Theorem fznn

Description: Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011) (Revised by Mario Carneiro, 18-Jun-2015)

	Ref	Expression
Assertion	fznn	⊢ ( 𝑁 ∈ ℤ → ( 𝐾 ∈ ( 1 ... 𝑁 ) ↔ ( 𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elfzuzb	⊢ ( 𝐾 ∈ ( 1 ... 𝑁 ) ↔ ( 𝐾 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) )
2		elnnuz	⊢ ( 𝐾 ∈ ℕ ↔ 𝐾 ∈ ( ℤ≥ ‘ 1 ) )
3	2	anbi1i	⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) ↔ ( 𝐾 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) )
4	1 3	bitr4i	⊢ ( 𝐾 ∈ ( 1 ... 𝑁 ) ↔ ( 𝐾 ∈ ℕ ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) )
5		nnz	⊢ ( 𝐾 ∈ ℕ → 𝐾 ∈ ℤ )
6		eluz	⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ↔ 𝐾 ≤ 𝑁 ) )
7	5 6	sylan	⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ↔ 𝐾 ≤ 𝑁 ) )
8	7	ancoms	⊢ ( ( 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℕ ) → ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ↔ 𝐾 ≤ 𝑁 ) )
9	8	pm5.32da	⊢ ( 𝑁 ∈ ℤ → ( ( 𝐾 ∈ ℕ ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) ↔ ( 𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁 ) ) )
10	4 9	syl5bb	⊢ ( 𝑁 ∈ ℤ → ( 𝐾 ∈ ( 1 ... 𝑁 ) ↔ ( 𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁 ) ) )