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Metamath Proof Explorer

Theorem ablsubadd23

Description: Commutative/associative law for addition and subtraction in abelian groups. ( subadd23d analog.) (Contributed by AV, 2-Mar-2025)

Ref Expression
Hypotheses ablsubadd.b 
|- B = ( Base ` G )
ablsubadd.p 
|- .+ = ( +g ` G )
ablsubadd.m 
|- .- = ( -g ` G )
Assertion ablsubadd23 
|- ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) .+ Z ) = ( X .+ ( Z .- Y ) ) )

Proof

Step Hyp Ref Expression
1 ablsubadd.b 
 |-  B = ( Base ` G )
2 ablsubadd.p 
 |-  .+ = ( +g ` G )
3 ablsubadd.m 
 |-  .- = ( -g ` G )
4 3ancomb 
 |-  ( ( X e. B /\ Y e. B /\ Z e. B ) <-> ( X e. B /\ Z e. B /\ Y e. B ) )
5 4 biimpi 
 |-  ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( X e. B /\ Z e. B /\ Y e. B ) )
6 1 2 3 abladdsub 
 |-  ( ( G e. Abel /\ ( X e. B /\ Z e. B /\ Y e. B ) ) -> ( ( X .+ Z ) .- Y ) = ( ( X .- Y ) .+ Z ) )
7 5 6 sylan2 
 |-  ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) .- Y ) = ( ( X .- Y ) .+ Z ) )
8 ablgrp 
 |-  ( G e. Abel -> G e. Grp )
9 1 2 3 grpaddsubass 
 |-  ( ( G e. Grp /\ ( X e. B /\ Z e. B /\ Y e. B ) ) -> ( ( X .+ Z ) .- Y ) = ( X .+ ( Z .- Y ) ) )
10 8 5 9 syl2an 
 |-  ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) .- Y ) = ( X .+ ( Z .- Y ) ) )
11 7 10 eqtr3d 
 |-  ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) .+ Z ) = ( X .+ ( Z .- Y ) ) )