MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablpnpcan Structured version   Unicode version

Theorem ablpnpcan 16307
Description: Cancellation law for mixed addition and subtraction. (pnpcan 9646 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 16300 . . 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 1227 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .-  ( X  .+  Z ) )  =  ( ( X 
.-  X )  .+  ( Y  .-  Z ) ) )
10 ablgrp 16280 . . . . 5  |-  ( G  e.  Abel  ->  G  e. 
Grp )
111, 10syl 16 . . . 4  |-  ( ph  ->  G  e.  Grp )
12 eqid 2441 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
135, 12, 7grpsubid 15608 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .-  X
)  =  ( 0g
`  G ) )
1411, 2, 13syl2anc 661 . . 3  |-  ( ph  ->  ( X  .-  X
)  =  ( 0g
`  G ) )
1514oveq1d 6104 . 2  |-  ( ph  ->  ( ( X  .-  X )  .+  ( Y  .-  Z ) )  =  ( ( 0g
`  G )  .+  ( Y  .-  Z ) ) )
165, 7grpsubcl 15604 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  e.  B )
1711, 3, 4, 16syl3anc 1218 . . 3  |-  ( ph  ->  ( Y  .-  Z
)  e.  B )
185, 6, 12grplid 15566 . . 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 2477 1  |-  ( ph  ->  ( ( X  .+  Y )  .-  ( X  .+  Z ) )  =  ( Y  .-  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5416  (class class class)co 6089   Basecbs 14172   +g cplusg 14236   0gc0g 14376   Grpcgrp 15408   -gcsg 15411   Abelcabel 16276
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
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 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-0g 14378  df-mnd 15413  df-grp 15543  df-minusg 15544  df-sbg 15545  df-cmn 16277  df-abl 16278
This theorem is referenced by:  hdmaprnlem7N  35500
  Copyright terms: Public domain W3C validator