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Theorem grpinvid2 14531
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
grpinvid2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <-> 
( Y  .+  X
)  =  .0.  )
)

Proof of Theorem grpinvid2
StepHypRef Expression
1 oveq1 5865 . . . 4  |-  ( ( N `  X )  =  Y  ->  (
( N `  X
)  .+  X )  =  ( Y  .+  X ) )
21adantl 452 . . 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  =  ( inv g `  G )
73, 4, 5, 6grplinv 14528 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
873adant3 975 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
98adantr 451 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( ( N `
 X )  .+  X )  =  .0.  )
102, 9eqtr3d 2317 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( Y  .+  X )  =  .0.  )
113, 6grpinvcl 14527 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
123, 4, 5grplid 14512 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
(  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
1311, 12syldan 456 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
14133adant3 975 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
1514eqcomd 2288 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  X
)  =  (  .0.  .+  ( N `  X
) ) )
1615adantr 451 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( N `  X
)  =  (  .0.  .+  ( N `  X
) ) )
17 oveq1 5865 . . . 4  |-  ( ( Y  .+  X )  =  .0.  ->  (
( Y  .+  X
)  .+  ( N `  X ) )  =  (  .0.  .+  ( N `  X )
) )
1817adantl 452 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  (  .0.  .+  ( N `  X
) ) )
19 simprr 733 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  Y  e.  B )
20 simprl 732 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
2111adantrr 697 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( N `  X )  e.  B )
2219, 20, 213jca 1132 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( Y  e.  B  /\  X  e.  B  /\  ( N `  X )  e.  B ) )
233, 4grpass 14496 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( Y  e.  B  /\  X  e.  B  /\  ( N `  X
)  e.  B ) )  ->  ( ( Y  .+  X )  .+  ( N `  X ) )  =  ( Y 
.+  ( X  .+  ( N `  X ) ) ) )
2422, 23syldan 456 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( Y  .+  X
)  .+  ( N `  X ) )  =  ( Y  .+  ( X  .+  ( N `  X ) ) ) )
25243impb 1147 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  ( Y 
.+  ( X  .+  ( N `  X ) ) ) )
263, 4, 5, 6grprinv 14529 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
2726oveq2d 5874 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( Y  .+  ( X  .+  ( N `  X ) ) )  =  ( Y  .+  .0.  ) )
28273adant3 975 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .+  ( X  .+  ( N `  X ) ) )  =  ( Y  .+  .0.  ) )
293, 4, 5grprid 14513 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( Y  .+  .0.  )  =  Y )
30293adant2 974 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .+  .0.  )  =  Y )
3125, 28, 303eqtrd 2319 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  Y )
3231adantr 451 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  Y )
3316, 18, 323eqtr2d 2321 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( N `  X
)  =  Y )
3410, 33impbida 805 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <-> 
( Y  .+  X
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363
This theorem is referenced by:  grpinvcnv  14536  grpsubeq0  14552  prdsinvgd  14605  rngnegr  15381  psrneg  16145  pi1inv  18550  islindf4  27308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490
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