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Theorem grpnpcan 15597
Description: Cancellation law for subtraction (npcan 9607 analog). (Contributed by NM, 19-Apr-2014.)
Hypotheses
Ref Expression
grpsubadd.b  |-  B  =  ( Base `  G
)
grpsubadd.p  |-  .+  =  ( +g  `  G )
grpsubadd.m  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
grpnpcan  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  .+  Y
)  =  X )

Proof of Theorem grpnpcan
StepHypRef Expression
1 grpsubadd.b . . . . . 6  |-  B  =  ( Base `  G
)
2 eqid 2433 . . . . . 6  |-  ( invg `  G )  =  ( invg `  G )
31, 2grpinvcl 15563 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
433adant2 1000 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
5 grpsubadd.p . . . . 5  |-  .+  =  ( +g  `  G )
61, 5grpcl 15531 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B )  ->  ( X  .+  ( ( invg `  G ) `  Y
) )  e.  B
)
74, 6syld3an3 1256 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  (
( invg `  G ) `  Y
) )  e.  B
)
8 grpsubadd.m . . . 4  |-  .-  =  ( -g `  G )
91, 5, 2, 8grpsubval 15561 . . 3  |-  ( ( ( X  .+  (
( invg `  G ) `  Y
) )  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B )  ->  ( ( X 
.+  ( ( invg `  G ) `
 Y ) ) 
.-  ( ( invg `  G ) `
 Y ) )  =  ( ( X 
.+  ( ( invg `  G ) `
 Y ) ) 
.+  ( ( invg `  G ) `
 ( ( invg `  G ) `
 Y ) ) ) )
107, 4, 9syl2anc 654 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( ( invg `  G ) `  Y
) )  .-  (
( invg `  G ) `  Y
) )  =  ( ( X  .+  (
( invg `  G ) `  Y
) )  .+  (
( invg `  G ) `  (
( invg `  G ) `  Y
) ) ) )
111, 5, 8grppncan 15596 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B )  ->  ( ( X 
.+  ( ( invg `  G ) `
 Y ) ) 
.-  ( ( invg `  G ) `
 Y ) )  =  X )
124, 11syld3an3 1256 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( ( invg `  G ) `  Y
) )  .-  (
( invg `  G ) `  Y
) )  =  X )
131, 5, 2, 8grpsubval 15561 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
14133adant1 999 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
1514eqcomd 2438 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  (
( invg `  G ) `  Y
) )  =  ( X  .-  Y ) )
161, 2grpinvinv 15573 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
17163adant2 1000 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
1815, 17oveq12d 6098 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( ( invg `  G ) `  Y
) )  .+  (
( invg `  G ) `  (
( invg `  G ) `  Y
) ) )  =  ( ( X  .-  Y )  .+  Y
) )
1910, 12, 183eqtr3rd 2474 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  .+  Y
)  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 958    = wceq 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080   Basecbs 14157   +g cplusg 14221   Grpcgrp 15393   invgcminusg 15394   -gcsg 15396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-0g 14363  df-mnd 15398  df-grp 15525  df-minusg 15526  df-sbg 15527
This theorem is referenced by:  grpsubsub4  15598  grpnpncan  15600  grpnnncan2  15601  nsgconj  15694  conjghm  15757  conjnmz  15760  sylow2blem1  16099  ablpncan3  16286  lmodvnpcan  16923  coe1subfv  17618  ipsubdir  17913  ipsubdi  17914  mdetunilem9  18268  subgntr  19519  ghmcnp  19527  tgpt0  19531  r1pid  21516  archiabllem1a  26032  archiabllem2a  26035  ornglmulle  26126  orngrmulle  26127  kercvrlsm  29281  hbtlem5  29329
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