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Theorem tgpgrp 21929
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 2651 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
2 eqid 2651 . . 3 (invg𝐺) = (invg𝐺)
31, 2istgp 21928 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
43simp1bi 1096 1 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  cfv 5926  (class class class)co 6690  TopOpenctopn 16129  Grpcgrp 17469  invgcminusg 17470   Cn ccn 21076  TopMndctmd 21921  TopGrpctgp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-tgp 21924
This theorem is referenced by:  grpinvhmeo  21937  istgp2  21942  oppgtgp  21949  tgplacthmeo  21954  subgtgp  21956  subgntr  21957  opnsubg  21958  clssubg  21959  cldsubg  21961  tgpconncompeqg  21962  tgpconncomp  21963  snclseqg  21966  tgphaus  21967  tgpt1  21968  tgpt0  21969  qustgpopn  21970  qustgplem  21971  qustgphaus  21973  prdstgpd  21975  tsmsinv  21998  tsmssub  21999  tgptsmscls  22000  tsmsxplem1  22003  tsmsxplem2  22004
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