Theorem fprodnncl

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

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

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fprodnncl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℕ )
3		nnsscn	⊢ ℕ ⊆ ℂ
4	3	a1i	⊢ ( 𝜑 → ℕ ⊆ ℂ )
5		nnmulcl	⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑥 · 𝑦 ) ∈ ℕ )
6	5	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) → ( 𝑥 · 𝑦 ) ∈ ℕ )
7		1nn	⊢ 1 ∈ ℕ
8	7	a1i	⊢ ( 𝜑 → 1 ∈ ℕ )
9	4 6 1 2 8	fprodcllem	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ ℕ )