MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgpinv Unicode version

Theorem tgpinv 18036
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 18028 . 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 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   TopOpenctopn 13576   Grpcgrp 14612   inv gcminusg 14613    Cn ccn 17210  TopMndctmd 18021   TopGrpctgp 18022
This theorem is referenced by:  grpinvhmeo  18037  tgpsubcn  18041  tgpmulg  18044  oppgtgp  18049  subgtgp  18056  prdstgpd  18075  tsmsinv  18098  invrcn2  18130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-nul 4279
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023  df-tgp 18024
  Copyright terms: Public domain W3C validator