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Type | Label | Description |
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Statement | ||
Theorem | chpdifbndlem1 26701* | Lemma for chpdifbnd 26703. (Contributed by Mario Carneiro, 25-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 1 ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑧 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑧) · (log‘𝑧)) + Σ𝑚 ∈ (1...(⌊‘𝑧))((Λ‘𝑚) · (ψ‘(𝑧 / 𝑚)))) / 𝑧) − (2 · (log‘𝑧)))) ≤ 𝐵) & ⊢ 𝐶 = ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴))) & ⊢ (𝜑 → 𝑋 ∈ (1(,)+∞)) & ⊢ (𝜑 → 𝑌 ∈ (𝑋[,](𝐴 · 𝑋))) ⇒ ⊢ (𝜑 → ((ψ‘𝑌) − (ψ‘𝑋)) ≤ ((2 · (𝑌 − 𝑋)) + (𝐶 · (𝑋 / (log‘𝑋))))) | ||
Theorem | chpdifbndlem2 26702* | Lemma for chpdifbnd 26703. (Contributed by Mario Carneiro, 25-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 1 ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑧 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑧) · (log‘𝑧)) + Σ𝑚 ∈ (1...(⌊‘𝑧))((Λ‘𝑚) · (ψ‘(𝑧 / 𝑚)))) / 𝑧) − (2 · (log‘𝑧)))) ≤ 𝐵) & ⊢ 𝐶 = ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴))) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥))))) | ||
Theorem | chpdifbnd 26703* | A bound on the difference of nearby ψ values. Theorem 10.5.2 of [Shapiro], p. 427. (Contributed by Mario Carneiro, 25-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥))))) | ||
Theorem | logdivbnd 26704* | A bound on a sum of logs, used in pntlemk 26754. This is not as precise as logdivsum 26681 in its asymptotic behavior, but it is valid for all 𝑁 and does not require a limit value. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ ((((log‘𝑁) + 1)↑2) / 2)) | ||
Theorem | selberg3lem1 26705* | Introduce a log weighting on the summands of Σ𝑚 · 𝑛 ≤ 𝑥, Λ(𝑚)Λ(𝑛), the core of selberg2 26699 (written here as Σ𝑛 ≤ 𝑥, Λ(𝑛)ψ(𝑥 / 𝑛)). Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1)) | ||
Theorem | selberg3lem2 26706* | Lemma for selberg3 26707. Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1) | ||
Theorem | selberg3 26707* | Introduce a log weighting on the summands of Σ𝑚 · 𝑛 ≤ 𝑥, Λ(𝑚)Λ(𝑛), the core of selberg2 26699 (written here as Σ𝑛 ≤ 𝑥, Λ(𝑛)ψ(𝑥 / 𝑛)). Equation 10.6.7 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝑥 ∈ (1(,)+∞) ↦ (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1) | ||
Theorem | selberg4lem1 26708* | Lemma for selberg4 26709. Equation 10.4.20 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1)) | ||
Theorem | selberg4 26709* | The Selberg symmetry formula for products of three primes, instead of two. The sum here can also be written in the symmetric form Σ𝑖𝑗𝑘 ≤ 𝑥, Λ(𝑖)Λ(𝑗)Λ(𝑘); we eliminate one of the nested sums by using the definition of ψ(𝑥) = Σ𝑘 ≤ 𝑥, Λ(𝑘). This statement can thus equivalently be written ψ(𝑥)log↑2(𝑥) = 2Σ𝑖𝑗𝑘 ≤ 𝑥, Λ(𝑖)Λ(𝑗)Λ(𝑘) + 𝑂(𝑥log𝑥). Equation 10.4.23 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1) | ||
Theorem | pntrval 26710* | Define the residual of the second Chebyshev function. The goal is to have 𝑅(𝑥) ∈ 𝑜(𝑥), or 𝑅(𝑥) / 𝑥 ⇝𝑟 0. (Contributed by Mario Carneiro, 8-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ (𝐴 ∈ ℝ+ → (𝑅‘𝐴) = ((ψ‘𝐴) − 𝐴)) | ||
Theorem | pntrf 26711 | Functionality of the residual. Lemma for pnt 26762. (Contributed by Mario Carneiro, 8-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ 𝑅:ℝ+⟶ℝ | ||
Theorem | pntrmax 26712* | There is a bound on the residual valid for all 𝑥. (Contributed by Mario Carneiro, 9-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ ℝ+ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑐 | ||
Theorem | pntrsumo1 26713* | A bound on a sum over 𝑅. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1) | ||
Theorem | pntrsumbnd 26714* | A bound on a sum over 𝑅. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ ∃𝑐 ∈ ℝ+ ∀𝑚 ∈ ℤ (abs‘Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐 | ||
Theorem | pntrsumbnd2 26715* | A bound on a sum over 𝑅. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 14-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ ∃𝑐 ∈ ℝ+ ∀𝑘 ∈ ℕ ∀𝑚 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑘...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐 | ||
Theorem | selbergr 26716* | Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.2 of [Shapiro], p. 428. (Contributed by Mario Carneiro, 16-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ (𝑥 ∈ ℝ+ ↦ ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥)) ∈ 𝑂(1) | ||
Theorem | selberg3r 26717* | Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.8 of [Shapiro], p. 429. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ 𝑂(1) | ||
