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Metamath Proof Explorer

Theorem nnmulcld

Description: Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses nnge1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℕ )
nnmulcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℕ )
Assertion nnmulcld ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℕ )

Proof

Step Hyp Ref Expression
1 nnge1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℕ )
2 nnmulcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℕ )
3 nnmulcl ⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) ∈ ℕ )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℕ )