Theorem lsmelval

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

	Ref	Expression
Hypotheses	lsmelval.a	|- .+ = ( +g ` G )
	lsmelval.p	|- .(+) = ( LSSum ` G )
Assertion	lsmelval	|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( X e. ( T .(+) U ) <-> E. y e. T E. z e. U X = ( y .+ z ) ) )

Proof

Step	Hyp	Ref	Expression
1		lsmelval.a	|- .+ = ( +g ` G )
2		lsmelval.p	|- .(+) = ( LSSum ` G )
3		subgrcl	|- ( T e. ( SubGrp ` G ) -> G e. Grp )
4		eqid	|- ( Base ` G ) = ( Base ` G )
5	4	subgss	|- ( T e. ( SubGrp ` G ) -> T C_ ( Base ` G ) )
6	4	subgss	|- ( U e. ( SubGrp ` G ) -> U C_ ( Base ` G ) )
7	4 1 2	lsmelvalx	|- ( ( G e. Grp /\ T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) -> ( X e. ( T .(+) U ) <-> E. y e. T E. z e. U X = ( y .+ z ) ) )
8	3 5 6 7	syl2an3an	|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( X e. ( T .(+) U ) <-> E. y e. T E. z e. U X = ( y .+ z ) ) )