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Theorem grpinvid2 15580
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 6097 . . . 4  |-  ( ( N `  X )  =  Y  ->  (
( N `  X
)  .+  X )  =  ( Y  .+  X ) )
21adantl 463 . . 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 15577 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
873adant3 1003 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
98adantr 462 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( ( N `
 X )  .+  X )  =  .0.  )
102, 9eqtr3d 2475 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( Y  .+  X )  =  .0.  )
113, 6grpinvcl 15576 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
123, 4, 5grplid 15561 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
(  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
1311, 12syldan 467 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
14133adant3 1003 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
1514eqcomd 2446 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  X
)  =  (  .0.  .+  ( N `  X
) ) )
1615adantr 462 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( N `  X
)  =  (  .0.  .+  ( N `  X
) ) )
17 oveq1 6097 . . . 4  |-  ( ( Y  .+  X )  =  .0.  ->  (
( Y  .+  X
)  .+  ( N `  X ) )  =  (  .0.  .+  ( N `  X )
) )
1817adantl 463 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  (  .0.  .+  ( N `  X
) ) )
19 simprr 751 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  Y  e.  B )
20 simprl 750 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
2111adantrr 711 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( N `  X )  e.  B )
2219, 20, 213jca 1163 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( Y  e.  B  /\  X  e.  B  /\  ( N `  X )  e.  B ) )
233, 4grpass 15545 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( Y  e.  B  /\  X  e.  B  /\  ( N `  X
)  e.  B ) )  ->  ( ( Y  .+  X )  .+  ( N `  X ) )  =  ( Y 
.+  ( X  .+  ( N `  X ) ) ) )
2422, 23syldan 467 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( Y  .+  X
)  .+  ( N `  X ) )  =  ( Y  .+  ( X  .+  ( N `  X ) ) ) )
25243impb 1178 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  ( Y 
.+  ( X  .+  ( N `  X ) ) ) )
263, 4, 5, 6grprinv 15578 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
2726oveq2d 6106 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( Y  .+  ( X  .+  ( N `  X ) ) )  =  ( Y  .+  .0.  ) )
28273adant3 1003 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .+  ( X  .+  ( N `  X ) ) )  =  ( Y  .+  .0.  ) )
293, 4, 5grprid 15562 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( Y  .+  .0.  )  =  Y )
30293adant2 1002 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .+  .0.  )  =  Y )
3125, 28, 303eqtrd 2477 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  Y )
3231adantr 462 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  Y )
3316, 18, 323eqtr2d 2479 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( N `  X
)  =  Y )
3410, 33impbida 823 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 960    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   0gc0g 14374   Grpcgrp 15406   invgcminusg 15407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539
This theorem is referenced by:  grpinvcnv  15587  grpsubeq0  15605  prdsinvgd  15658  rngnegr  16676  psrneg  17449  islindf4  18226  pi1inv  20583  lindslinindimp2lem4  30836  lincresunit3  30856
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