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Theorem ablpnpcan 16956
Description: Cancellation law for mixed addition and subtraction. (pnpcan 9877 analog.) (Contributed by NM, 29-May-2015.)
Hypotheses
Ref Expression
ablsubadd.b  |-  B  =  ( Base `  G
)
ablsubadd.p  |-  .+  =  ( +g  `  G )
ablsubadd.m  |-  .-  =  ( -g `  G )
ablsubsub.g  |-  ( ph  ->  G  e.  Abel )
ablsubsub.x  |-  ( ph  ->  X  e.  B )
ablsubsub.y  |-  ( ph  ->  Y  e.  B )
ablsubsub.z  |-  ( ph  ->  Z  e.  B )
ablpnpcan.g  |-  ( ph  ->  G  e.  Abel )
ablpnpcan.x  |-  ( ph  ->  X  e.  B )
ablpnpcan.y  |-  ( ph  ->  Y  e.  B )
ablpnpcan.z  |-  ( ph  ->  Z  e.  B )
Assertion
Ref Expression
ablpnpcan  |-  ( ph  ->  ( ( X  .+  Y )  .-  ( X  .+  Z ) )  =  ( Y  .-  Z ) )

Proof of Theorem ablpnpcan
StepHypRef Expression
1 ablsubsub.g . . 3  |-  ( ph  ->  G  e.  Abel )
2 ablsubsub.x . . 3  |-  ( ph  ->  X  e.  B )
3 ablsubsub.y . . 3  |-  ( ph  ->  Y  e.  B )
4 ablsubsub.z . . 3  |-  ( ph  ->  Z  e.  B )
5 ablsubadd.b . . . 4  |-  B  =  ( Base `  G
)
6 ablsubadd.p . . . 4  |-  .+  =  ( +g  `  G )
7 ablsubadd.m . . . 4  |-  .-  =  ( -g `  G )
85, 6, 7ablsub4 16949 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .-  ( X  .+  Z ) )  =  ( ( X  .-  X )  .+  ( Y  .-  Z ) ) )
91, 2, 3, 2, 4, 8syl122anc 1237 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .-  ( X  .+  Z ) )  =  ( ( X 
.-  X )  .+  ( Y  .-  Z ) ) )
10 ablgrp 16929 . . . . 5  |-  ( G  e.  Abel  ->  G  e. 
Grp )
111, 10syl 16 . . . 4  |-  ( ph  ->  G  e.  Grp )
12 eqid 2457 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
135, 12, 7grpsubid 16248 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .-  X
)  =  ( 0g
`  G ) )
1411, 2, 13syl2anc 661 . . 3  |-  ( ph  ->  ( X  .-  X
)  =  ( 0g
`  G ) )
1514oveq1d 6311 . 2  |-  ( ph  ->  ( ( X  .-  X )  .+  ( Y  .-  Z ) )  =  ( ( 0g
`  G )  .+  ( Y  .-  Z ) ) )
165, 7grpsubcl 16244 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  e.  B )
1711, 3, 4, 16syl3anc 1228 . . 3  |-  ( ph  ->  ( Y  .-  Z
)  e.  B )
185, 6, 12grplid 16206 . . 3  |-  ( ( G  e.  Grp  /\  ( Y  .-  Z )  e.  B )  -> 
( ( 0g `  G )  .+  ( Y  .-  Z ) )  =  ( Y  .-  Z ) )
1911, 17, 18syl2anc 661 . 2  |-  ( ph  ->  ( ( 0g `  G )  .+  ( Y  .-  Z ) )  =  ( Y  .-  Z ) )
209, 15, 193eqtrd 2502 1  |-  ( ph  ->  ( ( X  .+  Y )  .-  ( X  .+  Z ) )  =  ( Y  .-  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711   0gc0g 14856   Grpcgrp 16179   -gcsg 16181   Abelcabl 16925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183  df-minusg 16184  df-sbg 16185  df-cmn 16926  df-abl 16927
This theorem is referenced by:  hdmaprnlem7N  37686
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