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Theorem grpinvnzcl 15720
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  =  ( invg `  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 3589 . . 3  |-  ( X  e.  ( B  \  {  .0.  } )  ->  X  e.  B )
2 grpinvnzcl.b . . . 4  |-  B  =  ( Base `  G
)
3 grpinvnzcl.n . . . 4  |-  N  =  ( invg `  G )
42, 3grpinvcl 15705 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
51, 4sylan2 474 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  B
)
6 eldifsn 4111 . . 3  |-  ( X  e.  ( B  \  {  .0.  } )  <->  ( X  e.  B  /\  X  =/= 
.0.  ) )
7 grpinvnzcl.z . . . . 5  |-  .0.  =  ( 0g `  G )
82, 7, 3grpinvnz 15719 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( N `  X
)  =/=  .0.  )
983expb 1189 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( N `  X )  =/=  .0.  )
106, 9sylan2b 475 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  =/=  .0.  )
11 eldifsn 4111 . 2  |-  ( ( N `  X )  e.  ( B  \  {  .0.  } )  <->  ( ( N `  X )  e.  B  /\  ( N `  X )  =/=  .0.  ) )
125, 10, 11sylanbrc 664 1  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  ( B  \  {  .0.  } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648    \ cdif 3436   {csn 3988   ` cfv 5529   Basecbs 14295   0gc0g 14500   Grpcgrp 15532   invgcminusg 15533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-0g 14502  df-mnd 15537  df-grp 15667  df-minusg 15668
This theorem is referenced by:  islindf4  18395  lindslinindsimp1  31143  lindslinindsimp2lem5  31148  baerlem5amN  35719  baerlem5bmN  35720  baerlem5abmN  35721  hdmap1neglem1N  35831
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