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Theorem tgpinv 18105
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 18097 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) ) )
43simp3bi 974 1  |-  ( G  e.  TopGrp  ->  I  e.  ( J  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   TopOpenctopn 13639   Grpcgrp 14675   inv gcminusg 14676    Cn ccn 17278  TopMndctmd 18090   TopGrpctgp 18091
This theorem is referenced by:  grpinvhmeo  18106  tgpsubcn  18110  tgpmulg  18113  oppgtgp  18118  subgtgp  18125  prdstgpd  18144  tsmsinv  18167  invrcn2  18199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-tgp 18093
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