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Metamath Proof Explorer

Theorem pcndvds2

Description: The remainder after dividing out all factors of P is not divisible by P . (Contributed by Mario Carneiro, 23-Feb-2014)

		Ref	Expression
	Assertion	pcndvds2	$⊢ (P \in ℙ \land N \in ℕ) \to \neg P ∥ (\frac{N}{P^{P pCnt N}})$

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		pczndvds2	$⊢ (P \in ℙ \land (N \in ℤ \land N \neq 0)) \to \neg P ∥ (\frac{N}{P^{P pCnt N}})$
5	3 4	sylan2	$⊢ (P \in ℙ \land N \in ℕ) \to \neg P ∥ (\frac{N}{P^{P pCnt N}})$