Theorem nnehalf

Description: The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021)

	Ref	Expression
Assertion	nnehalf	⊢ ( ( 𝑁 ∈ ℕ ∧ 2 ∥ 𝑁 ) → ( 𝑁 / 2 ) ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		nnnn0	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ0 )
2		nn0ehalf	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 2 ∥ 𝑁 ) → ( 𝑁 / 2 ) ∈ ℕ0 )
3	1 2	sylan	⊢ ( ( 𝑁 ∈ ℕ ∧ 2 ∥ 𝑁 ) → ( 𝑁 / 2 ) ∈ ℕ0 )
4		nn0enne	⊢ ( 𝑁 ∈ ℕ → ( ( 𝑁 / 2 ) ∈ ℕ0 ↔ ( 𝑁 / 2 ) ∈ ℕ ) )
5	4	adantr	⊢ ( ( 𝑁 ∈ ℕ ∧ 2 ∥ 𝑁 ) → ( ( 𝑁 / 2 ) ∈ ℕ0 ↔ ( 𝑁 / 2 ) ∈ ℕ ) )
6	3 5	mpbid	⊢ ( ( 𝑁 ∈ ℕ ∧ 2 ∥ 𝑁 ) → ( 𝑁 / 2 ) ∈ ℕ )