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Theorem grpinvnz 15958
Description: The inverse of a nonzero group element is not zero. (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
grpinvnz  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( N `  X
)  =/=  .0.  )

Proof of Theorem grpinvnz
StepHypRef Expression
1 fveq2 5871 . . . . . 6  |-  ( ( N `  X )  =  .0.  ->  ( N `  ( N `  X ) )  =  ( N `  .0.  ) )
21adantl 466 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( N `  X )  =  .0.  )  ->  ( N `  ( N `  X
) )  =  ( N `  .0.  )
)
3 grpinvnzcl.b . . . . . . 7  |-  B  =  ( Base `  G
)
4 grpinvnzcl.n . . . . . . 7  |-  N  =  ( invg `  G )
53, 4grpinvinv 15954 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  ( N `  X )
)  =  X )
65adantr 465 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( N `  X )  =  .0.  )  ->  ( N `  ( N `  X
) )  =  X )
7 grpinvnzcl.z . . . . . . 7  |-  .0.  =  ( 0g `  G )
87, 4grpinvid 15950 . . . . . 6  |-  ( G  e.  Grp  ->  ( N `  .0.  )  =  .0.  )
98ad2antrr 725 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( N `  X )  =  .0.  )  ->  ( N `  .0.  )  =  .0.  )
102, 6, 93eqtr3d 2516 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( N `  X )  =  .0.  )  ->  X  =  .0.  )
1110ex 434 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  =  .0. 
->  X  =  .0.  ) )
1211necon3d 2691 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  =/=  .0.  ->  ( N `  X
)  =/=  .0.  )
)
13123impia 1193 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( N `  X
)  =/=  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5593   Basecbs 14502   0gc0g 14707   Grpcgrp 15902   invgcminusg 15903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-grp 15906  df-minusg 15907
This theorem is referenced by:  grpinvnzcl  15959
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