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Theorem tgpgrp 22614
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
StepHypRef Expression
1 eqid 2818 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
2 eqid 2818 . . 3 (invg𝐺) = (invg𝐺)
31, 2istgp 22613 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
43simp1bi 1137 1 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cfv 6348  (class class class)co 7145  TopOpenctopn 16683  Grpcgrp 18041  invgcminusg 18042   Cn ccn 21760  TopMndctmd 22606  TopGrpctgp 22607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-tgp 22609
This theorem is referenced by:  grpinvhmeo  22622  istgp2  22627  oppgtgp  22634  tgplacthmeo  22639  subgtgp  22641  subgntr  22642  opnsubg  22643  clssubg  22644  cldsubg  22646  tgpconncompeqg  22647  tgpconncomp  22648  snclseqg  22651  tgphaus  22652  tgpt1  22653  tgpt0  22654  qustgpopn  22655  qustgplem  22656  qustgphaus  22658  prdstgpd  22660  tsmsinv  22683  tsmssub  22684  tgptsmscls  22685  tsmsxplem1  22688  tsmsxplem2  22689
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