Theorem 1elfzo1ceilhalf1

Description: 1 is in the half-open integer range from 1 to the ceiling of half of an integer greater than two is greater than one. (Contributed by AV, 2-Nov-2025)

	Ref	Expression
Assertion	1elfzo1ceilhalf1	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 1 ∈ ( 1 ..^ ( ⌈ ‘ ( 𝑁 / 2 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		1nn	⊢ 1 ∈ ℕ
2	1	a1i	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 1 ∈ ℕ )
3		eluzge3nn	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 𝑁 ∈ ℕ )
4		ceilhalfnn	⊢ ( 𝑁 ∈ ℕ → ( ⌈ ‘ ( 𝑁 / 2 ) ) ∈ ℕ )
5	3 4	syl	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → ( ⌈ ‘ ( 𝑁 / 2 ) ) ∈ ℕ )
6		ceilhalfgt1	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 1 < ( ⌈ ‘ ( 𝑁 / 2 ) ) )
7		elfzo1	⊢ ( 1 ∈ ( 1 ..^ ( ⌈ ‘ ( 𝑁 / 2 ) ) ) ↔ ( 1 ∈ ℕ ∧ ( ⌈ ‘ ( 𝑁 / 2 ) ) ∈ ℕ ∧ 1 < ( ⌈ ‘ ( 𝑁 / 2 ) ) ) )
8	2 5 6 7	syl3anbrc	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 1 ∈ ( 1 ..^ ( ⌈ ‘ ( 𝑁 / 2 ) ) ) )