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Theorem grpinvid1 15947
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 6302 . . . 4  |-  ( ( N `  X )  =  Y  ->  ( X  .+  ( N `  X ) )  =  ( X  .+  Y
) )
21adantl 466 . . 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 15946 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
873adant3 1016 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
98adantr 465 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( X  .+  ( N `  X ) )  =  .0.  )
102, 9eqtr3d 2510 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( X  .+  Y )  =  .0.  )
11 oveq2 6302 . . . 4  |-  ( ( X  .+  Y )  =  .0.  ->  (
( N `  X
)  .+  ( X  .+  Y ) )  =  ( ( N `  X )  .+  .0.  ) )
1211adantl 466 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  ( ( N `
 X )  .+  .0.  ) )
133, 4, 5, 6grplinv 15945 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
1413oveq1d 6309 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  (  .0.  .+  Y ) )
15143adant3 1016 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  (  .0.  .+  Y ) )
163, 6grpinvcl 15944 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
1716adantrr 716 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( N `  X )  e.  B )
18 simprl 755 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
19 simprr 756 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  Y  e.  B )
2017, 18, 193jca 1176 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( N `  X
)  e.  B  /\  X  e.  B  /\  Y  e.  B )
)
213, 4grpass 15913 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( ( N `  X )  e.  B  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( ( N `  X )  .+  X
)  .+  Y )  =  ( ( N `
 X )  .+  ( X  .+  Y ) ) )
2220, 21syldan 470 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( ( N `  X )  .+  X
)  .+  Y )  =  ( ( N `
 X )  .+  ( X  .+  Y ) ) )
23223impb 1192 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  ( ( N `  X ) 
.+  ( X  .+  Y ) ) )
2415, 23eqtr3d 2510 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  ( ( N `  X ) 
.+  ( X  .+  Y ) ) )
253, 4, 5grplid 15929 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  Y )
26253adant2 1015 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  Y )
2724, 26eqtr3d 2510 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  Y )
2827adantr 465 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  Y )
293, 4, 5grprid 15930 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3016, 29syldan 470 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
31303adant3 1016 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3231adantr 465 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3312, 28, 323eqtr3rd 2517 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( N `  X
)  =  Y )
3410, 33impbida 830 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 973    = wceq 1379    e. wcel 1767   ` cfv 5593  (class class class)co 6294   Basecbs 14502   +g cplusg 14567   0gc0g 14707   Grpcgrp 15902   invgcminusg 15903
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
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 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-grp 15906  df-minusg 15907
This theorem is referenced by:  grpinvid  15950  grpinvcnv  15955  grpinvadd  15965  subginv  16057  qusinv  16109  ghminv  16123  symginv  16276  frgpinv  16632  ringnegl  17089  lmodindp1  17508  lmodvsinv2  17531  cnfldneg  18291  zringinvg  18365  mdetunilem6  18965  invrvald  19024  dchrinv  23379  baerlem3lem1  36797
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