Theorem lsmval

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

	Ref	Expression
Hypotheses	lsmval.v	|- B = ( Base ` G )
	lsmval.a	|- .+ = ( +g ` G )
	lsmval.p	|- .(+) = ( LSSum ` G )
Assertion	lsmval	|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( T .(+) U ) = ran ( x e. T , y e. U |-> ( x .+ y ) ) )

Proof

Step	Hyp	Ref	Expression
1		lsmval.v	|- B = ( Base ` G )
2		lsmval.a	|- .+ = ( +g ` G )
3		lsmval.p	|- .(+) = ( LSSum ` G )
4		subgrcl	|- ( T e. ( SubGrp ` G ) -> G e. Grp )
5	1	subgss	|- ( T e. ( SubGrp ` G ) -> T C_ B )
6	1	subgss	|- ( U e. ( SubGrp ` G ) -> U C_ B )
7	1 2 3	lsmvalx	|- ( ( G e. Grp /\ T C_ B /\ U C_ B ) -> ( T .(+) U ) = ran ( x e. T , y e. U |-> ( x .+ y ) ) )
8	4 5 6 7	syl2an3an	|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( T .(+) U ) = ran ( x e. T , y e. U |-> ( x .+ y ) ) )