Theorem fprodclf

Description: Closure of a finite product of complex numbers. A version of fprodcl using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	fprodclf.kph	|- F/ k ph
	fprodclf.a	|- ( ph -> A e. Fin )
	fprodclf.b	|- ( ( ph /\ k e. A ) -> B e. CC )
Assertion	fprodclf	|- ( ph -> prod_ k e. A B e. CC )

Proof

Step	Hyp	Ref	Expression
1		fprodclf.kph	|- F/ k ph
2		fprodclf.a	|- ( ph -> A e. Fin )
3		fprodclf.b	|- ( ( ph /\ k e. A ) -> B e. CC )
4		ssidd	|- ( ph -> CC C_ CC )
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		1cnd	|- ( ph -> 1 e. CC )
8	1 4 6 2 3 7	fprodcllemf	|- ( ph -> prod_ k e. A B e. CC )