Theorem | selberg4r 26718* | Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.11 of [Shapiro], p. 430. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1) | ||
Theorem | selberg34r 26719* | The sum of selberg3r 26717 and selberg4r 26718. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) ∈ 𝑂(1) | ||
Theorem | pntsval 26720* | Define the "Selberg function", whose asymptotic behavior is the content of selberg 26696. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) ⇒ ⊢ (𝐴 ∈ ℝ → (𝑆‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝐴 / 𝑛))))) | ||
Theorem | pntsf 26721* | Functionality of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) ⇒ ⊢ 𝑆:ℝ⟶ℝ | ||
Theorem | selbergs 26722* | Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) ⇒ ⊢ (𝑥 ∈ ℝ+ ↦ (((𝑆‘𝑥) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1) | ||
Theorem | selbergsb 26723* | Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) ⇒ ⊢ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1[,)+∞)(abs‘(((𝑆‘𝑥) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐 | ||
Theorem | pntsval2 26724* | The Selberg function can be expressed using the convolution product of the von Mangoldt function with itself. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) ⇒ ⊢ (𝐴 ∈ ℝ → (𝑆‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(((Λ‘𝑛) · (log‘𝑛)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))) | ||
Theorem | pntrlog2bndlem1 26725* | The sum of selberg3r 26717 and selberg4r 26718. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) & ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥)) ∈ ≤𝑂(1) | ||
Theorem | pntrlog2bndlem2 26726* | Lemma for pntrlog2bnd 26732. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) & ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ+ (ψ‘𝑦) ≤ (𝐴 · 𝑦)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1)) | ||
Theorem | pntrlog2bndlem3 26727* | Lemma for pntrlog2bnd 26732. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) & ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘(((𝑆‘𝑦) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) − (abs‘(𝑅‘(𝑥 / (𝑛 + 1))))) · ((𝑆‘𝑛) − (2 · (𝑛 · (log‘𝑛))))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1)) | ||
Theorem | pntrlog2bndlem4 26728* | Lemma for pntrlog2bnd 26732. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) & ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0)) ⇒ ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥)) ∈ ≤𝑂(1) | ||
Theorem | pntrlog2bndlem5 26729* | Lemma for pntrlog2bnd 26732. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) & ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0)) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ+ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1)) | ||
Theorem | pntrlog2bndlem6a 26730* | Lemma for pntrlog2bndlem6 26731. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) & ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0)) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ+ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 1 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) = ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)))) | ||
Theorem | pntrlog2bndlem6 26731* | Lemma for pntrlog2bnd 26732. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.) |
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) & ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0)) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ+ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 1 ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1)) | ||
Theorem | pntrlog2bnd 26732* | A bound on 𝑅(𝑥)log↑2(𝑥). Equation 10.6.15 of [Shapiro], p. 431. (Contributed by Mario Carneiro, 1-Jun-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1(,)+∞)((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) ≤ 𝑐) | ||
Theorem | pntpbnd1a 26733* | Lemma for pntpbnd 26736. (Contributed by Mario Carneiro, 11-Apr-2016.) Replace reference to OLD theorem. (Revised by Wolf Lammen, 8-Sep-2020.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) & ⊢ 𝑋 = (exp‘(2 / 𝐸)) & ⊢ (𝜑 → 𝑌 ∈ (𝑋(,)+∞)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝑌 < 𝑁 ∧ 𝑁 ≤ (𝐾 · 𝑌))) & ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (abs‘((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁)))) ⇒ ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ≤ 𝐸) | ||
