Theorem tgpinv 24223

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 24215	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
4	3	simp3bi 1165	1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ‘cfv 6538  (class class class)co 7412  TopOpenctopn 17475  Grpcgrp 19001  inv_gcminusg 19002   Cn ccn 23362  TopMndctmd 24208  TopGrpctgp 24209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415  df-tgp 24211

This theorem is referenced by:  grpinvhmeo  24224  tgpsubcn  24228  tgpmulg  24231  oppgtgp  24236  subgtgp  24243  prdstgpd  24263  tsmsinv  24286  invrcn2  24318