Theorem nnnn0addcl

Description: A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005) (Proof shortened by Mario Carneiro, 16-May-2014)

	Ref	Expression
Assertion	nnnn0addcl	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 + 𝑁 ) ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		elnn0	⊢ ( 𝑁 ∈ ℕ0 ↔ ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) )
2		nnaddcl	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝑀 + 𝑁 ) ∈ ℕ )
3		oveq2	⊢ ( 𝑁 = 0 → ( 𝑀 + 𝑁 ) = ( 𝑀 + 0 ) )
4		nncn	⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℂ )
5	4	addid1d	⊢ ( 𝑀 ∈ ℕ → ( 𝑀 + 0 ) = 𝑀 )
6	3 5	sylan9eqr	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 = 0 ) → ( 𝑀 + 𝑁 ) = 𝑀 )
7		simpl	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 = 0 ) → 𝑀 ∈ ℕ )
8	6 7	eqeltrd	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 = 0 ) → ( 𝑀 + 𝑁 ) ∈ ℕ )
9	2 8	jaodan	⊢ ( ( 𝑀 ∈ ℕ ∧ ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) → ( 𝑀 + 𝑁 ) ∈ ℕ )
10	1 9	sylan2b	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 + 𝑁 ) ∈ ℕ )