgphaus  24274  tgpt1  24275  tgpt0  24276  qustgpopn  24277  qustgplem  24278  qustgphaus  24280  prdstgpd  24282  tsmsinv  24305  tsmssub  24306  tgptsmscls  24307  tsmsxplem1  24310  tsmsxplem2  24311