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	⊢ Ⅎ 𝑘 𝜑
	fprodcl.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fprodrecl.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
Assertion	fprodreclf	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		fprodreclf.kph	⊢ Ⅎ 𝑘 𝜑
2		fprodcl.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
3		fprodrecl.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
4		ax-resscn	⊢ ℝ ⊆ ℂ
5	4	a1i	⊢ ( 𝜑 → ℝ ⊆ ℂ )
6		remulcl	⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 · 𝑦 ) ∈ ℝ )
7	6	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( 𝑥 · 𝑦 ) ∈ ℝ )
8		1red	⊢ ( 𝜑 → 1 ∈ ℝ )
9	1 5 7 2 3 8	fprodcllemf	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ ℝ )