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Theorem grpinvid2 15587
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 6098 . . . 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 15584 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
873adant3 1008 . . . 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 2477 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( Y  .+  X )  =  .0.  )
113, 6grpinvcl 15583 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
123, 4, 5grplid 15568 . . . . . . 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 1008 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
1514eqcomd 2448 . . . 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 6098 . . . 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 1168 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( Y  e.  B  /\  X  e.  B  /\  ( N `  X )  e.  B ) )
233, 4grpass 15552 . . . . . . 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 1183 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  ( Y 
.+  ( X  .+  ( N `  X ) ) ) )
263, 4, 5, 6grprinv 15585 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
2726oveq2d 6107 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( Y  .+  ( X  .+  ( N `  X ) ) )  =  ( Y  .+  .0.  ) )
28273adant3 1008 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .+  ( X  .+  ( N `  X ) ) )  =  ( Y  .+  .0.  ) )
293, 4, 5grprid 15569 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( Y  .+  .0.  )  =  Y )
30293adant2 1007 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .+  .0.  )  =  Y )
3125, 28, 303eqtrd 2479 . . . 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 2481 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( N `  X
)  =  Y )
3410, 33impbida 828 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 965    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   0gc0g 14378   Grpcgrp 15410   invgcminusg 15411
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546
This theorem is referenced by:  grpinvcnv  15594  grpsubeq0  15612  prdsinvgd  15665  rngnegr  16686  psrneg  17471  islindf4  18267  pi1inv  20624  lindslinindimp2lem4  30995  lincresunit3  31015
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