Theorem pccld

Description: Closure of the prime power function. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	pccld.1	$$⊢\phi \to P\in \mathbb{P}$$
	pccld.2	$$⊢\phi \to N\in \mathbb{N}$$
Assertion	pccld	$$⊢\phi \to P\mathrm{pCnt}N\in {\mathbb{N}}_0$$

Proof

Step	Hyp	Ref	Expression
1		pccld.1	$$⊢\phi \to P\in \mathbb{P}$$
2		pccld.2	$$⊢\phi \to N\in \mathbb{N}$$
3		pccl	$$⊢\left(P\in \mathbb{P}\wedge N\in \mathbb{N}\right)\to P\mathrm{pCnt}N\in {\mathbb{N}}_0$$
4	1 2 3	syl2anc	$$⊢\phi \to P\mathrm{pCnt}N\in {\mathbb{N}}_0$$