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Theorem grpinvnzcl 16324
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 3562 . . 3  |-  ( X  e.  ( B  \  {  .0.  } )  ->  X  e.  B )
2 grpinvnzcl.b . . . 4  |-  B  =  ( Base `  G
)
3 grpinvnzcl.n . . . 4  |-  N  =  ( invg `  G )
42, 3grpinvcl 16309 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
51, 4sylan2 472 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  B
)
6 eldifsn 4094 . . 3  |-  ( X  e.  ( B  \  {  .0.  } )  <->  ( X  e.  B  /\  X  =/= 
.0.  ) )
7 grpinvnzcl.z . . . . 5  |-  .0.  =  ( 0g `  G )
82, 7, 3grpinvnz 16323 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( N `  X
)  =/=  .0.  )
983expb 1196 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( N `  X )  =/=  .0.  )
106, 9sylan2b 473 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  =/=  .0.  )
11 eldifsn 4094 . 2  |-  ( ( N `  X )  e.  ( B  \  {  .0.  } )  <->  ( ( N `  X )  e.  B  /\  ( N `  X )  =/=  .0.  ) )
125, 10, 11sylanbrc 662 1  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  ( B  \  {  .0.  } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840    =/= wne 2596    \ cdif 3408   {csn 3969   ` cfv 5523   Basecbs 14731   0gc0g 14944   Grpcgrp 16267   invgcminusg 16268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-0g 14946  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-grp 16271  df-minusg 16272
This theorem is referenced by:  islindf4  19055  baerlem5amN  34700  baerlem5bmN  34701  baerlem5abmN  34702  hdmap1neglem1N  34812  lindslinindsimp1  38502  lindslinindsimp2lem5  38507
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