Theorem tgpgrp 22688

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 2823	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2823	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 22687	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1141	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6357  (class class class)co 7158  TopOpenctopn 16697  Grpcgrp 18105  inv_gcminusg 18106   Cn ccn 21834  TopMndctmd 22680  TopGrpctgp 22681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161  df-tgp 22683

This theorem is referenced by:  grpinvhmeo  22696  istgp2  22701  oppgtgp  22708  tgplacthmeo  22713  subgtgp  22715  subgntr  22717  opnsubg  22718  clssubg  22719  cldsubg  22721  tgpconncompeqg  22722  tgpconncomp  22723  snclseqg  22726  tgphaus  22727  tgpt1  22728  tgpt0  22729  qustgpopn  22730  qustgplem  22731  qustgphaus  22733  prdstgpd  22735  tsmsinv  22758  tsmssub  22759  tgptsmscls  22760  tsmsxplem1  22763  tsmsxplem2  22764