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Theorem grppnpcan2 14887
Description: Cancellation law for mixed addition and subtraction. (pnpcan2 9346 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
grpsubadd.b  |-  B  =  ( Base `  G
)
grpsubadd.p  |-  .+  =  ( +g  `  G )
grpsubadd.m  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
grppnpcan2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Z
)  .-  ( Y  .+  Z ) )  =  ( X  .-  Y
) )

Proof of Theorem grppnpcan2
StepHypRef Expression
1 simpl 445 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  G  e.  Grp )
2 grpsubadd.b . . . . 5  |-  B  =  ( Base `  G
)
3 grpsubadd.p . . . . 5  |-  .+  =  ( +g  `  G )
42, 3grpcl 14823 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .+  Z
)  e.  B )
543adant3r2 1164 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .+  Z )  e.  B )
6 simpr3 966 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
7 simpr2 965 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
8 grpsubadd.m . . . 4  |-  .-  =  ( -g `  G )
92, 3, 8grpsubsub4 14886 . . 3  |-  ( ( G  e.  Grp  /\  ( ( X  .+  Z )  e.  B  /\  Z  e.  B  /\  Y  e.  B
) )  ->  (
( ( X  .+  Z )  .-  Z
)  .-  Y )  =  ( ( X 
.+  Z )  .-  ( Y  .+  Z ) ) )
101, 5, 6, 7, 9syl13anc 1187 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  .+  Z )  .-  Z
)  .-  Y )  =  ( ( X 
.+  Z )  .-  ( Y  .+  Z ) ) )
112, 3, 8grppncan 14884 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Z )  .-  Z
)  =  X )
12113adant3r2 1164 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Z
)  .-  Z )  =  X )
1312oveq1d 6099 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  .+  Z )  .-  Z
)  .-  Y )  =  ( X  .-  Y ) )
1410, 13eqtr3d 2472 1  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Z
)  .-  ( Y  .+  Z ) )  =  ( X  .-  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   Basecbs 13474   +g cplusg 13534   Grpcgrp 14690   -gcsg 14693
This theorem is referenced by:  ngprcan  18661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818  df-sbg 14819
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