Theorem fprodnn0cl

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

	Ref	Expression
Hypotheses	fprodcl.1	|- ( ph -> A e. Fin )
	fprodnn0cl.2	|- ( ( ph /\ k e. A ) -> B e. NN0 )
Assertion	fprodnn0cl	|- ( ph -> prod_ k e. A B e. NN0 )

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	|- ( ph -> A e. Fin )
2		fprodnn0cl.2	|- ( ( ph /\ k e. A ) -> B e. NN0 )
3		nn0sscn	|- NN0 C_ CC
4	3	a1i	|- ( ph -> NN0 C_ CC )
5		nn0mulcl	|- ( ( x e. NN0 /\ y e. NN0 ) -> ( x x. y ) e. NN0 )
6	5	adantl	|- ( ( ph /\ ( x e. NN0 /\ y e. NN0 ) ) -> ( x x. y ) e. NN0 )
7		1nn0	|- 1 e. NN0
8	7	a1i	|- ( ph -> 1 e. NN0 )
9	4 6 1 2 8	fprodcllem	|- ( ph -> prod_ k e. A B e. NN0 )