Theorem lmodvsubadd

Description: Relationship between vector subtraction and addition. ( hvsubadd analog.) (Contributed by NM, 31-Mar-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmod4.v	|- V = ( Base ` W )
	lmod4.p	|- .+ = ( +g ` W )
	lmodvaddsub4.m	|- .- = ( -g ` W )
Assertion	lmodvsubadd	|- ( ( W e. LMod /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) = C <-> ( B .+ C ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		lmod4.v	|- V = ( Base ` W )
2		lmod4.p	|- .+ = ( +g ` W )
3		lmodvaddsub4.m	|- .- = ( -g ` W )
4		lmodabl	|- ( W e. LMod -> W e. Abel )
5	1 2 3	ablsubadd	|- ( ( W e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) = C <-> ( B .+ C ) = A ) )
6	4 5	sylan	|- ( ( W e. LMod /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) = C <-> ( B .+ C ) = A ) )