Theorem tgpinv 24410

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 24402	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
4	3	simp3bi 1165	1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6539  (class class class)co 7421  TopOpenctopn 17598  Grpcgrp 19150  inv_gcminusg 19151   Cn ccn 23548  TopMndctmd 24395  TopGrpctgp 24396

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6547  df-ov 7424  df-tgp 24398

This theorem is used by:  grpinvhmeo  24411  tgpsubcn  24415  tgpmulg  24418  oppgtgp  24423  subgtgp  24430  prdstgpd  24450  tsmsinv  24473  invrcn2  24505