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Theorem grpinvnzcl 14792
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 3414 . . 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 14779 . . 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 3872 . . 3  |-  ( X  e.  ( B  \  {  .0.  } )  <->  ( X  e.  B  /\  X  =/= 
.0.  ) )
7 grpinvnzcl.z . . . . 5  |-  .0.  =  ( 0g `  G )
82, 7, 3grpinvnz 14791 . . . 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 3872 . 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 1649    e. wcel 1717    =/= wne 2552    \ cdif 3262   {csn 3759   ` cfv 5396   Basecbs 13398   0gc0g 13652   Grpcgrp 14614   inv gcminusg 14615
This theorem is referenced by:  islindf4  26979  baerlem5amN  31833  baerlem5bmN  31834  baerlem5abmN  31835  hdmap1neglem1N  31945
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-riota 6487  df-0g 13656  df-mnd 14619  df-grp 14741  df-minusg 14742
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