Theorem pcndvds

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

	Ref	Expression
Assertion	pcndvds	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ) → ¬ ( 𝑃 ↑ ( ( 𝑃 pCnt 𝑁 ) + 1 ) ) ∥ 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		nnz	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ )
2		nnne0	⊢ ( 𝑁 ∈ ℕ → 𝑁 ≠ 0 )
3	1 2	jca	⊢ ( 𝑁 ∈ ℕ → ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) )
4		pczndvds	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ¬ ( 𝑃 ↑ ( ( 𝑃 pCnt 𝑁 ) + 1 ) ) ∥ 𝑁 )
5	3 4	sylan2	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ) → ¬ ( 𝑃 ↑ ( ( 𝑃 pCnt 𝑁 ) + 1 ) ) ∥ 𝑁 )