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Theorem tgpinv 20876
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 20868 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) ) )
43simp3bi 1014 1  |-  ( G  e.  TopGrp  ->  I  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   ` cfv 5569  (class class class)co 6278   TopOpenctopn 15036   Grpcgrp 16377   invgcminusg 16378    Cn ccn 20018  TopMndctmd 20861   TopGrpctgp 20862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4525
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-ov 6281  df-tgp 20864
This theorem is referenced by:  grpinvhmeo  20877  tgpsubcn  20881  tgpmulg  20884  oppgtgp  20889  subgtgp  20896  prdstgpd  20915  tsmsinv  20942  invrcn2  20974
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