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 e. NN0 -> N e. ( 0 ... ( 2 x. N ) ) )

Proof

Step	Hyp	Ref	Expression
1		nn0ge0	|- ( N e. NN0 -> 0 <_ N )
2		nn0le2x	|- ( N e. NN0 -> N <_ ( 2 x. N ) )
3		nn0z	|- ( N e. NN0 -> N e. ZZ )
4		0zd	|- ( N e. NN0 -> 0 e. ZZ )
5		2z	|- 2 e. ZZ
6		zmulcl	|- ( ( 2 e. ZZ /\ N e. ZZ ) -> ( 2 x. N ) e. ZZ )
7	5 3 6	sylancr	|- ( N e. NN0 -> ( 2 x. N ) e. ZZ )
8		elfz	|- ( ( N e. ZZ /\ 0 e. ZZ /\ ( 2 x. N ) e. ZZ ) -> ( N e. ( 0 ... ( 2 x. N ) ) <-> ( 0 <_ N /\ N <_ ( 2 x. N ) ) ) )
9	3 4 7 8	syl3anc	|- ( N e. NN0 -> ( N e. ( 0 ... ( 2 x. N ) ) <-> ( 0 <_ N /\ N <_ ( 2 x. N ) ) ) )
10	1 2 9	mpbir2and	|- ( N e. NN0 -> N e. ( 0 ... ( 2 x. N ) ) )