Theorem nnaddcld

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

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

Proof

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