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Theorem tgptmd 23582
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 2732 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
2 eqid 2732 . . 3 (invg𝐺) = (invg𝐺)
31, 2istgp 23580 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
43simp2bi 1146 1 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cfv 6543  (class class class)co 7408  TopOpenctopn 17366  Grpcgrp 18818  invgcminusg 18819   Cn ccn 22727  TopMndctmd 23573  TopGrpctgp 23574
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-tgp 23576
This theorem is referenced by:  tgptps  23583  tgpcn  23587  tgpsubcn  23593  tgpmulg  23596  oppgtgp  23601  tgplacthmeo  23606  subgtgp  23608  clsnsg  23613  tgpt0  23622  prdstgpd  23628  tsmssub  23652  tsmsxp  23658  trgtmd2  23672  nlmtlm  24210  qqhcn  32966
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