Theorem fprodrecl

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

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

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fprodrecl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
3		ax-resscn	⊢ ℝ ⊆ ℂ
4	3	a1i	⊢ ( 𝜑 → ℝ ⊆ ℂ )
5		remulcl	⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 · 𝑦 ) ∈ ℝ )
6	5	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( 𝑥 · 𝑦 ) ∈ ℝ )
7		1red	⊢ ( 𝜑 → 1 ∈ ℝ )
8	4 6 1 2 7	fprodcllem	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ ℝ )