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Theorem tgpinv 17768
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  =  ( inv g `  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  =  ( inv g `  G )
31, 2istgp 17760 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) ) )
43simp3bi 972 1  |-  ( G  e.  TopGrp  ->  I  e.  ( J  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   TopOpenctopn 13326   Grpcgrp 14362   inv gcminusg 14363    Cn ccn 16954  TopMndctmd 17753   TopGrpctgp 17754
This theorem is referenced by:  grpinvhmeo  17769  tgpsubcn  17773  tgpmulg  17776  oppgtgp  17781  subgtgp  17788  prdstgpd  17807  tsmsinv  17830  invrcn2  17862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-tgp 17756
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