Theorem lsmelvalx

Description: Subspace sum membership (for a group or vector space). Extended domain version of lsmelval . (Contributed by NM, 28-Jan-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	lsmelvalx	|- ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( X e. ( T .(+) U ) <-> E. y e. T E. z e. U X = ( y .+ z ) ) )

Proof

Step	Hyp	Ref	Expression
1		lsmfval.v	|- B = ( Base ` G )
2		lsmfval.a	|- .+ = ( +g ` G )
3		lsmfval.s	|- .(+) = ( LSSum ` G )
4	1 2 3	lsmvalx	|- ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( T .(+) U ) = ran ( y e. T , z e. U |-> ( y .+ z ) ) )
5	4	eleq2d	|- ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( X e. ( T .(+) U ) <-> X e. ran ( y e. T , z e. U |-> ( y .+ z ) ) ) )
6		eqid	|- ( y e. T , z e. U |-> ( y .+ z ) ) = ( y e. T , z e. U |-> ( y .+ z ) )
7		ovex	|- ( y .+ z ) e. _V
8	6 7	elrnmpo	|- ( X e. ran ( y e. T , z e. U |-> ( y .+ z ) ) <-> E. y e. T E. z e. U X = ( y .+ z ) )
9	5 8	bitrdi	|- ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( X e. ( T .(+) U ) <-> E. y e. T E. z e. U X = ( y .+ z ) ) )