Theorem fprodzcl

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

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

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fprodzcl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℤ )
3		zsscn	⊢ ℤ ⊆ ℂ
4	3	a1i	⊢ ( 𝜑 → ℤ ⊆ ℂ )
5		zmulcl	⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) → ( 𝑥 · 𝑦 ) ∈ ℤ )
6	5	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( 𝑥 · 𝑦 ) ∈ ℤ )
7		1zzd	⊢ ( 𝜑 → 1 ∈ ℤ )
8	4 6 1 2 7	fprodcllem	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ )