This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ablsubsub

Description: Law for double subtraction. (Contributed by NM, 7-Apr-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 )
Assertion ablsubsub 
|- ( ph -> ( X .- ( Y .- Z ) ) = ( ( X .- 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 ablgrp 
 |-  ( G e. Abel -> G e. Grp )
9 4 8 syl 
 |-  ( ph -> G e. Grp )
10 1 2 3 grpsubsub 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .- ( Y .- Z ) ) = ( X .+ ( Z .- Y ) ) )
11 9 5 6 7 10 syl13anc 
 |-  ( ph -> ( X .- ( Y .- Z ) ) = ( X .+ ( Z .- Y ) ) )
12 1 2 3 grpaddsubass 
 |-  ( ( G e. Grp /\ ( X e. B /\ Z e. B /\ Y e. B ) ) -> ( ( X .+ Z ) .- Y ) = ( X .+ ( Z .- Y ) ) )
13 9 5 7 6 12 syl13anc 
 |-  ( ph -> ( ( X .+ Z ) .- Y ) = ( X .+ ( Z .- Y ) ) )
14 1 2 3 abladdsub 
 |-  ( ( G e. Abel /\ ( X e. B /\ Z e. B /\ Y e. B ) ) -> ( ( X .+ Z ) .- Y ) = ( ( X .- Y ) .+ Z ) )
15 4 5 7 6 14 syl13anc 
 |-  ( ph -> ( ( X .+ Z ) .- Y ) = ( ( X .- Y ) .+ Z ) )
16 11 13 15 3eqtr2d 
 |-  ( ph -> ( X .- ( Y .- Z ) ) = ( ( X .- Y ) .+ Z ) )