Theorem fprodreclf

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

	Ref	Expression
Hypotheses	fprodreclf.kph	|- F/ k ph
	fprodcl.a	|- ( ph -> A e. Fin )
	fprodrecl.b	|- ( ( ph /\ k e. A ) -> B e. RR )
Assertion	fprodreclf	|- ( ph -> prod_ k e. A B e. RR )

Proof

Step	Hyp	Ref	Expression
1		fprodreclf.kph	|- F/ k ph
2		fprodcl.a	|- ( ph -> A e. Fin )
3		fprodrecl.b	|- ( ( ph /\ k e. A ) -> B e. RR )
4		ax-resscn	|- RR C_ CC
5	4	a1i	|- ( ph -> RR C_ CC )
6		remulcl	|- ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR )
7	6	adantl	|- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
8		1red	|- ( ph -> 1 e. RR )
9	1 5 7 2 3 8	fprodcllemf	|- ( ph -> prod_ k e. A B e. RR )