Theorem lsmelvalix

Description: Subspace sum membership (for a group or vector space). (Contributed by NM, 4-Feb-2014) (Revised by Mario Carneiro, 19-Apr-2016)

	Ref	Expression
Hypotheses	lsmfval.v	|- B = ( Base ` G )
	lsmfval.a	|- .+ = ( +g ` G )
	lsmfval.s	|- .(+) = ( LSSum ` G )
Assertion	lsmelvalix	|- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( X e. T /\ Y e. U ) ) -> ( X .+ Y ) e. ( T .(+) U ) )

Proof

Step	Hyp	Ref	Expression
1		lsmfval.v	|- B = ( Base ` G )
2		lsmfval.a	|- .+ = ( +g ` G )
3		lsmfval.s	|- .(+) = ( LSSum ` G )
4		eqid	|- ( X .+ Y ) = ( X .+ Y )
5		rspceov	|- ( ( X e. T /\ Y e. U /\ ( X .+ Y ) = ( X .+ Y ) ) -> E. x e. T E. y e. U ( X .+ Y ) = ( x .+ y ) )
6	4 5	mp3an3	|- ( ( X e. T /\ Y e. U ) -> E. x e. T E. y e. U ( X .+ Y ) = ( x .+ y ) )
7	1 2 3	lsmelvalx	|- ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( ( X .+ Y ) e. ( T .(+) U ) <-> E. x e. T E. y e. U ( X .+ Y ) = ( x .+ y ) ) )
8	7	biimpar	|- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ E. x e. T E. y e. U ( X .+ Y ) = ( x .+ y ) ) -> ( X .+ Y ) e. ( T .(+) U ) )
9	6 8	sylan2	|- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( X e. T /\ Y e. U ) ) -> ( X .+ Y ) e. ( T .(+) U ) )