Theorem tgpgrp 22252

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 2825	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2825	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 22251	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1181	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2166  ‘cfv 6123  (class class class)co 6905  TopOpenctopn 16435  Grpcgrp 17776  inv_gcminusg 17777   Cn ccn 21399  TopMndctmd 22244  TopGrpctgp 22245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-nul 5013

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-tgp 22247

This theorem is referenced by:  grpinvhmeo  22260  istgp2  22265  oppgtgp  22272  tgplacthmeo  22277  subgtgp  22279  subgntr  22280  opnsubg  22281  clssubg  22282  cldsubg  22284  tgpconncompeqg  22285  tgpconncomp  22286  snclseqg  22289  tgphaus  22290  tgpt1  22291  tgpt0  22292  qustgpopn  22293  qustgplem  22294  qustgphaus  22296  prdstgpd  22298  tsmsinv  22321  tsmssub  22322  tgptsmscls  22323  tsmsxplem1  22326  tsmsxplem2  22327