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Theorem tgptmd 22617
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 2821 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
2 eqid 2821 . . 3 (invg𝐺) = (invg𝐺)
31, 2istgp 22615 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
43simp2bi 1138 1 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cfv 6349  (class class class)co 7145  TopOpenctopn 16685  Grpcgrp 18043  invgcminusg 18044   Cn ccn 21762  TopMndctmd 22608  TopGrpctgp 22609
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 2793  ax-nul 5202
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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7148  df-tgp 22611
This theorem is referenced by:  tgptps  22618  tgpcn  22622  tgpsubcn  22628  tgpmulg  22631  oppgtgp  22636  tgplacthmeo  22641  subgtgp  22643  clsnsg  22647  tgpt0  22656  prdstgpd  22662  tsmssub  22686  tsmsxp  22692  trgtmd2  22706  nlmtlm  23232  qqhcn  31132
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