Theorem | pntpbnd1 26734* | Lemma for pntpbnd 26736. (Contributed by Mario Carneiro, 11-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) & ⊢ 𝑋 = (exp‘(2 / 𝐸)) & ⊢ (𝜑 → 𝑌 ∈ (𝑋(,)+∞)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑦 ∈ (𝑖...𝑗)((𝑅‘𝑦) / (𝑦 · (𝑦 + 1)))) ≤ 𝐴) & ⊢ 𝐶 = (𝐴 + 2) & ⊢ (𝜑 → 𝐾 ∈ ((exp‘(𝐶 / 𝐸))[,)+∞)) & ⊢ (𝜑 → ¬ ∃𝑦 ∈ ℕ ((𝑌 < 𝑦 ∧ 𝑦 ≤ (𝐾 · 𝑌)) ∧ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐸)) ⇒ ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘𝑌) + 1)...(⌊‘(𝐾 · 𝑌)))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝐴) | ||
Theorem | pntpbnd2 26735* | Lemma for pntpbnd 26736. (Contributed by Mario Carneiro, 11-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) & ⊢ 𝑋 = (exp‘(2 / 𝐸)) & ⊢ (𝜑 → 𝑌 ∈ (𝑋(,)+∞)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑦 ∈ (𝑖...𝑗)((𝑅‘𝑦) / (𝑦 · (𝑦 + 1)))) ≤ 𝐴) & ⊢ 𝐶 = (𝐴 + 2) & ⊢ (𝜑 → 𝐾 ∈ ((exp‘(𝐶 / 𝐸))[,)+∞)) & ⊢ (𝜑 → ¬ ∃𝑦 ∈ ℕ ((𝑌 < 𝑦 ∧ 𝑦 ≤ (𝐾 · 𝑌)) ∧ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐸)) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | pntpbnd 26736* | Lemma for pnt 26762. Establish smallness of 𝑅 at a point. Lemma 10.6.1 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 10-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ ∃𝑐 ∈ ℝ+ ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘(𝑐 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒) | ||
Theorem | pntibndlem1 26737 | Lemma for pntibnd 26741. (Contributed by Mario Carneiro, 10-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ 𝐿 = ((1 / 4) / (𝐴 + 3)) ⇒ ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) | ||
Theorem | pntibndlem2a 26738* | Lemma for pntibndlem2 26739. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ 𝐿 = ((1 / 4) / (𝐴 + 3)) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ 𝐾 = (exp‘(𝐵 / (𝐸 / 2))) & ⊢ 𝐶 = ((2 · 𝐵) + (log‘2)) & ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) & ⊢ (𝜑 → 𝑍 ∈ ℝ+) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁[,]((1 + (𝐿 · 𝐸)) · 𝑁))) → (𝑢 ∈ ℝ ∧ 𝑁 ≤ 𝑢 ∧ 𝑢 ≤ ((1 + (𝐿 · 𝐸)) · 𝑁))) | ||
Theorem | pntibndlem2 26739* | Lemma for pntibnd 26741. The main work, after eliminating all the quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ 𝐿 = ((1 / 4) / (𝐴 + 3)) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ 𝐾 = (exp‘(𝐵 / (𝐸 / 2))) & ⊢ 𝐶 = ((2 · 𝐵) + (log‘2)) & ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) & ⊢ (𝜑 → 𝑍 ∈ ℝ+) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](2 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝑇 · (𝑥 / (log‘𝑥))))) & ⊢ 𝑋 = ((exp‘(𝑇 / (𝐸 / 4))) + 𝑍) & ⊢ (𝜑 → 𝑀 ∈ ((exp‘(𝐶 / 𝐸))[,)+∞)) & ⊢ (𝜑 → 𝑌 ∈ (𝑋(,)+∞)) & ⊢ (𝜑 → ((𝑌 < 𝑁 ∧ 𝑁 ≤ ((𝑀 / 2) · 𝑌)) ∧ (abs‘((𝑅‘𝑁) / 𝑁)) ≤ (𝐸 / 2))) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ ℝ+ ((𝑌 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑀 · 𝑌)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) | ||
Theorem | pntibndlem3 26740* | Lemma for pntibnd 26741. Package up pntibndlem2 26739 in quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ 𝐿 = ((1 / 4) / (𝐴 + 3)) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ 𝐾 = (exp‘(𝐵 / (𝐸 / 2))) & ⊢ 𝐶 = ((2 · 𝐵) + (log‘2)) & ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) & ⊢ (𝜑 → 𝑍 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑚 ∈ (𝐾[,)+∞)∀𝑣 ∈ (𝑍(,)+∞)∃𝑖 ∈ ℕ ((𝑣 < 𝑖 ∧ 𝑖 ≤ (𝑚 · 𝑣)) ∧ (abs‘((𝑅‘𝑖) / 𝑖)) ≤ (𝐸 / 2))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘(𝐶 / 𝐸))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) | ||
Theorem | pntibnd 26741* | Lemma for pnt 26762. Establish smallness of 𝑅 on an interval. Lemma 10.6.2 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 10-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) ⇒ ⊢ ∃𝑐 ∈ ℝ+ ∃𝑙 ∈ (0(,)1)∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘(𝑐 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝑙 · 𝑒)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝑙 · 𝑒)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝑒) | ||
Theorem | pntlemd 26742 | Lemma for pnt 26762. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝐴 is C^*, 𝐵 is c1, 𝐿 is λ, 𝐷 is c2, and 𝐹 is c3. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) ⇒ ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) | ||
Theorem | pntlemc 26743* | Lemma for pnt 26762. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑈 is α, 𝐸 is ε, and 𝐾 is K. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) ⇒ ⊢ (𝜑 → (𝐸 ∈ ℝ+ ∧ 𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+))) | ||
