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		nn0le2x	$$⊢N\in {\mathbb{N}}_0\to N\le 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}$$
3		nn0z	$$⊢N\in {\mathbb{N}}_0\to N\in \mathbb{Z}$$
4		0zd	$$⊢N\in {\mathbb{N}}_0\to 0\in \mathbb{Z}$$
5		2z	$$⊢2\in \mathbb{Z}$$
6		zmulcl	$$⊢\left(2\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\to 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\in \mathbb{Z}$$
7	5 3 6	sylancr	$$⊢N\in {\mathbb{N}}_0\to 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\in \mathbb{Z}$$
8		elfz	$$⊢\left(N\in \mathbb{Z}\wedge 0\in \mathbb{Z}\wedge 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\in \mathbb{Z}\right)\to \left(N\in \left(0\dots 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\right)\leftrightarrow \left(0\le N\wedge N\le 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\right)\right)$$
9	3 4 7 8	syl3anc	$$⊢N\in {\mathbb{N}}_0\to \left(N\in \left(0\dots 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\right)\leftrightarrow \left(0\le N\wedge N\le 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\right)\right)$$
10	1 2 9	mpbir2and	$$⊢N\in {\mathbb{N}}_0\to N\in \left(0\dots 2\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\right)$$