P. 254 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pntrsumbnd2 25301*	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 25302*	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 25303*	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 25304*	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 25305*	The sum of selberg3r 25303 and selberg4r 25304. (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 25306*	Define the "Selberg function", whose asymptotic behavior is the content of selberg 25282. (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 25307*	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 25308*	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 25309*	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 25310*	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 25311*	The sum of selberg3r 25303 and selberg4r 25304. (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 25312*	Lemma for pntrlog2bnd 25318. 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 25313*	Lemma for pntrlog2bnd 25318. 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 25314*	Lemma for pntrlog2bnd 25318. 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 25315*	Lemma for pntrlog2bnd 25318. 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 25316*	Lemma for pntrlog2bndlem6 25317. (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 25317*	Lemma for pntrlog2bnd 25318. 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 25318*	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 25319*	Lemma for pntpbnd 25322. (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 25320*	Lemma for pntpbnd 25322. (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 25321*	Lemma for pntpbnd 25322. (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 25322*	Lemma for pnt 25348. 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 25323	Lemma for pntibnd 25327. (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 25324*	Lemma for pntibndlem2 25325. (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 25325*	Lemma for pntibnd 25327. 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 25326*	Lemma for pntibnd 25327. Package up pntibndlem2 25325 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 25327*	Lemma for pnt 25348. 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 25328	Lemma for pnt 25348. 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 25329*	Lemma for pnt 25348. 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 25330*	Lemma for pnt 25348. 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 25331*	Lemma for pnt 25348. 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 25332*	Lemma for pnt 25348. 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 25333*	Lemma for pnt 25348. 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 25334*	Lemma for pnt 25348. 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 25335*	Lemma for pntlemj 25337. (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 25336*	Lemma for pntlemj 25337. (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 25337*	Lemma for pnt 25348. The induction step. Using pntibnd 25327, 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 25338*	Lemma for pnt 25348. Eliminate some assumptions from pntlemj 25337. (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 25339*	Lemma for pnt 25348. Add up the pieces in pntlemi 25338 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 25340*	Lemma for pnt 25348. 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 25341*	Lemma for pnt 25348. 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 25342*	Lemma for pnt 25348. Package up pntlemo 25341 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 25343*	Lemma for pnt 25348. 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 25344*	Lemma for pnt 25348. 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 25345*	Lemma for pnt 25348. 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 25346	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 25347	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 25348	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
14.4.14  Ostrowski's theorem
Theorem	abvcxp 25349*	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 25350*	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 25351*	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 25352*	Lemma for ostth2 25371, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 25371. 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 25353	The base set of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ ℚ = (Base‘𝑄)
Theorem	qdrng 25354	The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ 𝑄 ∈ DivRing
Theorem	qrng0 25355	The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ 0 = (0g‘𝑄)
Theorem	qrng1 25356	The unit element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ 1 = (1r‘𝑄)
Theorem	qrngneg 25357	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 25358	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 25359	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 25360	Induct the product rule abvmul 18877 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 25361*	Lemma for ostth 25373. 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 25362*	Lemma for ostth 25373. Refine ostthlem1 25361 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 25363	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 25364*	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 25365*	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 25366*	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 25367*	- Lemma for ostth 25373: 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 18877 of the absolute value, 𝐹 is equal to 1 on all the integers, and ostthlem1 25361 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 25368*	Lemma for ostth2 25371. (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 25369*	Lemma for ostth2 25371. (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 25370*	Lemma for ostth2 25371. (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 25371*	- Lemma for ostth 25373: 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 25372*	- Lemma for ostth 25373: 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 25373*	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))))
PART 15  ELEMENTARY GEOMETRY 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 the definitions df-trkg 25397 et sequentes). This allows us to treat a Tarski structure as a special kind of extensible structure (see df-struct 15906). 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. A similar representation 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 http://www.michaelbeeson.com/research/FormalTarski/. 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 trud 1533 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 476, the theorem as stated, and ex 449. 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 25397), of "Euclidean dimensionless Tarski structures" (df-trkge 25395) and of "Tarski structures of dimension no less than N" (df-trkgld 25396). 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.