Theorem | pntlema 26744* | Lemma for pnt 26762. Closure for the constants used in the proof. The mammoth expression 𝑊 is a number large enough to satisfy all the lower bounds needed for 𝑍. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑌 is x2, 𝑋 is x1, 𝐶 is the big-O constant in Equation 10.6.29 of [Shapiro], p. 435, and 𝑊 is the unnamed lower bound of "for sufficiently large x" in Equation 10.6.34 of [Shapiro], p. 436. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) ⇒ ⊢ (𝜑 → 𝑊 ∈ ℝ+) | ||
Theorem | pntlemb 26745* | Lemma for pnt 26762. Unpack all the lower bounds contained in 𝑊, in the form they will be used. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑍 is x. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞)) ⇒ ⊢ (𝜑 → (𝑍 ∈ ℝ+ ∧ (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)) ∧ ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))))) | ||
Theorem | pntlemg 26746* | Lemma for pnt 26762. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑀 is j^* and 𝑁 is ĵ. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞)) & ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) & ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) ⇒ ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀))) | ||
Theorem | pntlemh 26747* | Lemma for pnt 26762. Bounds on the subintervals in the induction. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞)) & ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) & ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) ⇒ ⊢ ((𝜑 ∧ 𝐽 ∈ (𝑀...𝑁)) → (𝑋 < (𝐾↑𝐽) ∧ (𝐾↑𝐽) ≤ (√‘𝑍))) | ||
Theorem | pntlemn 26748* | Lemma for pnt 26762. The "naive" base bound, which we will slightly improve. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞)) & ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) & ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) ⇒ ⊢ ((𝜑 ∧ (𝐽 ∈ ℕ ∧ 𝐽 ≤ (𝑍 / 𝑌))) → 0 ≤ (((𝑈 / 𝐽) − (abs‘((𝑅‘(𝑍 / 𝐽)) / 𝑍))) · (log‘𝐽))) | ||
Theorem | pntlemq 26749* | Lemma for pntlemj 26751. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞)) & ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) & ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) & ⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) & ⊢ 𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝐽)))) & ⊢ (𝜑 → 𝑉 ∈ ℝ+) & ⊢ (𝜑 → (((𝐾↑𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽))) ∧ ∀𝑢 ∈ (𝑉[,]((1 + (𝐿 · 𝐸)) · 𝑉))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) & ⊢ (𝜑 → 𝐽 ∈ (𝑀..^𝑁)) & ⊢ 𝐼 = (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉))) ⇒ ⊢ (𝜑 → 𝐼 ⊆ 𝑂) | ||
Theorem | pntlemr 26750* | Lemma for pntlemj 26751. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞)) & ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) & ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) & ⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) & ⊢ 𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝐽)))) & ⊢ (𝜑 → 𝑉 ∈ ℝ+) & ⊢ (𝜑 → (((𝐾↑𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽))) ∧ ∀𝑢 ∈ (𝑉[,]((1 + (𝐿 · 𝐸)) · 𝑉))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) & ⊢ (𝜑 → 𝐽 ∈ (𝑀..^𝑁)) & ⊢ 𝐼 = (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉))) ⇒ ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ ((♯‘𝐼) · ((𝑈 − 𝐸) · ((log‘(𝑍 / 𝑉)) / (𝑍 / 𝑉))))) | ||
Theorem | pntlemj 26751* | Lemma for pnt 26762. The induction step. Using pntibnd 26741, we find an interval in 𝐾↑𝐽...𝐾↑(𝐽 + 1) which is sufficiently large and has a much smaller value, 𝑅(𝑧) / 𝑧 ≤ 𝐸 (instead of our original bound 𝑅(𝑧) / 𝑧 ≤ 𝑈). (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞)) & ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) & ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) & ⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) & ⊢ 𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝐽)))) & ⊢ (𝜑 → 𝑉 ∈ ℝ+) & ⊢ (𝜑 → (((𝐾↑𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽))) ∧ ∀𝑢 ∈ (𝑉[,]((1 + (𝐿 · 𝐸)) · 𝑉))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) & ⊢ (𝜑 → 𝐽 ∈ (𝑀..^𝑁)) & ⊢ 𝐼 = (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉))) ⇒ ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛 ∈ 𝑂 (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) | ||
Theorem | pntlemi 26752* | Lemma for pnt 26762. Eliminate some assumptions from pntlemj 26751. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞)) & ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) & ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) & ⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) & ⊢ 𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝐽)))) ⇒ ⊢ ((𝜑 ∧ 𝐽 ∈ (𝑀..^𝑁)) → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛 ∈ 𝑂 (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) | ||
