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Theorem tgpinv 20312
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  e.  TopGrp  ->  I  e.  ( J  Cn  J ) )

Proof of Theorem tgpinv
StepHypRef Expression
1 tgpcn.j . . 3  |-  J  =  ( TopOpen `  G )
2 tgpinv.5 . . 3  |-  I  =  ( invg `  G )
31, 2istgp 20304 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) ) )
43simp3bi 1008 1  |-  ( G  e.  TopGrp  ->  I  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   TopOpenctopn 14666   Grpcgrp 15716   invgcminusg 15717    Cn ccn 19484  TopMndctmd 20297   TopGrpctgp 20298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-nul 4569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587  df-ov 6278  df-tgp 20300
This theorem is referenced by:  grpinvhmeo  20313  tgpsubcn  20317  tgpmulg  20320  oppgtgp  20325  subgtgp  20332  prdstgpd  20351  tsmsinv  20378  invrcn2  20410
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