Theorem ablsubadd

Description: Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014)

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

Proof

Step	Hyp	Ref	Expression
1		ablsubadd.b	|- B = ( Base ` G )
2		ablsubadd.p	|- .+ = ( +g ` G )
3		ablsubadd.m	|- .- = ( -g ` G )
4		ablgrp	|- ( G e. Abel -> G e. Grp )
5	1 2 3	grpsubadd	|- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) = Z <-> ( Z .+ Y ) = X ) )
6	4 5	sylan	|- ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) = Z <-> ( Z .+ Y ) = X ) )
7	1 2	ablcom	|- ( ( G e. Abel /\ Y e. B /\ Z e. B ) -> ( Y .+ Z ) = ( Z .+ Y ) )
8	7	3adant3r1	|- ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .+ Z ) = ( Z .+ Y ) )
9	8	eqeq1d	|- ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( Y .+ Z ) = X <-> ( Z .+ Y ) = X ) )
10	6 9	bitr4d	|- ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) = Z <-> ( Y .+ Z ) = X ) )