Theorem tgptmd 24023

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 2736	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2736	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24021	. 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 2113  ‘cfv 6492  (class class class)co 7358  TopOpenctopn 17341  Grpcgrp 18863  inv_gcminusg 18864   Cn ccn 23168  TopMndctmd 24014  TopGrpctgp 24015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-tgp 24017

This theorem is referenced by:  tgptps  24024  tgpcn  24028  tgpsubcn  24034  tgpmulg  24037  oppgtgp  24042  tgplacthmeo  24047  subgtgp  24049  clsnsg  24054  tgpt0  24063  prdstgpd  24069  tsmssub  24093  tsmsxp  24099  trgtmd2  24113  nlmtlm  24638  qqhcn  34148