Theorem tgpinv 23180

Description: In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.)

Hypotheses

Ref	Expression
tgpcn.j	⊢ 𝐽 = (TopOpen‘𝐺)
tgpinv.5	⊢ 𝐼 = (invg‘𝐺)

Assertion

Ref	Expression
tgpinv	⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))

Proof of Theorem tgpinv

Step	Hyp	Ref	Expression
1		tgpcn.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
2		tgpinv.5	. . 3 ⊢ 𝐼 = (invg‘𝐺)
3	1, 2	istgp 23172	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
4	3	simp3bi 1145	1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ‘cfv 6423  (class class class)co 7260  TopOpenctopn 17076  Grpcgrp 18521  inv_gcminusg 18522   Cn ccn 22319  TopMndctmd 23165  TopGrpctgp 23166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3429  df-sbc 3717  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5076  df-iota 6381  df-fv 6431  df-ov 7263  df-tgp 23168

This theorem is referenced by:  grpinvhmeo  23181  tgpsubcn  23185  tgpmulg  23188  oppgtgp  23193  subgtgp  23200  prdstgpd  23220  tsmsinv  23243  invrcn2  23275