Theorem nn0mulcld

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

	Ref	Expression
Hypotheses	nn0red.1	⊢ ( 𝜑 → 𝐴 ∈ ℕ0 )
	nn0addcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℕ0 )
Assertion	nn0mulcld	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		nn0red.1	⊢ ( 𝜑 → 𝐴 ∈ ℕ0 )
2		nn0addcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℕ0 )
3		nn0mulcl	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 · 𝐵 ) ∈ ℕ0 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℕ0 )