Theorem fzctr

Description: Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014) (Revised by Mario Carneiro, 2-Aug-2014)

	Ref	Expression
Assertion	fzctr	⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ( 0 ... ( 2 · 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nn0ge0	⊢ ( 𝑁 ∈ ℕ0 → 0 ≤ 𝑁 )
2		nn0le2x	⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ≤ ( 2 · 𝑁 ) )
3		nn0z	⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ )
4		0zd	⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ℤ )
5		2z	⊢ 2 ∈ ℤ
6		zmulcl	⊢ ( ( 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 2 · 𝑁 ) ∈ ℤ )
7	5 3 6	sylancr	⊢ ( 𝑁 ∈ ℕ0 → ( 2 · 𝑁 ) ∈ ℤ )
8		elfz	⊢ ( ( 𝑁 ∈ ℤ ∧ 0 ∈ ℤ ∧ ( 2 · 𝑁 ) ∈ ℤ ) → ( 𝑁 ∈ ( 0 ... ( 2 · 𝑁 ) ) ↔ ( 0 ≤ 𝑁 ∧ 𝑁 ≤ ( 2 · 𝑁 ) ) ) )
9	3 4 7 8	syl3anc	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 ∈ ( 0 ... ( 2 · 𝑁 ) ) ↔ ( 0 ≤ 𝑁 ∧ 𝑁 ≤ ( 2 · 𝑁 ) ) ) )
10	1 2 9	mpbir2and	⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ( 0 ... ( 2 · 𝑁 ) ) )