Theorem fprodnn0cl

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

	Ref	Expression
Hypotheses	fprodcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fprodnn0cl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℕ0 )
Assertion	fprodnn0cl	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fprodnn0cl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℕ0 )
3		nn0sscn	⊢ ℕ0 ⊆ ℂ
4	3	a1i	⊢ ( 𝜑 → ℕ0 ⊆ ℂ )
5		nn0mulcl	⊢ ( ( 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) → ( 𝑥 · 𝑦 ) ∈ ℕ0 )
6	5	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) ) → ( 𝑥 · 𝑦 ) ∈ ℕ0 )
7		1nn0	⊢ 1 ∈ ℕ0
8	7	a1i	⊢ ( 𝜑 → 1 ∈ ℕ0 )
9	4 6 1 2 8	fprodcllem	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ ℕ0 )