15.1  Definition and Tarski's Axioms of Geometry
Syntax	cstrkg 25374	Extends class notation with the class of Tarski geometries.
class TarskiG
Syntax	cstrkgc 25375	Extends class notation with the class of geometries fulfilling the congruence axioms.
class TarskiGC
Syntax	cstrkgb 25376	Extends class notation with the class of geometries fulfilling the betweenness axioms.
class TarskiGB
Syntax	cstrkgcb 25377	Extends class notation with the class of geometries fulfilling the congruence and betweenness axioms.
class TarskiGCB
Syntax	cstrkgld 25378	Extends class notation with the relation for geometries fulfilling the lower dimension axioms.
class DimTarskiG≥
Syntax	cstrkge 25379	Extends class notation with the class of geometries fulfilling Euclid's axiom.
class TarskiGE
Syntax	citv 25380	Declare the syntax for the Interval (segment) index extractor.
class Itv
Syntax	clng 25381	Declare the syntax for the Line function.
class LineG
Definition	df-itv 25382	Define the Interval (segment) index extractor for Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.)
⊢ Itv = Slot ;16
Definition	df-lng 25383	Define the line index extractor for geometries. (Contributed by Thierry Arnoux, 27-Mar-2019.)
⊢ LineG = Slot ;17
Theorem	itvndx 25384	Index value of the Interval (segment) slot. Use ndxarg 15929. (Contributed by Thierry Arnoux, 24-Aug-2017.)
⊢ (Itv‘ndx) = ;16
Theorem	lngndx 25385	Index value of the "line" slot. Use ndxarg 15929. (Contributed by Thierry Arnoux, 27-Mar-2019.)
⊢ (LineG‘ndx) = ;17
Theorem	itvid 25386	Utility theorem: index-independent form of df-itv 25382. (Contributed by Thierry Arnoux, 24-Aug-2017.)
⊢ Itv = Slot (Itv‘ndx)
Theorem	lngid 25387	Utility theorem: index-independent form of df-lng 25383. (Contributed by Thierry Arnoux, 27-Mar-2019.)
⊢ LineG = Slot (LineG‘ndx)
Theorem	trkgstr 25388	Functionality of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝑈⟩, ⟨(dist‘ndx), 𝐷⟩, ⟨(Itv‘ndx), 𝐼⟩}    ⇒   ⊢ 𝑊 Struct ⟨1, ;16⟩
Theorem	trkgbas 25389	The base set of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝑈⟩, ⟨(dist‘ndx), 𝐷⟩, ⟨(Itv‘ndx), 𝐼⟩}    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝑈 = (Base‘𝑊))
Theorem	trkgdist 25390	The measure of a distance in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝑈⟩, ⟨(dist‘ndx), 𝐷⟩, ⟨(Itv‘ndx), 𝐼⟩}    ⇒   ⊢ (𝐷 ∈ 𝑉 → 𝐷 = (dist‘𝑊))
Theorem	trkgitv 25391	The congruence relation in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝑈⟩, ⟨(dist‘ndx), 𝐷⟩, ⟨(Itv‘ndx), 𝐼⟩}    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐼 = (Itv‘𝑊))
Definition	df-trkgc 25392*	Define the class of geometries fulfilling the congruence axioms of reflexivity, identity and transitivity. These are axioms A1 to A3 of [Schwabhauser] p. 10. With our distance based notation for congruence, transitivity of congruence boils down to transitivity of equality and is already given by eqtr 2670, so it is not listed in this definition. (Contributed by Thierry Arnoux, 24-Aug-2017.)
⊢ TarskiGC = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
Definition	df-trkgb 25393*	Define the class of geometries fulfilling the 3 betweenness axioms in Tarski's Axiomatization of Geometry: identity, Axiom A6 of [Schwabhauser] p. 11, axiom of Pasch, Axiom A7 of [Schwabhauser] p. 12, and continuity, Axiom A11 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 24-Aug-2017.)
⊢ TarskiGB = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝∀𝑡 ∈ 𝒫 𝑝(∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦)))}
Definition	df-trkgcb 25394*	Define the class of geometries fulfilling the five segment axiom, Axiom A5 of [Schwabhauser] p. 11, and segment construction axiom, Axiom A4 of [Schwabhauser] p. 11. (Contributed by Thierry Arnoux, 14-Mar-2019.)
