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Theorem tgpinv 22685
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 = (invg𝐺)
Assertion
Ref Expression
tgpinv (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))

Proof of Theorem tgpinv
StepHypRef Expression
1 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
2 tgpinv.5 . . 3 𝐼 = (invg𝐺)
31, 2istgp 22677 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
43simp3bi 1141 1 (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cfv 6348  (class class class)co 7148  TopOpenctopn 16687  Grpcgrp 18095  invgcminusg 18096   Cn ccn 21824  TopMndctmd 22670  TopGrpctgp 22671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-nul 5201
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7151  df-tgp 22673
This theorem is referenced by:  grpinvhmeo  22686  tgpsubcn  22690  tgpmulg  22693  oppgtgp  22698  subgtgp  22705  prdstgpd  22725  tsmsinv  22748  invrcn2  22780
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