Theorem nnmulcli

Description: Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014)

	Ref	Expression
Hypotheses	nnmulcli.1	⊢ 𝐴 ∈ ℕ
	nnmulcli.2	⊢ 𝐵 ∈ ℕ
Assertion	nnmulcli	⊢ ( 𝐴 · 𝐵 ) ∈ ℕ

Proof

Step	Hyp	Ref	Expression
1		nnmulcli.1	⊢ 𝐴 ∈ ℕ
2		nnmulcli.2	⊢ 𝐵 ∈ ℕ
3		nnmulcl	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) ∈ ℕ )
4	1 2 3	mp2an	⊢ ( 𝐴 · 𝐵 ) ∈ ℕ