Theorem fprodrecl

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

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

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	|- ( ph -> A e. Fin )
2		fprodrecl.2	|- ( ( ph /\ k e. A ) -> B e. RR )
3		ax-resscn	|- RR C_ CC
4	3	a1i	|- ( ph -> RR C_ CC )
5		remulcl	|- ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR )
6	5	adantl	|- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
7		1red	|- ( ph -> 1 e. RR )
8	4 6 1 2 7	fprodcllem	|- ( ph -> prod_ k e. A B e. RR )