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Metamath Proof Explorer

Theorem ablsubsub4

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 ablsubsub4 
|- ( 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 3 grpsubcl 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) e. B )
11 9 5 6 10 syl3anc 
 |-  ( ph -> ( X .- Y ) e. B )
12 eqid 
 |-  ( invg ` G ) = ( invg ` G )
13 1 2 12 3 grpsubval 
 |-  ( ( ( X .- Y ) e. B /\ Z e. B ) -> ( ( X .- Y ) .- Z ) = ( ( X .- Y ) .+ ( ( invg ` G ) ` Z ) ) )
14 11 7 13 syl2anc 
 |-  ( ph -> ( ( X .- Y ) .- Z ) = ( ( X .- Y ) .+ ( ( invg ` G ) ` Z ) ) )
15 1 12 grpinvcl 
 |-  ( ( G e. Grp /\ Z e. B ) -> ( ( invg ` G ) ` Z ) e. B )
16 9 7 15 syl2anc 
 |-  ( ph -> ( ( invg ` G ) ` Z ) e. B )
17 1 2 3 4 5 6 16 ablsubsub 
 |-  ( ph -> ( X .- ( Y .- ( ( invg ` G ) ` Z ) ) ) = ( ( X .- Y ) .+ ( ( invg ` G ) ` Z ) ) )
18 1 2 3 12 9 6 7 grpsubinv 
 |-  ( ph -> ( Y .- ( ( invg ` G ) ` Z ) ) = ( Y .+ Z ) )
19 18 oveq2d 
 |-  ( ph -> ( X .- ( Y .- ( ( invg ` G ) ` Z ) ) ) = ( X .- ( Y .+ Z ) ) )
20 14 17 19 3eqtr2d 
 |-  ( ph -> ( ( X .- Y ) .- Z ) = ( X .- ( Y .+ Z ) ) )