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Metamath Proof Explorer

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 bitrid ⊢ ( 𝑁 ∈ ℤ → ( 𝐾 ∈ ( 1 ... 𝑁 ) ↔ ( 𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁 ) ) )