Theorem fprodzcl

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

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

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	|- ( ph -> A e. Fin )
2		fprodzcl.2	|- ( ( ph /\ k e. A ) -> B e. ZZ )
3		zsscn	|- ZZ C_ CC
4	3	a1i	|- ( ph -> ZZ C_ CC )
5		zmulcl	|- ( ( x e. ZZ /\ y e. ZZ ) -> ( x x. y ) e. ZZ )
6	5	adantl	|- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x x. y ) e. ZZ )
7		1zzd	|- ( ph -> 1 e. ZZ )
8	4 6 1 2 7	fprodcllem	|- ( ph -> prod_ k e. A B e. ZZ )