Theorem fprodrpcl

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

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

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	|- ( ph -> A e. Fin )
2		fprodrpcl.2	|- ( ( ph /\ k e. A ) -> B e. RR+ )
3		rpssre	|- RR+ C_ RR
4		ax-resscn	|- RR C_ CC
5	3 4	sstri	|- RR+ C_ CC
6	5	a1i	|- ( ph -> RR+ C_ CC )
7		rpmulcl	|- ( ( x e. RR+ /\ y e. RR+ ) -> ( x x. y ) e. RR+ )
8	7	adantl	|- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( x x. y ) e. RR+ )
9		1rp	|- 1 e. RR+
10	9	a1i	|- ( ph -> 1 e. RR+ )
11	6 8 1 2 10	fprodcllem	|- ( ph -> prod_ k e. A B e. RR+ )