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Theorem grpinvnzcl 14855
Description: The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Hypotheses
Ref Expression
grpinvnzcl.b  |-  B  =  ( Base `  G
)
grpinvnzcl.z  |-  .0.  =  ( 0g `  G )
grpinvnzcl.n  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
grpinvnzcl  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  ( B  \  {  .0.  } ) )

Proof of Theorem grpinvnzcl
StepHypRef Expression
1 eldifi 3461 . . 3  |-  ( X  e.  ( B  \  {  .0.  } )  ->  X  e.  B )
2 grpinvnzcl.b . . . 4  |-  B  =  ( Base `  G
)
3 grpinvnzcl.n . . . 4  |-  N  =  ( inv g `  G )
42, 3grpinvcl 14842 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
51, 4sylan2 461 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  B
)
6 eldifsn 3919 . . 3  |-  ( X  e.  ( B  \  {  .0.  } )  <->  ( X  e.  B  /\  X  =/= 
.0.  ) )
7 grpinvnzcl.z . . . . 5  |-  .0.  =  ( 0g `  G )
82, 7, 3grpinvnz 14854 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( N `  X
)  =/=  .0.  )
983expb 1154 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( N `  X )  =/=  .0.  )
106, 9sylan2b 462 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  =/=  .0.  )
11 eldifsn 3919 . 2  |-  ( ( N `  X )  e.  ( B  \  {  .0.  } )  <->  ( ( N `  X )  e.  B  /\  ( N `  X )  =/=  .0.  ) )
125, 10, 11sylanbrc 646 1  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  ( B  \  {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309   {csn 3806   ` cfv 5446   Basecbs 13461   0gc0g 13715   Grpcgrp 14677   inv gcminusg 14678
This theorem is referenced by:  islindf4  27266  baerlem5amN  32441  baerlem5bmN  32442  baerlem5abmN  32443  hdmap1neglem1N  32553
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  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-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-riota 6541  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805
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