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Theorem grpinvid1 15579
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
grpinvid1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <-> 
( X  .+  Y
)  =  .0.  )
)

Proof of Theorem grpinvid1
StepHypRef Expression
1 oveq2 6098 . . . 4  |-  ( ( N `  X )  =  Y  ->  ( X  .+  ( N `  X ) )  =  ( X  .+  Y
) )
21adantl 463 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( X  .+  ( N `  X ) )  =  ( X 
.+  Y ) )
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, 6grprinv 15578 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
873adant3 1003 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
98adantr 462 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( X  .+  ( N `  X ) )  =  .0.  )
102, 9eqtr3d 2475 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( X  .+  Y )  =  .0.  )
11 oveq2 6098 . . . 4  |-  ( ( X  .+  Y )  =  .0.  ->  (
( N `  X
)  .+  ( X  .+  Y ) )  =  ( ( N `  X )  .+  .0.  ) )
1211adantl 463 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  ( ( N `
 X )  .+  .0.  ) )
133, 4, 5, 6grplinv 15577 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
1413oveq1d 6105 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  (  .0.  .+  Y ) )
15143adant3 1003 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  (  .0.  .+  Y ) )
163, 6grpinvcl 15576 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
1716adantrr 711 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( N `  X )  e.  B )
18 simprl 750 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
19 simprr 751 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  Y  e.  B )
2017, 18, 193jca 1163 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( N `  X
)  e.  B  /\  X  e.  B  /\  Y  e.  B )
)
213, 4grpass 15545 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( ( N `  X )  e.  B  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( ( N `  X )  .+  X
)  .+  Y )  =  ( ( N `
 X )  .+  ( X  .+  Y ) ) )
2220, 21syldan 467 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( ( N `  X )  .+  X
)  .+  Y )  =  ( ( N `
 X )  .+  ( X  .+  Y ) ) )
23223impb 1178 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  ( ( N `  X ) 
.+  ( X  .+  Y ) ) )
2415, 23eqtr3d 2475 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  ( ( N `  X ) 
.+  ( X  .+  Y ) ) )
253, 4, 5grplid 15561 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  Y )
26253adant2 1002 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  Y )
2724, 26eqtr3d 2475 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  Y )
2827adantr 462 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  Y )
293, 4, 5grprid 15562 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3016, 29syldan 467 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
31303adant3 1003 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3231adantr 462 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3312, 28, 323eqtr3rd 2482 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( N `  X
)  =  Y )
3410, 33impbida 823 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <-> 
( X  .+  Y
)  =  .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 2263  df-mo 2264  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:  grpinvid  15582  grpinvcnv  15587  grpinvadd  15597  subginv  15681  divsinv  15733  ghminv  15747  symginv  15900  frgpinv  16254  rngnegl  16675  lmodindp1  17073  lmodvsinv2  17096  cnfldneg  17742  zringinvg  17813  mdetunilem6  18323  invrvald  18382  dchrinv  22543  baerlem3lem1  35040
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