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	$$⊢N\in {\mathbb{N}}_0\to N\in \left(0\dots 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\right)$$

Proof

Step	Hyp	Ref	Expression
1		nn0ge0	$$⊢N\in {\mathbb{N}}_0\to 0\le N$$
2		nn0re	$$⊢N\in {\mathbb{N}}_0\to N\in ℝ$$
3		nn0addge1	$$⊢\left(N\in ℝ\wedge N\in {\mathbb{N}}_0\right)\to N\le N+N$$
4	2 3	mpancom	$$⊢N\in {\mathbb{N}}_0\to N\le N+N$$
5		nn0cn	$$⊢N\in {\mathbb{N}}_0\to N\in ℂ$$
6	5	2timesd	$$⊢N\in {\mathbb{N}}_0\to 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}=N+N$$
7	4 6	breqtrrd	$$⊢N\in {\mathbb{N}}_0\to N\le 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}$$
8		nn0z	$$⊢N\in {\mathbb{N}}_0\to N\in \mathbb{Z}$$
9		0zd	$$⊢N\in {\mathbb{N}}_0\to 0\in \mathbb{Z}$$
10		2z	$$⊢2\in \mathbb{Z}$$
11		zmulcl	$$⊢\left(2\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\to 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\in \mathbb{Z}$$
12	10 8 11	sylancr	$$⊢N\in {\mathbb{N}}_0\to 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\in \mathbb{Z}$$
13		elfz	$$⊢\left(N\in \mathbb{Z}\wedge 0\in \mathbb{Z}\wedge 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\in \mathbb{Z}\right)\to \left(N\in \left(0\dots 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\right)\leftrightarrow \left(0\le N\wedge N\le 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\right)\right)$$
14	8 9 12 13	syl3anc	$$⊢N\in {\mathbb{N}}_0\to \left(N\in \left(0\dots 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\right)\leftrightarrow \left(0\le N\wedge N\le 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\right)\right)$$
15	1 7 14	mpbir2and	$$⊢N\in {\mathbb{N}}_0\to N\in \left(0\dots 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\right)$$