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Metamath Proof Explorer

Theorem nn0addcld

Description: Closure of addition 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 nn0addcld ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℕ0 )

Proof

Step Hyp Ref Expression
1 nn0red.1 ⊢ ( 𝜑 → 𝐴 ∈ ℕ0 )
2 nn0addcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℕ0 )
3 nn0addcl ⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 + 𝐵 ) ∈ ℕ0 )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℕ0 )