Theorem fprodrpcl

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

	Ref	Expression
Hypotheses	fprodcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fprodrpcl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ+ )
Assertion	fprodrpcl	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fprodrpcl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ+ )
3		rpssre	⊢ ℝ+ ⊆ ℝ
4		ax-resscn	⊢ ℝ ⊆ ℂ
5	3 4	sstri	⊢ ℝ+ ⊆ ℂ
6	5	a1i	⊢ ( 𝜑 → ℝ+ ⊆ ℂ )
7		rpmulcl	⊢ ( ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) → ( 𝑥 · 𝑦 ) ∈ ℝ+ )
8	7	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) ) → ( 𝑥 · 𝑦 ) ∈ ℝ+ )
9		1rp	⊢ 1 ∈ ℝ+
10	9	a1i	⊢ ( 𝜑 → 1 ∈ ℝ+ )
11	6 8 1 2 10	fprodcllem	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ ℝ+ )