Theorem | pntlemf 26753* | Lemma for pnt 26762. Add up the pieces in pntlemi 26752 to get an estimate slightly better than the naive lower bound 0. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞)) & ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) & ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) & ⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) ⇒ ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) | ||
Theorem | pntlemk 26754* | Lemma for pnt 26762. Evaluate the naive part of the estimate. (Contributed by Mario Carneiro, 14-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞)) & ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) & ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) & ⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) ⇒ ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍))) | ||
Theorem | pntlemo 26755* | Lemma for pnt 26762. Combine all the estimates to establish a smaller eventual bound on 𝑅(𝑍) / 𝑍. (Contributed by Mario Carneiro, 14-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞)) & ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) & ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) & ⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) & ⊢ (𝜑 → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅‘𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶) ⇒ ⊢ (𝜑 → (abs‘((𝑅‘𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3)))) | ||
Theorem | pntleme 26756* | Lemma for pnt 26762. Package up pntlemo 26755 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) & ⊢ (𝜑 → ∀𝑘 ∈ (𝐾[,)+∞)∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)) & ⊢ (𝜑 → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅‘𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3)))) | ||
Theorem | pntlem3 26757* | Lemma for pnt 26762. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 8-Apr-2016.) (Proof shortened by AV, 27-Sep-2020.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴) & ⊢ 𝑇 = {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡} & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑇) → (𝑢 − (𝐶 · (𝑢↑3))) ∈ 𝑇) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ⇝𝑟 1) | ||
Theorem | pntlemp 26758* | Lemma for pnt 26762. Wrapping up more quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘(𝐵 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝑒)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝑒)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝑒)) & ⊢ (𝜑 → 𝑈 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ 𝐸 = (𝑈 / 𝐷) & ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) & ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3)))) | ||
Theorem | pntleml 26759* | Lemma for pnt 26762. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 14-Apr-2016.) |
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) & ⊢ 𝐷 = (𝐴 + 1) & ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) & ⊢ (𝜑 → ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘(𝐵 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝑒)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝑒)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝑒)) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ⇝𝑟 1) | ||
Theorem | pnt3 26760 | The Prime Number Theorem, version 3: the second Chebyshev function tends asymptotically to 𝑥. (Contributed by Mario Carneiro, 1-Jun-2016.) |
⊢ (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ⇝𝑟 1 | ||
Theorem | pnt2 26761 | The Prime Number Theorem, version 2: the first Chebyshev function tends asymptotically to 𝑥. (Contributed by Mario Carneiro, 1-Jun-2016.) |
⊢ (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ⇝𝑟 1 | ||
Theorem | pnt 26762 | The Prime Number Theorem: the number of prime numbers less than 𝑥 tends asymptotically to 𝑥 / log(𝑥) as 𝑥 goes to infinity. This is Metamath 100 proof #5. (Contributed by Mario Carneiro, 1-Jun-2016.) |
⊢ (𝑥 ∈ (1(,)+∞) ↦ ((π‘𝑥) / (𝑥 / (log‘𝑥)))) ⇝𝑟 1 | ||
Theorem | abvcxp 26763* | Raising an absolute value to a power less than one yields another absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ ((𝐹‘𝑥)↑𝑐𝑆)) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑆 ∈ (0(,]1)) → 𝐺 ∈ 𝐴) | ||
Theorem | padicfval 26764* | Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) ⇒ ⊢ (𝑃 ∈ ℙ → (𝐽‘𝑃) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑃↑-(𝑃 pCnt 𝑥))))) | ||
Theorem | padicval 26765* | Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) ⇒ ⊢ ((𝑃 ∈ ℙ ∧ 𝑋 ∈ ℚ) → ((𝐽‘𝑃)‘𝑋) = if(𝑋 = 0, 0, (𝑃↑-(𝑃 pCnt 𝑋)))) | ||
