Metamath Proof Explorer


Theorem nn0mulcld

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

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

Proof

Step Hyp Ref Expression
1 nn0red.1 ( 𝜑𝐴 ∈ ℕ0 )
2 nn0addcld.2 ( 𝜑𝐵 ∈ ℕ0 )
3 nn0mulcl ( ( 𝐴 ∈ ℕ0𝐵 ∈ ℕ0 ) → ( 𝐴 · 𝐵 ) ∈ ℕ0 )
4 1 2 3 syl2anc ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℕ0 )