Theorem tgpgrp 24056

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 24055	. 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 6493  (class class class)co 7361  TopOpenctopn 17378  Grpcgrp 18903  inv_gcminusg 18904   Cn ccn 23202  TopMndctmd 24048  TopGrpctgp 24049

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 5242

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 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6449  df-fv 6501  df-ov 7364  df-tgp 24051

This theorem is referenced by:  grpinvhmeo  24064  istgp2  24069  oppgtgp  24076  tgplacthmeo  24081  subgtgp  24083  subgntr  24085  opnsubg  24086  clssubg  24087  cldsubg  24089  tgpconncompeqg  24090  tgpconncomp  24091  snclseqg  24094  tgphaus  24095  tgpt1  24096  tgpt0  24097  qustgpopn  24098  qustgplem  24099  qustgphaus  24101  prdstgpd  24103  tsmsinv  24126  tsmssub  24127  tgptsmscls  24128  tsmsxplem1  24131  tsmsxplem2  24132