Theorem tgpgrp 22683

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 2798	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2798	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 22682	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1142	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2111  ‘cfv 6324  (class class class)co 7135  TopOpenctopn 16687  Grpcgrp 18095  inv_gcminusg 18096   Cn ccn 21829  TopMndctmd 22675  TopGrpctgp 22676

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-tgp 22678

This theorem is referenced by:  grpinvhmeo  22691  istgp2  22696  oppgtgp  22703  tgplacthmeo  22708  subgtgp  22710  subgntr  22712  opnsubg  22713  clssubg  22714  cldsubg  22716  tgpconncompeqg  22717  tgpconncomp  22718  snclseqg  22721  tgphaus  22722  tgpt1  22723  tgpt0  22724  qustgpopn  22725  qustgplem  22726  qustgphaus  22728  prdstgpd  22730  tsmsinv  22753  tsmssub  22754  tgptsmscls  22755  tsmsxplem1  22758  tsmsxplem2  22759