Theorem pcndvds

Description: Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014)

	Ref	Expression
Assertion	pcndvds	$$⊢\left(P\in \mathbb{P}\wedge N\in \mathbb{N}\right)\to \neg P^{\left(P\mathrm{pCnt}N\right)+1}\parallel N$$

Proof

Step	Hyp	Ref	Expression
1		nnz	$$⊢N\in \mathbb{N}\to N\in \mathbb{Z}$$
2		nnne0	$$⊢N\in \mathbb{N}\to N\ne 0$$
3	1 2	jca	$$⊢N\in \mathbb{N}\to \left(N\in \mathbb{Z}\wedge N\ne 0\right)$$
4		pczndvds	$$⊢\left(P\in \mathbb{P}\wedge \left(N\in \mathbb{Z}\wedge N\ne 0\right)\right)\to \neg P^{\left(P\mathrm{pCnt}N\right)+1}\parallel N$$
5	3 4	sylan2	$$⊢\left(P\in \mathbb{P}\wedge N\in \mathbb{N}\right)\to \neg P^{\left(P\mathrm{pCnt}N\right)+1}\parallel N$$