Theorem tgptmd 24204

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 2769	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2769	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24202	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp2bi 1162	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)

Syntax hints:   → wi 4   ∈ wcel 2149  ‘cfv 6537  (class class class)co 7411  TopOpenctopn 17473  Grpcgrp 18999  inv_gcminusg 19000   Cn ccn 23349  TopMndctmd 24195  TopGrpctgp 24196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414  df-tgp 24198

This theorem is referenced by:  tgptps  24205  tgpcn  24209  tgpsubcn  24215  tgpmulg  24218  oppgtgp  24223  tgplacthmeo  24228  subgtgp  24230  clsnsg  24235  tgpt0  24244  prdstgpd  24250  tsmssub  24274  tsmsxp  24280  trgtmd2  24294  nlmtlm  24819  qqhcn  34325