Theorem | ostth2lem1 26766* | Lemma for ostth2 26785, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 26785. If a power is upper bounded by a linear term, the exponent must be less than one. Or in big-O notation, 𝑛 ∈ 𝑜(𝐴↑𝑛) for any 1 < 𝐴. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ≤ (𝑛 · 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 1) | ||
Theorem | qrngbas 26767 | The base set of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) ⇒ ⊢ ℚ = (Base‘𝑄) | ||
Theorem | qdrng 26768 | The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) ⇒ ⊢ 𝑄 ∈ DivRing | ||
Theorem | qrng0 26769 | The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) ⇒ ⊢ 0 = (0g‘𝑄) | ||
Theorem | qrng1 26770 | The unit element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) ⇒ ⊢ 1 = (1r‘𝑄) | ||
Theorem | qrngneg 26771 | The additive inverse in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) ⇒ ⊢ (𝑋 ∈ ℚ → ((invg‘𝑄)‘𝑋) = -𝑋) | ||
Theorem | qrngdiv 26772 | The division operation in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) ⇒ ⊢ ((𝑋 ∈ ℚ ∧ 𝑌 ∈ ℚ ∧ 𝑌 ≠ 0) → (𝑋(/r‘𝑄)𝑌) = (𝑋 / 𝑌)) | ||
Theorem | qabvle 26773 | By using induction on 𝑁, we show a long-range inequality coming from the triangle inequality. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐹‘𝑁) ≤ 𝑁) | ||
Theorem | qabvexp 26774 | Induct the product rule abvmul 20089 to find the absolute value of a power. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑀 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐹‘(𝑀↑𝑁)) = ((𝐹‘𝑀)↑𝑁)) | ||
Theorem | ostthlem1 26775* | Lemma for ostth 26787. If two absolute values agree on the positive integers greater than one, then they agree for all rational numbers and thus are equal as functions. (Contributed by Mario Carneiro, 9-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ (𝜑 → 𝐹 ∈ 𝐴) & ⊢ (𝜑 → 𝐺 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝐹‘𝑛) = (𝐺‘𝑛)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | ostthlem2 26776* | Lemma for ostth 26787. Refine ostthlem1 26775 so that it is sufficient to only show equality on the primes. (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 20-Jun-2015.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ (𝜑 → 𝐹 ∈ 𝐴) & ⊢ (𝜑 → 𝐺 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝐹‘𝑝) = (𝐺‘𝑝)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | qabsabv 26777 | The regular absolute value function on the rationals is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 9-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) ⇒ ⊢ (abs ↾ ℚ) ∈ 𝐴 | ||
Theorem | padicabv 26778* | The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ 𝐹 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥)))) ⇒ ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝐹 ∈ 𝐴) | ||
Theorem | padicabvf 26779* | The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) ⇒ ⊢ 𝐽:ℙ⟶𝐴 | ||
Theorem | padicabvcxp 26780* | All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) ⇒ ⊢ ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℝ+) → (𝑦 ∈ ℚ ↦ (((𝐽‘𝑃)‘𝑦)↑𝑐𝑅)) ∈ 𝐴) | ||
Theorem | ostth1 26781* | - Lemma for ostth 26787: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If 𝐹 is equal to 1 on the primes, then by complete induction and the multiplicative property abvmul 20089 of the absolute value, 𝐹 is equal to 1 on all the integers, and ostthlem1 26775 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) & ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) & ⊢ (𝜑 → 𝐹 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ¬ 1 < (𝐹‘𝑛)) & ⊢ (𝜑 → ∀𝑛 ∈ ℙ ¬ (𝐹‘𝑛) < 1) ⇒ ⊢ (𝜑 → 𝐹 = 𝐾) | ||
Theorem | ostth2lem2 26782* | Lemma for ostth2 26785. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) & ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) & ⊢ (𝜑 → 𝐹 ∈ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 1 < (𝐹‘𝑁)) & ⊢ 𝑅 = ((log‘(𝐹‘𝑁)) / (log‘𝑁)) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘2)) & ⊢ 𝑆 = ((log‘(𝐹‘𝑀)) / (log‘𝑀)) & ⊢ 𝑇 = if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ ℕ0 ∧ 𝑌 ∈ (0...((𝑀↑𝑋) − 1))) → (𝐹‘𝑌) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))) | ||
Theorem | ostth2lem3 26783* | Lemma for ostth2 26785. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) & ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) & ⊢ (𝜑 → 𝐹 ∈ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 1 < (𝐹‘𝑁)) & ⊢ 𝑅 = ((log‘(𝐹‘𝑁)) / (log‘𝑁)) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘2)) & ⊢ 𝑆 = ((log‘(𝐹‘𝑀)) / (log‘𝑀)) & ⊢ 𝑇 = if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) & ⊢ 𝑈 = ((log‘𝑁) / (log‘𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ ℕ) → (((𝐹‘𝑁) / (𝑇↑𝑐𝑈))↑𝑋) ≤ (𝑋 · ((𝑀 · 𝑇) · (𝑈 + 1)))) | ||
