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Theorem tgptmd 23453
Description: A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tgptmd (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)

Proof of Theorem tgptmd
StepHypRef Expression
1 eqid 2733 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
2 eqid 2733 . . 3 (invg𝐺) = (invg𝐺)
31, 2istgp 23451 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
43simp2bi 1147 1 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cfv 6500  (class class class)co 7361  TopOpenctopn 17311  Grpcgrp 18756  invgcminusg 18757   Cn ccn 22598  TopMndctmd 23444  TopGrpctgp 23445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364  df-tgp 23447
This theorem is referenced by:  tgptps  23454  tgpcn  23458  tgpsubcn  23464  tgpmulg  23467  oppgtgp  23472  tgplacthmeo  23477  subgtgp  23479  clsnsg  23484  tgpt0  23493  prdstgpd  23499  tsmssub  23523  tsmsxp  23529  trgtmd2  23543  nlmtlm  24081  qqhcn  32636
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