Theorem fprodcl

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

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

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	|- ( ph -> A e. Fin )
2		fprodcl.2	|- ( ( ph /\ k e. A ) -> B e. CC )
3		ssidd	|- ( ph -> CC C_ CC )
4		mulcl	|- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
5	4	adantl	|- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
6		1cnd	|- ( ph -> 1 e. CC )
7	3 5 1 2 6	fprodcllem	|- ( ph -> prod_ k e. A B e. CC )