Theorem tgptmd 24035

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 2737	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2737	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24033	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp2bi 1147	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6500  (class class class)co 7368  TopOpenctopn 17353  Grpcgrp 18875  inv_gcminusg 18876   Cn ccn 23180  TopMndctmd 24026  TopGrpctgp 24027

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-tgp 24029

This theorem is referenced by:  tgptps  24036  tgpcn  24040  tgpsubcn  24046  tgpmulg  24049  oppgtgp  24054  tgplacthmeo  24059  subgtgp  24061  clsnsg  24066  tgpt0  24075  prdstgpd  24081  tsmssub  24105  tsmsxp  24111  trgtmd2  24125  nlmtlm  24650  qqhcn  34169