Theorem tgpinv 24109

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 24101	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
4	3	simp3bi 1146	1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431  TopOpenctopn 17468  Grpcgrp 18964  inv_gcminusg 18965   Cn ccn 23248  TopMndctmd 24094  TopGrpctgp 24095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-tgp 24097

This theorem is referenced by:  grpinvhmeo  24110  tgpsubcn  24114  tgpmulg  24117  oppgtgp  24122  subgtgp  24129  prdstgpd  24149  tsmsinv  24172  invrcn2  24204