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Metamath Proof Explorer

Theorem pczndvds2

Description: The remainder after dividing out all factors of P is not divisible by P . (Contributed by Mario Carneiro, 9-Sep-2014)

Ref Expression
Assertion pczndvds2 ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ¬ 𝑃 ∥ ( 𝑁 / ( 𝑃 ↑ ( 𝑃 pCnt 𝑁 ) ) ) )

Proof

Step Hyp Ref Expression
1 prmuz2 ⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ( ℤ≥ ‘ 2 ) )
2 eqid ⊢ { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } = { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 }
3 eqid ⊢ sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) = sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < )
4 2 3 pcprendvds2 ⊢ ( ( 𝑃 ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ¬ 𝑃 ∥ ( 𝑁 / ( 𝑃 ↑ sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) ) ) )
5 1 4 sylan ⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ¬ 𝑃 ∥ ( 𝑁 / ( 𝑃 ↑ sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) ) ) )
6 3 pczpre ⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ( 𝑃 pCnt 𝑁 ) = sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) )
7 6 oveq2d ⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ( 𝑃 ↑ ( 𝑃 pCnt 𝑁 ) ) = ( 𝑃 ↑ sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) ) )
8 7 oveq2d ⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ( 𝑁 / ( 𝑃 ↑ ( 𝑃 pCnt 𝑁 ) ) ) = ( 𝑁 / ( 𝑃 ↑ sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) ) ) )
9 8 breq2d ⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ( 𝑃 ∥ ( 𝑁 / ( 𝑃 ↑ ( 𝑃 pCnt 𝑁 ) ) ) ↔ 𝑃 ∥ ( 𝑁 / ( 𝑃 ↑ sup ( { 𝑛 ∈ ℕ0 ∣ ( 𝑃 ↑ 𝑛 ) ∥ 𝑁 } , ℝ , < ) ) ) ) )
10 5 9 mtbird ⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ¬ 𝑃 ∥ ( 𝑁 / ( 𝑃 ↑ ( 𝑃 pCnt 𝑁 ) ) ) )