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Theorem tgpinv 23459
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 = (TopOpenβ€˜πΊ)
tgpinv.5 𝐼 = (invgβ€˜πΊ)
Assertion
Ref Expression
tgpinv (𝐺 ∈ TopGrp β†’ 𝐼 ∈ (𝐽 Cn 𝐽))

Proof of Theorem tgpinv
StepHypRef Expression
1 tgpcn.j . . 3 𝐽 = (TopOpenβ€˜πΊ)
2 tgpinv.5 . . 3 𝐼 = (invgβ€˜πΊ)
31, 2istgp 23451 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
43simp3bi 1148 1 (𝐺 ∈ TopGrp β†’ 𝐼 ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  β€˜cfv 6500  (class class class)co 7361  TopOpenctopn 17311  Grpcgrp 18756  invgcminusg 18757   Cn ccn 22598  TopMndctmd 23444  TopGrpctgp 23445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364  df-tgp 23447
This theorem is referenced by:  grpinvhmeo  23460  tgpsubcn  23464  tgpmulg  23467  oppgtgp  23472  subgtgp  23479  prdstgpd  23499  tsmsinv  23522  invrcn2  23554
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