Theorem tgptmd 24399

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 2761	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2761	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24397	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp2bi 1164	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ‘cfv 6539  (class class class)co 7421  TopOpenctopn 17593  Grpcgrp 19145  inv_gcminusg 19146   Cn ccn 23543  TopMndctmd 24390  TopGrpctgp 24391

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6495  df-fv 6547  df-ov 7424  df-tgp 24393

This theorem is used by:  tgptps  24400  tgpcn  24404  tgpsubcn  24410  tgpmulg  24413  oppgtgp  24418  tgplacthmeo  24423  subgtgp  24425  clsnsg  24430  tgpt0  24439  prdstgpd  24445  tsmssub  24469  tsmsxp  24475  trgtmd2  24489  nlmtlm  25014  qqhcn  34624