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Theorem grpinvid1 14781
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  =  ( inv g `  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 6029 . . . 4  |-  ( ( N `  X )  =  Y  ->  ( X  .+  ( N `  X ) )  =  ( X  .+  Y
) )
21adantl 453 . . 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  =  ( inv g `  G )
73, 4, 5, 6grprinv 14780 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
873adant3 977 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
98adantr 452 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( X  .+  ( N `  X ) )  =  .0.  )
102, 9eqtr3d 2422 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( X  .+  Y )  =  .0.  )
11 oveq2 6029 . . . 4  |-  ( ( X  .+  Y )  =  .0.  ->  (
( N `  X
)  .+  ( X  .+  Y ) )  =  ( ( N `  X )  .+  .0.  ) )
1211adantl 453 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  ( ( N `
 X )  .+  .0.  ) )
133, 4, 5, 6grplinv 14779 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
1413oveq1d 6036 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  (  .0.  .+  Y ) )
15143adant3 977 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  (  .0.  .+  Y ) )
163, 6grpinvcl 14778 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
1716adantrr 698 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( N `  X )  e.  B )
18 simprl 733 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
19 simprr 734 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  Y  e.  B )
2017, 18, 193jca 1134 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( N `  X
)  e.  B  /\  X  e.  B  /\  Y  e.  B )
)
213, 4grpass 14747 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( ( N `  X )  e.  B  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( ( N `  X )  .+  X
)  .+  Y )  =  ( ( N `
 X )  .+  ( X  .+  Y ) ) )
2220, 21syldan 457 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( ( N `  X )  .+  X
)  .+  Y )  =  ( ( N `
 X )  .+  ( X  .+  Y ) ) )
23223impb 1149 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  ( ( N `  X ) 
.+  ( X  .+  Y ) ) )
2415, 23eqtr3d 2422 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  ( ( N `  X ) 
.+  ( X  .+  Y ) ) )
253, 4, 5grplid 14763 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  Y )
26253adant2 976 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  Y )
2724, 26eqtr3d 2422 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  Y )
2827adantr 452 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  Y )
293, 4, 5grprid 14764 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3016, 29syldan 457 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
31303adant3 977 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3231adantr 452 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3312, 28, 323eqtr3rd 2429 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( N `  X
)  =  Y )
3410, 33impbida 806 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <-> 
( X  .+  Y
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   Basecbs 13397   +g cplusg 13457   0gc0g 13651   Grpcgrp 14613   inv gcminusg 14614
This theorem is referenced by:  grpinvid  14784  grpinvcnv  14787  grpinvadd  14795  subginv  14879  divsinv  14927  ghminv  14941  symginv  15033  frgpinv  15324  rngnegl  15631  lmodindp1  16018  lmodvsinv2  16041  cnfldneg  16651  dchrinv  20913  baerlem3lem1  31823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-riota 6486  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741
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