Theorem | ostth2lem4 26784* | Lemma for ostth2 26785. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) & ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) & ⊢ (𝜑 → 𝐹 ∈ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 1 < (𝐹‘𝑁)) & ⊢ 𝑅 = ((log‘(𝐹‘𝑁)) / (log‘𝑁)) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘2)) & ⊢ 𝑆 = ((log‘(𝐹‘𝑀)) / (log‘𝑀)) & ⊢ 𝑇 = if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) & ⊢ 𝑈 = ((log‘𝑁) / (log‘𝑀)) ⇒ ⊢ (𝜑 → (1 < (𝐹‘𝑀) ∧ 𝑅 ≤ 𝑆)) | ||
Theorem | ostth2 26785* | - Lemma for ostth 26787: regular case. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) & ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) & ⊢ (𝜑 → 𝐹 ∈ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 1 < (𝐹‘𝑁)) & ⊢ 𝑅 = ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑎))) | ||
Theorem | ostth3 26786* | - Lemma for ostth 26787: p-adic case. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) & ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) & ⊢ (𝜑 → 𝐹 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ¬ 1 < (𝐹‘𝑛)) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → (𝐹‘𝑃) < 1) & ⊢ 𝑅 = -((log‘(𝐹‘𝑃)) / (log‘𝑃)) & ⊢ 𝑆 = if((𝐹‘𝑃) ≤ (𝐹‘𝑝), (𝐹‘𝑝), (𝐹‘𝑃)) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ ℝ+ 𝐹 = (𝑦 ∈ ℚ ↦ (((𝐽‘𝑃)‘𝑦)↑𝑐𝑎))) | ||
Theorem | ostth 26787* | Ostrowski's theorem, which classifies all absolute values on ℚ. Any such absolute value must either be the trivial absolute value 𝐾, a constant exponent 0 < 𝑎 ≤ 1 times the regular absolute value, or a positive exponent times the p-adic absolute value. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐴 = (AbsVal‘𝑄) & ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) & ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) ⇒ ⊢ (𝐹 ∈ 𝐴 ↔ (𝐹 = 𝐾 ∨ ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑎)) ∨ ∃𝑎 ∈ ℝ+ ∃𝑔 ∈ ran 𝐽 𝐹 = (𝑦 ∈ ℚ ↦ ((𝑔‘𝑦)↑𝑐𝑎)))) | ||
This part develops elementary geometry based on Tarski's axioms, following [Schwabhauser]. Tarski's geometry is a first-order theory with one sort, the "points". It has two primitive notions, the ternary predicate of "betweenness" and the quaternary predicate of "congruence". To adapt this theory to the framework of set.mm, and to be able to talk of *a* Tarski structure as a space satisfying the given axioms, we use the following definition, stated informally: A Tarski structure 𝑓 is a set (of points) (Base‘𝑓) together with functions (Itv‘𝑓) and (dist‘𝑓) on ((Base‘𝑓) × (Base‘𝑓)) satisfying certain axioms (given in Definitions df-trkg 26814 et sequentes). This allows us to treat a Tarski structure as a special kind of extensible structure (see df-struct 16848). The translation to and from Tarski's treatment is as follows (given, again, informally). Suppose that one is given an extensible structure 𝑓. One defines a betweenness ternary predicate Btw by positing that, for any 𝑥, 𝑦, 𝑧 ∈ (Base‘𝑓), one has "Btw 𝑥𝑦𝑧 " if and only if 𝑦 ∈ 𝑥(Itv‘𝑓)𝑧, and a congruence quaternary predicate Congr by positing that, for any 𝑥, 𝑦, 𝑧, 𝑡 ∈ (Base‘𝑓), one has "Congr 𝑥𝑦𝑧𝑡 " if and only if 𝑥(dist‘𝑓)𝑦 = 𝑧(dist‘𝑓)𝑡. It is easy to check that if 𝑓 satisfies our Tarski axioms, then Btw and Congr satisfy Tarski's Tarski axioms when (Base‘𝑓) is interpreted as the universe of discourse. Conversely, suppose that one is given a set 𝑎, a ternary predicate Btw, and a quaternary predicate Congr. One defines the extensible structure 𝑓 such that (Base‘𝑓) is 𝑎, and (Itv‘𝑓) is the function which associates with each 〈𝑥, 𝑦〉 ∈ (𝑎 × 𝑎) the set of points 𝑧 ∈ 𝑎 such that "Btw 𝑥𝑧𝑦", and (dist‘𝑓) is the function which associates with each 〈𝑥, 𝑦〉 ∈ (𝑎 × 𝑎) the set of ordered pairs 〈𝑧, 𝑡〉 ∈ (𝑎 × 𝑎) such that "Congr 𝑥𝑦𝑧𝑡". It is easy to check that if Btw and Congr satisfy Tarski's Tarski axioms when 𝑎 is interpreted as the universe of discourse, then 𝑓 satisfies our Tarski axioms. We intentionally choose to represent congruence (without loss of generality) as 𝑥(dist‘𝑓)𝑦 = 𝑧(dist‘𝑓)𝑡 instead of "Congr 𝑥𝑦𝑧𝑡", as it is more convenient. It is always possible to define dist for any particular geometry to produce equal results when conguence is desired, and in many cases there is an obvious interpretation of "distance" between two points that can be useful in other situations. Encoding congruence as an equality of distances makes it easier to use these theorems in cases where there is a preferred