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Metamath Proof Explorer

Theorem ablpnpcan

Description: Cancellation law for mixed addition and subtraction. ( pnpcan analog.) (Contributed by NM, 29-May-2015)

Ref Expression
Hypotheses 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 ablpnpcan 
|- ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( Y .- Z ) )

Proof

Step Hyp Ref Expression
1 ablsubadd.b 
 |-  B = ( Base ` G )
2 ablsubadd.p 
 |-  .+ = ( +g ` G )
3 ablsubadd.m 
 |-  .- = ( -g ` G )
4 ablsubsub.g 
 |-  ( ph -> G e. Abel )
5 ablsubsub.x 
 |-  ( ph -> X e. B )
6 ablsubsub.y 
 |-  ( ph -> Y e. B )
7 ablsubsub.z 
 |-  ( ph -> Z e. B )
8 ablpnpcan.g 
 |-  ( ph -> G e. Abel )
9 ablpnpcan.x 
 |-  ( ph -> X e. B )
10 ablpnpcan.y 
 |-  ( ph -> Y e. B )
11 ablpnpcan.z 
 |-  ( ph -> Z e. B )
12 1 2 3 ablsub4 
 |-  ( ( G e. Abel /\ ( X e. B /\ Y e. B ) /\ ( X e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( ( X .- X ) .+ ( Y .- Z ) ) )
13 4 5 6 5 7 12 syl122anc 
 |-  ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( ( X .- X ) .+ ( Y .- Z ) ) )
14 ablgrp 
 |-  ( G e. Abel -> G e. Grp )
15 4 14 syl 
 |-  ( ph -> G e. Grp )
16 eqid 
 |-  ( 0g ` G ) = ( 0g ` G )
17 1 16 3 grpsubid 
 |-  ( ( G e. Grp /\ X e. B ) -> ( X .- X ) = ( 0g ` G ) )
18 15 5 17 syl2anc 
 |-  ( ph -> ( X .- X ) = ( 0g ` G ) )
19 18 oveq1d 
 |-  ( ph -> ( ( X .- X ) .+ ( Y .- Z ) ) = ( ( 0g ` G ) .+ ( Y .- Z ) ) )
20 1 3 grpsubcl 
 |-  ( ( G e. Grp /\ Y e. B /\ Z e. B ) -> ( Y .- Z ) e. B )
21 15 6 7 20 syl3anc 
 |-  ( ph -> ( Y .- Z ) e. B )
22 1 2 16 grplid 
 |-  ( ( G e. Grp /\ ( Y .- Z ) e. B ) -> ( ( 0g ` G ) .+ ( Y .- Z ) ) = ( Y .- Z ) )
23 15 21 22 syl2anc 
 |-  ( ph -> ( ( 0g ` G ) .+ ( Y .- Z ) ) = ( Y .- Z ) )
24 13 19 23 3eqtrd 
 |-  ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( Y .- Z ) )