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Theorem grpnpcan 14573
Description: Cancellation law for subtraction (npcan 9076 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 2296 . . . . . 6  |-  ( inv g `  G )  =  ( inv g `  G )
31, 2grpinvcl 14543 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  Y
)  e.  B )
433adant2 974 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  Y
)  e.  B )
5 grpsubadd.p . . . . 5  |-  .+  =  ( +g  `  G )
61, 5grpcl 14511 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( ( inv g `  G ) `  Y
)  e.  B )  ->  ( X  .+  ( ( inv g `  G ) `  Y
) )  e.  B
)
74, 6syld3an3 1227 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  (
( inv g `  G ) `  Y
) )  e.  B
)
8 grpsubadd.m . . . 4  |-  .-  =  ( -g `  G )
91, 5, 2, 8grpsubval 14541 . . 3  |-  ( ( ( X  .+  (
( inv g `  G ) `  Y
) )  e.  B  /\  ( ( inv g `  G ) `  Y
)  e.  B )  ->  ( ( X 
.+  ( ( inv g `  G ) `
 Y ) ) 
.-  ( ( inv g `  G ) `
 Y ) )  =  ( ( X 
.+  ( ( inv g `  G ) `
 Y ) ) 
.+  ( ( inv g `  G ) `
 ( ( inv g `  G ) `
 Y ) ) ) )
107, 4, 9syl2anc 642 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( ( inv g `  G ) `  Y
) )  .-  (
( inv g `  G ) `  Y
) )  =  ( ( X  .+  (
( inv g `  G ) `  Y
) )  .+  (
( inv g `  G ) `  (
( inv g `  G ) `  Y
) ) ) )
111, 5, 8grppncan 14572 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( ( inv g `  G ) `  Y
)  e.  B )  ->  ( ( X 
.+  ( ( inv g `  G ) `
 Y ) ) 
.-  ( ( inv g `  G ) `
 Y ) )  =  X )
124, 11syld3an3 1227 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( ( inv g `  G ) `  Y
) )  .-  (
( inv g `  G ) `  Y
) )  =  X )
131, 5, 2, 8grpsubval 14541 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( inv g `  G ) `
 Y ) ) )
14133adant1 973 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( inv g `  G ) `
 Y ) ) )
1514eqcomd 2301 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  (
( inv g `  G ) `  Y
) )  =  ( X  .-  Y ) )
161, 2grpinvinv 14551 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  (
( inv g `  G ) `  Y
) )  =  Y )
17163adant2 974 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  (
( inv g `  G ) `  Y
) )  =  Y )
1815, 17oveq12d 5892 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( ( inv g `  G ) `  Y
) )  .+  (
( inv g `  G ) `  (
( inv g `  G ) `  Y
) ) )  =  ( ( X  .-  Y )  .+  Y
) )
1910, 12, 183eqtr3rd 2337 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  .+  Y
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   inv gcminusg 14379   -gcsg 14381
This theorem is referenced by:  grpsubsub4  14574  grpnpncan  14576  grpnnncan2  14577  nsgconj  14666  conjghm  14729  conjnmz  14732  sylow2blem1  14947  ablpncan3  15134  lmodvnpcan  15695  coe1subfv  16359  ipsubdir  16562  ipsubdi  16563  subgntr  17805  ghmcnp  17813  tgpt0  17817  r1pid  19561  kercvrlsm  27284  hbtlem5  27435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507
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