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Theorem grpnpcan 15924
Description: Cancellation law for subtraction (npcan 9818 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 2460 . . . . . 6  |-  ( invg `  G )  =  ( invg `  G )
31, 2grpinvcl 15889 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
433adant2 1010 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
5 grpsubadd.p . . . . 5  |-  .+  =  ( +g  `  G )
61, 5grpcl 15857 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B )  ->  ( X  .+  ( ( invg `  G ) `  Y
) )  e.  B
)
74, 6syld3an3 1268 . . 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 15887 . . 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 15923 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B )  ->  ( ( X 
.+  ( ( invg `  G ) `
 Y ) ) 
.-  ( ( invg `  G ) `
 Y ) )  =  X )
124, 11syld3an3 1268 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( ( invg `  G ) `  Y
) )  .-  (
( invg `  G ) `  Y
) )  =  X )
131, 5, 2, 8grpsubval 15887 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
14133adant1 1009 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
1514eqcomd 2468 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  (
( invg `  G ) `  Y
) )  =  ( X  .-  Y ) )
161, 2grpinvinv 15899 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
17163adant2 1010 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
1815, 17oveq12d 6293 . 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 2510 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 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   Grpcgrp 15716   invgcminusg 15717   -gcsg 15719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853
This theorem is referenced by:  grpsubsub4  15925  grpnpncan  15927  grpnnncan2  15929  nsgconj  16022  conjghm  16085  conjnmz  16088  sylow2blem1  16429  ablpncan3  16616  lmodvnpcan  17340  coe1subfv  18071  ipsubdir  18437  ipsubdi  18438  mdetunilem9  18882  subgntr  20333  ghmcnp  20341  tgpt0  20345  r1pid  22288  archiabllem1a  27383  archiabllem2a  27386  ornglmulle  27444  orngrmulle  27445  kercvrlsm  30622  hbtlem5  30670
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