distance function. We prove that representing a congruence relationship using a distance in the form 𝑥(dist‘𝑓)𝑦 = 𝑧(dist‘𝑓)𝑡 causes no loss of generality in tgjustc1 26836 and tgjustc2 26837, which in turn are supported by tgjustf 26834 and tgjustr 26835. A similar representation of congruence (using a "distance" function) is used in Axiom A1 of [Beeson2016] p. 5, which discusses how a large number of formalized proofs were found in Tarskian Geometry using OTTER. Their detailed proofs in Tarski Geometry, along with other information, are available at https://www.michaelbeeson.com/research/FormalTarski/ 26835. Most theorems are in deduction form, as this is a very general, simple, and convenient format to use in Metamath. An assertion in deduction form can be easily converted into an assertion in inference form (removing the antecedents 𝜑 →) by insert a ⊤ → in each hypothesis, using a1i 11, then using mptru 1546 to remove the final ⊤ → prefix. In some cases we represent, without loss of generality, an implication antecedent in [Schwabhauser] as a hypothesis. The implication can be retrieved from the by using simpr 485, the theorem as stated, and ex 413. For descriptions of individual axioms, we refer to the specific definitions below. A particular feature of Tarski's axioms is modularity, so by using various subsets of the set of axioms, we can define the classes of "absolute dimensionless Tarski structures" (df-trkg 26814), of "Euclidean dimensionless Tarski structures" (df-trkge 26812) and of "Tarski structures of dimension no less than N" (df-trkgld 26813). In this system, angles are not a primitive notion, but instead a derived notion (see df-cgra 27169 and iscgra 27170). To maintain its simplicity, in this system congruence between shapes (a finite sequence of points) is the case where corresponding segments between all corresponding points are congruent. This includes triangles (a shape of 3 distinct points). Note that this definition has no direct regard for angles. For more details and rationale, see df-cgrg 26872. The first section is devoted to the definitions of these various structures. The second section ("Tarskian geometry") develops the synthetic treatment of geometry. The remaining sections prove that real Euclidean spaces and complex Hilbert spaces, with intended interpretations, are Euclidean Tarski structures. Most of the work in this part is due to Thierry Arnoux, with earlier work by Mario Carneiro and Scott Fenton. See also the credits in the comment of each statement. | ||
Syntax | cstrkg 26788 | Extends class notation with the class of Tarski geometries. |
class TarskiG | ||
Syntax | cstrkgc 26789 | Extends class notation with the class of geometries fulfilling the congruence axioms. |
class TarskiGC | ||
Syntax | cstrkgb 26790 | Extends class notation with the class of geometries fulfilling the betweenness axioms. |
class TarskiGB | ||
Syntax | cstrkgcb 26791 | Extends class notation with the class of geometries fulfilling the congruence and betweenness axioms. |
class TarskiGCB | ||
Syntax | cstrkgld 26792 | Extends class notation with the relation for geometries fulfilling the lower dimension axioms. |
class DimTarskiG≥ | ||
Syntax | cstrkge 26793 | Extends class notation with the class of geometries fulfilling Euclid's axiom. |
class TarskiGE | ||
Syntax | citv 26794 | Declare the syntax for the Interval (segment) index extractor. |
class Itv | ||
Syntax | clng 26795 | Declare the syntax for the Line function. |
class LineG | ||
Definition | df-itv 26796 | Define the Interval (segment) index extractor for Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) Use its index-independent form itvid 26800 instead. (New usage is discouraged.) |
⊢ Itv = Slot ;16 | ||
Definition | df-lng 26797 | Define the line index extractor for geometries. (Contributed by Thierry Arnoux, 27-Mar-2019.) Use its index-independent form lngid 26801 instead. (New usage is discouraged.) |
⊢ LineG = Slot ;17 | ||
Theorem | itvndx 26798 | Index value of the Interval (segment) slot. Use ndxarg 16897. (Contributed by Thierry Arnoux, 24-Aug-2017.) (New usage is discouraged.) |
⊢ (Itv‘ndx) = ;16 | ||
Theorem | lngndx 26799 | Index value of the "line" slot. Use ndxarg 16897. (Contributed by Thierry Arnoux, 27-Mar-2019.) (New usage is discouraged.) |
⊢ (LineG‘ndx) = ;17 | ||
Theorem | itvid 26800 | Utility theorem: index-independent form of df-itv 26796. (Contributed by Thierry Arnoux, 24-Aug-2017.) |
⊢ Itv = Slot (Itv‘ndx) |
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