Theorem tgptmd 21693

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 2610	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2610	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 21691	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp2bi 1070	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)

Syntax hints:   → wi 4   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  TopOpenctopn 15905  Grpcgrp 17245  inv_gcminusg 17246   Cn ccn 20838  TopMndctmd 21684  TopGrpctgp 21685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-tgp 21687

This theorem is referenced by:  tgptps  21694  tgpcn  21698  tgpsubcn  21704  tgpmulg  21707  oppgtgp  21712  tgplacthmeo  21717  subgtgp  21719  clsnsg  21723  tgpt0  21732  prdstgpd  21738  tsmssub  21762  tsmsxp  21768  trgtmd2  21782  nlmtlm  22308  qqhcn  29363