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Theorem grpinvid2 15971
Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)
Hypotheses
Ref Expression
grpinv.b  |-  B  =  ( Base `  G
)
grpinv.p  |-  .+  =  ( +g  `  G )
grpinv.u  |-  .0.  =  ( 0g `  G )
grpinv.n  |-  N  =  ( invg `  G )
Assertion
Ref Expression
grpinvid2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <-> 
( Y  .+  X
)  =  .0.  )
)

Proof of Theorem grpinvid2
StepHypRef Expression
1 oveq1 6302 . . . 4  |-  ( ( N `  X )  =  Y  ->  (
( N `  X
)  .+  X )  =  ( Y  .+  X ) )
21adantl 466 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( ( N `
 X )  .+  X )  =  ( Y  .+  X ) )
3 grpinv.b . . . . . 6  |-  B  =  ( Base `  G
)
4 grpinv.p . . . . . 6  |-  .+  =  ( +g  `  G )
5 grpinv.u . . . . . 6  |-  .0.  =  ( 0g `  G )
6 grpinv.n . . . . . 6  |-  N  =  ( invg `  G )
73, 4, 5, 6grplinv 15968 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
873adant3 1016 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
98adantr 465 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( ( N `
 X )  .+  X )  =  .0.  )
102, 9eqtr3d 2510 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( Y  .+  X )  =  .0.  )
113, 6grpinvcl 15967 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
123, 4, 5grplid 15952 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
(  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
1311, 12syldan 470 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
14133adant3 1016 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
1514eqcomd 2475 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  X
)  =  (  .0.  .+  ( N `  X
) ) )
1615adantr 465 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( N `  X
)  =  (  .0.  .+  ( N `  X
) ) )
17 oveq1 6302 . . . 4  |-  ( ( Y  .+  X )  =  .0.  ->  (
( Y  .+  X
)  .+  ( N `  X ) )  =  (  .0.  .+  ( N `  X )
) )
1817adantl 466 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  (  .0.  .+  ( N `  X
) ) )
19 simprr 756 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  Y  e.  B )
20 simprl 755 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
2111adantrr 716 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( N `  X )  e.  B )
2219, 20, 213jca 1176 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( Y  e.  B  /\  X  e.  B  /\  ( N `  X )  e.  B ) )
233, 4grpass 15936 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( Y  e.  B  /\  X  e.  B  /\  ( N `  X
)  e.  B ) )  ->  ( ( Y  .+  X )  .+  ( N `  X ) )  =  ( Y 
.+  ( X  .+  ( N `  X ) ) ) )
2422, 23syldan 470 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( Y  .+  X
)  .+  ( N `  X ) )  =  ( Y  .+  ( X  .+  ( N `  X ) ) ) )
25243impb 1192 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  ( Y 
.+  ( X  .+  ( N `  X ) ) ) )
263, 4, 5, 6grprinv 15969 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
2726oveq2d 6311 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( Y  .+  ( X  .+  ( N `  X ) ) )  =  ( Y  .+  .0.  ) )
28273adant3 1016 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .+  ( X  .+  ( N `  X ) ) )  =  ( Y  .+  .0.  ) )
293, 4, 5grprid 15953 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( Y  .+  .0.  )  =  Y )
30293adant2 1015 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .+  .0.  )  =  Y )
3125, 28, 303eqtrd 2512 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  Y )
3231adantr 465 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  Y )
3316, 18, 323eqtr2d 2514 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( N `  X
)  =  Y )
3410, 33impbida 830 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <-> 
( Y  .+  X
)  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   0gc0g 14712   Grpcgrp 15925   invgcminusg 15926
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
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 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930
This theorem is referenced by:  grpinvcnv  15978  grpsubeq0  15996  prdsinvgd  16052  rngnegr  17113  psrneg  17923  islindf4  18742  pi1inv  21420  lindslinindimp2lem4  32544  lincresunit3  32564
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