Theorem tgpgrp 24061

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 2739	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2739	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24060	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1151	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2119  ‘cfv 6485  (class class class)co 7356  TopOpenctopn 17375  Grpcgrp 18900  inv_gcminusg 18901   Cn ccn 23207  TopMndctmd 24053  TopGrpctgp 24054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-tgp 24056

This theorem is referenced by:  grpinvhmeo  24069  istgp2  24074  oppgtgp  24081  tgplacthmeo  24086  subgtgp  24088  subgntr  24090  opnsubg  24091  clssubg  24092  cldsubg  24094  tgpconncompeqg  24095  tgpconncomp  24096  snclseqg  24099  tgphaus  24100  tgpt1  24101  tgpt0  24102  qustgpopn  24103  qustgplem  24104  qustgphaus  24106  prdstgpd  24108  tsmsinv  24131  tsmssub  24132  tgptsmscls  24133  tsmsxplem1  24136  tsmsxplem2  24137