Theorem lsmelvalmi

Description: Membership of vector subtraction in subgroup sum. (Contributed by NM, 27-Apr-2015) (Revised by Mario Carneiro, 19-Apr-2016)

	Ref	Expression
Hypotheses	lsmelvalm.m	|- .- = ( -g ` G )
	lsmelvalm.p	|- .(+) = ( LSSum ` G )
	lsmelvalm.t	|- ( ph -> T e. ( SubGrp ` G ) )
	lsmelvalm.u	|- ( ph -> U e. ( SubGrp ` G ) )
	lsmelvalmi.x	|- ( ph -> X e. T )
	lsmelvalmi.y	|- ( ph -> Y e. U )
Assertion	lsmelvalmi	|- ( ph -> ( X .- Y ) e. ( T .(+) U ) )

Proof

Step	Hyp	Ref	Expression
1		lsmelvalm.m	|- .- = ( -g ` G )
2		lsmelvalm.p	|- .(+) = ( LSSum ` G )
3		lsmelvalm.t	|- ( ph -> T e. ( SubGrp ` G ) )
4		lsmelvalm.u	|- ( ph -> U e. ( SubGrp ` G ) )
5		lsmelvalmi.x	|- ( ph -> X e. T )
6		lsmelvalmi.y	|- ( ph -> Y e. U )
7		eqidd	|- ( ph -> ( X .- Y ) = ( X .- Y ) )
8		rspceov	|- ( ( X e. T /\ Y e. U /\ ( X .- Y ) = ( X .- Y ) ) -> E. x e. T E. y e. U ( X .- Y ) = ( x .- y ) )
9	5 6 7 8	syl3anc	|- ( ph -> E. x e. T E. y e. U ( X .- Y ) = ( x .- y ) )
10	1 2 3 4	lsmelvalm	|- ( ph -> ( ( X .- Y ) e. ( T .(+) U ) <-> E. x e. T E. y e. U ( X .- Y ) = ( x .- y ) ) )
11	9 10	mpbird	|- ( ph -> ( X .- Y ) e. ( T .(+) U ) )