Theorem tgptmd 24103

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 2735	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2735	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24101	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp2bi 1145	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)

Syntax hints:   → wi 4   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431  TopOpenctopn 17468  Grpcgrp 18964  inv_gcminusg 18965   Cn ccn 23248  TopMndctmd 24094  TopGrpctgp 24095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-tgp 24097

This theorem is referenced by:  tgptps  24104  tgpcn  24108  tgpsubcn  24114  tgpmulg  24117  oppgtgp  24122  tgplacthmeo  24127  subgtgp  24129  clsnsg  24134  tgpt0  24143  prdstgpd  24149  tsmssub  24173  tsmsxp  24179  trgtmd2  24193  nlmtlm  24731  qqhcn  33954