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Theorem tgpgrp 21692
 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 2610 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
2 eqid 2610 . . 3 (invg𝐺) = (invg𝐺)
31, 2istgp 21691 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
43simp1bi 1069 1 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  TopOpenctopn 15905  Grpcgrp 17245  invgcminusg 17246   Cn ccn 20838  TopMndctmd 21684  TopGrpctgp 21685 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-tgp 21687 This theorem is referenced by:  grpinvhmeo  21700  istgp2  21705  oppgtgp  21712  tgplacthmeo  21717  subgtgp  21719  subgntr  21720  opnsubg  21721  clssubg  21722  cldsubg  21724  tgpconcompeqg  21725  tgpconcomp  21726  snclseqg  21729  tgphaus  21730  tgpt1  21731  tgpt0  21732  qustgpopn  21733  qustgplem  21734  qustgphaus  21736  prdstgpd  21738  tsmsinv  21761  tsmssub  21762  tgptsmscls  21763  tsmsxplem1  21766  tsmsxplem2  21767
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