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Metamath Proof Explorer

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 ) ) )