Theorem prodfmul

Description: The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017)

	Ref	Expression
Hypotheses	prodfmul.1	|- ( ph -> N e. ( ZZ>= ` M ) )
	prodfmul.2	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
	prodfmul.3	|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. CC )
	prodfmul.4	|- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
Assertion	prodfmul	|- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) x. ( seq M ( x. , G ) ` N ) ) )

Proof

Step	Hyp	Ref	Expression
1		prodfmul.1	|- ( ph -> N e. ( ZZ>= ` M ) )
2		prodfmul.2	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
3		prodfmul.3	|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. CC )
4		prodfmul.4	|- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
5		mulcl	|- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
6	5	adantl	|- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
7		mulcom	|- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) = ( y x. x ) )
8	7	adantl	|- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) = ( y x. x ) )
9		mulass	|- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x x. y ) x. z ) = ( x x. ( y x. z ) ) )
10	9	adantl	|- ( ( ph /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( ( x x. y ) x. z ) = ( x x. ( y x. z ) ) )
11	6 8 10 1 2 3 4	seqcaopr	|- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) x. ( seq M ( x. , G ) ` N ) ) )