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Theorem tgpgrp 22672
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 2824 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
2 eqid 2824 . . 3 (invg𝐺) = (invg𝐺)
31, 2istgp 22671 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
43simp1bi 1142 1 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  cfv 6336  (class class class)co 7138  TopOpenctopn 16684  Grpcgrp 18092  invgcminusg 18093   Cn ccn 21818  TopMndctmd 22664  TopGrpctgp 22665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5191
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-iota 6295  df-fv 6344  df-ov 7141  df-tgp 22667
This theorem is referenced by:  grpinvhmeo  22680  istgp2  22685  oppgtgp  22692  tgplacthmeo  22697  subgtgp  22699  subgntr  22701  opnsubg  22702  clssubg  22703  cldsubg  22705  tgpconncompeqg  22706  tgpconncomp  22707  snclseqg  22710  tgphaus  22711  tgpt1  22712  tgpt0  22713  qustgpopn  22714  qustgplem  22715  qustgphaus  22717  prdstgpd  22719  tsmsinv  22742  tsmssub  22743  tgptsmscls  22744  tsmsxplem1  22747  tsmsxplem2  22748
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