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Theorem grpnpcan 15608
Description: Cancellation law for subtraction (npcan 9611 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 2438 . . . . . 6  |-  ( invg `  G )  =  ( invg `  G )
31, 2grpinvcl 15574 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
433adant2 1007 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
5 grpsubadd.p . . . . 5  |-  .+  =  ( +g  `  G )
61, 5grpcl 15542 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B )  ->  ( X  .+  ( ( invg `  G ) `  Y
) )  e.  B
)
74, 6syld3an3 1263 . . 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 15572 . . 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 661 . 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 15607 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B )  ->  ( ( X 
.+  ( ( invg `  G ) `
 Y ) ) 
.-  ( ( invg `  G ) `
 Y ) )  =  X )
124, 11syld3an3 1263 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( ( invg `  G ) `  Y
) )  .-  (
( invg `  G ) `  Y
) )  =  X )
131, 5, 2, 8grpsubval 15572 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
14133adant1 1006 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
1514eqcomd 2443 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  (
( invg `  G ) `  Y
) )  =  ( X  .-  Y ) )
161, 2grpinvinv 15584 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
17163adant2 1007 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
1815, 17oveq12d 6104 . 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 2479 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 965    = wceq 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   Grpcgrp 15402   invgcminusg 15403   -gcsg 15405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-0g 14372  df-mnd 15407  df-grp 15536  df-minusg 15537  df-sbg 15538
This theorem is referenced by:  grpsubsub4  15609  grpnpncan  15611  grpnnncan2  15612  nsgconj  15705  conjghm  15768  conjnmz  15771  sylow2blem1  16110  ablpncan3  16297  lmodvnpcan  16979  coe1subfv  17700  ipsubdir  18051  ipsubdi  18052  mdetunilem9  18406  subgntr  19657  ghmcnp  19665  tgpt0  19669  r1pid  21611  archiabllem1a  26176  archiabllem2a  26179  ornglmulle  26241  orngrmulle  26242  kercvrlsm  29407  hbtlem5  29455
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