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Theorem grpinvnz 15602
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 5696 . . . . . 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 15598 . . . . . 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 15594 . . . . . 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 2483 . . . 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 2651 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  =/=  .0.  ->  ( N `  X
)  =/=  .0.  )
)
13123impia 1184 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   ` cfv 5423   Basecbs 14179   0gc0g 14383   Grpcgrp 15415   invgcminusg 15416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551
This theorem is referenced by:  grpinvnzcl  15603
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