Theorem fprodnncl

Description: Closure of a finite product of positive integers. (Contributed by Scott Fenton, 14-Dec-2017)

	Ref	Expression
Hypotheses	fprodcl.1	|- ( ph -> A e. Fin )
	fprodnncl.2	|- ( ( ph /\ k e. A ) -> B e. NN )
Assertion	fprodnncl	|- ( ph -> prod_ k e. A B e. NN )

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	|- ( ph -> A e. Fin )
2		fprodnncl.2	|- ( ( ph /\ k e. A ) -> B e. NN )
3		nnsscn	|- NN C_ CC
4	3	a1i	|- ( ph -> NN C_ CC )
5		nnmulcl	|- ( ( x e. NN /\ y e. NN ) -> ( x x. y ) e. NN )
6	5	adantl	|- ( ( ph /\ ( x e. NN /\ y e. NN ) ) -> ( x x. y ) e. NN )
7		1nn	|- 1 e. NN
8	7	a1i	|- ( ph -> 1 e. NN )
9	4 6 1 2 8	fprodcllem	|- ( ph -> prod_ k e. A B e. NN )