Theorem tgptmd 23989

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 2731	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2731	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 23987	. 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 2111  ‘cfv 6476  (class class class)co 7341  TopOpenctopn 17320  Grpcgrp 18841  inv_gcminusg 18842   Cn ccn 23134  TopMndctmd 23980  TopGrpctgp 23981

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 2113  ax-9 2121  ax-ext 2703  ax-nul 5239

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-ov 7344  df-tgp 23983

This theorem is referenced by:  tgptps  23990  tgpcn  23994  tgpsubcn  24000  tgpmulg  24003  oppgtgp  24008  tgplacthmeo  24013  subgtgp  24015  clsnsg  24020  tgpt0  24029  prdstgpd  24035  tsmssub  24059  tsmsxp  24065  trgtmd2  24079  nlmtlm  24604  qqhcn  33996