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Metamath Proof Explorer

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