Theorem tgptmd 23803

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

Step	Hyp	Ref	Expression
1		eqid 2732	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2732	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 23801	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp2bi 1146	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)

Syntax hints:   → wi 4   ∈ wcel 2106  ‘cfv 6543  (class class class)co 7411  TopOpenctopn 17371  Grpcgrp 18855  inv_gcminusg 18856   Cn ccn 22948  TopMndctmd 23794  TopGrpctgp 23795

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 7414  df-tgp 23797

This theorem is referenced by:  tgptps  23804  tgpcn  23808  tgpsubcn  23814  tgpmulg  23817  oppgtgp  23822  tgplacthmeo  23827  subgtgp  23829  clsnsg  23834  tgpt0  23843  prdstgpd  23849  tsmssub  23873  tsmsxp  23879  trgtmd2  23893  nlmtlm  24431  qqhcn  33257