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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 = (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 23172 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
43simp3bi 1145 1 (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cfv 6423  (class class class)co 7260  TopOpenctopn 17076  Grpcgrp 18521  invgcminusg 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
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