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Metamath Proof Explorer

Theorem pcdvds

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

		Ref	Expression
	Assertion	pcdvds	$⊢ (P \in ℙ \land N \in ℕ) \to P^{P pCnt N} ∥ N$

Step	Hyp	Ref	Expression
1		nnz	$⊢ N \in ℕ \to N \in ℤ$
2		nnne0	$⊢ N \in ℕ \to N \neq 0$
3	1 2	jca	$⊢ N \in ℕ \to (N \in ℤ \land N \neq 0)$
4		pczdvds	$⊢ (P \in ℙ \land (N \in ℤ \land N \neq 0)) \to P^{P pCnt N} ∥ N$
5	3 4	sylan2	$⊢ (P \in ℙ \land N \in ℕ) \to P^{P pCnt N} ∥ N$