Theorem pczndvds

Description: Defining property of the prime count function. (Contributed by Mario Carneiro, 3-Oct-2014)

	Ref	Expression
Assertion	pczndvds	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ¬ ( 𝑃 ↑ ( ( 𝑃 pCnt 𝑁 ) + 1 ) ) ∥ 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) = sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < )
2	1	pczpre	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ( 𝑃 pCnt 𝑁 ) = sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) )
3	2	oveq1d	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ( ( 𝑃 pCnt 𝑁 ) + 1 ) = ( sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) + 1 ) )
4	3	oveq2d	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ( 𝑃 ↑ ( ( 𝑃 pCnt 𝑁 ) + 1 ) ) = ( 𝑃 ↑ ( sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) + 1 ) ) )
5		prmuz2	⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ( ℤ≥ ‘ 2 ) )
6		eqid	⊢ { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } = { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 }
7	6 1	pcprendvds	⊢ ( ( 𝑃 ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ¬ ( 𝑃 ↑ ( sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) + 1 ) ) ∥ 𝑁 )
8	5 7	sylan	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ¬ ( 𝑃 ↑ ( sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) + 1 ) ) ∥ 𝑁 )
9	4 8	eqnbrtrd	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ¬ ( 𝑃 ↑ ( ( 𝑃 pCnt 𝑁 ) + 1 ) ) ∥ 𝑁 )