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Theorem grpinvnzcl 14540
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 3298 . . 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 14527 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
51, 4sylan2 460 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  B
)
6 eldifsn 3749 . . 3  |-  ( X  e.  ( B  \  {  .0.  } )  <->  ( X  e.  B  /\  X  =/= 
.0.  ) )
7 grpinvnzcl.z . . . . 5  |-  .0.  =  ( 0g `  G )
82, 7, 3grpinvnz 14539 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( N `  X
)  =/=  .0.  )
983expb 1152 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( N `  X )  =/=  .0.  )
106, 9sylan2b 461 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  =/=  .0.  )
11 eldifsn 3749 . 2  |-  ( ( N `  X )  e.  ( B  \  {  .0.  } )  <->  ( ( N `  X )  e.  B  /\  ( N `  X )  =/=  .0.  ) )
125, 10, 11sylanbrc 645 1  |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  ( B  \  {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363
This theorem is referenced by:  grpinvnzclOLD  26766  islindf4  27308  baerlem5amN  31906  baerlem5bmN  31907  baerlem5abmN  31908  hdmap1neglem1N  32018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490
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