⊢ TarskiGCB = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑎 ∈ 𝑝 ∀𝑏 ∈ 𝑝 ∀𝑐 ∈ 𝑝 ∀𝑣 ∈ 𝑝 (((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥𝑖𝑧) ∧ 𝑏 ∈ (𝑎𝑖𝑐)) ∧ (((𝑥𝑑𝑦) = (𝑎𝑑𝑏) ∧ (𝑦𝑑𝑧) = (𝑏𝑑𝑐)) ∧ ((𝑥𝑑𝑢) = (𝑎𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑏𝑑𝑣)))) → (𝑧𝑑𝑢) = (𝑐𝑑𝑣)) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑎 ∈ 𝑝 ∀𝑏 ∈ 𝑝 ∃𝑧 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑧) ∧ (𝑦𝑑𝑧) = (𝑎𝑑𝑏)))}
Definition	df-trkge 25395*	Define the class of geometries fulfilling Euclid's axiom, Axiom A10 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 14-Mar-2019.)
⊢ TarskiGE = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖]∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ 𝑝 ∃𝑏 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏)))}
Definition	df-trkgld 25396*	Define the class of geometries fulfilling the lower dimension axiom for dimension 𝑛. For such geometries, there are three non-colinear points that are equidistant from 𝑛 − 1 distinct points. Derived from remarks in Tarski's System of Geometry, Alfred Tarski and Steven Givant, Bulletin of Symbolic Logic, Volume 5, Number 2 (1999), 175-214. (Contributed by Scott Fenton, 22-Apr-2013.) (Revised by Thierry Arnoux, 23-Nov-2019.)
⊢ DimTarskiG≥ = {⟨𝑔, 𝑛⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]∃𝑓(𝑓:(1..^𝑛)–1-1→𝑝 ∧ ∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 ∃𝑧 ∈ 𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓‘𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓‘𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓‘𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))}
Definition	df-trkg 25397*	Define the class of Tarski geometries. A Tarski geometry is a set of points, equipped with a betweenness relation (denoting that a point lies on a line segment between two other points) and a congruence relation (denoting equality of line segment lengths). Here, we are using the following: for congruence, (𝑥 − 𝑦) = (𝑧 − 𝑤) where − = (dist‘𝑊) for betweenness, 𝑦 ∈ (𝑥𝐼𝑧), where 𝐼 = (Itv‘𝑊) With this definition, the axiom A2 is actually equivalent to the transitivity of addition, eqtrd 2685. Tarski originally had more axioms, but later reduced his list to 11: A1 A kind of reflexivity for the congruence relation (TarskiGC) A2 Transitivity for the congruence relation (TarskiGC) A3 Identity for the congruence relation (TarskiGC) A4 Axiom of segment construction (TarskiGCB) A5 5-segment axiom (TarskiGCB) A6 Identity for the betweenness relation (TarskiGB) A7 Axiom of Pasch (TarskiGB) A8 Lower dimension axiom (DimTarskiG≥‘2) A9 Upper dimension axiom (V ∖ (DimTarskiG≥‘3)) A10 Euclid's axiom (TarskiGE) A11 Axiom of continuity (TarskiGB) Our definition is split into 5 parts: congruence axioms TarskiGC (which metric spaces fulfill) betweenness axioms TarskiGB congruence and betweenness axioms TarskiGCB upper and lower dimension axioms DimTarskiG≥ axiom of Euclid / parallel postulate TarskiGE So our definition of a Tarskian Geometry includes the 3 axioms for the quaternary congruence relation (A1, A2, A3), the 3 axioms for the ternary betweenness relation (A6, A7, A11), and the 2 axioms of compatibility of the congruence and the betweenness relations (A4,A5). It does not include Euclid's axiom A10, nor the 2-dimensional axioms A8 (Lower dimension axiom) and A9 (Upper dimension axiom) so the number of dimensions of the geometry it formalizes is not constrained. Considering A2 as one of the 3 axioms for the quaternary congruence relation is somewhat conventional, because the transitivity of the congruence relation is automatically given by our choice to take the distance as this congruence relation in our definition of Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) (Revised by Thierry Arnoux, 27-Apr-2019.)
⊢ TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
Theorem	istrkgc 25398*	Property of being a Tarski geometry - congruence part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    ⇒   ⊢ (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦))))
Theorem	istrkgb 25399*	Property of being a Tarski geometry - betweenness part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    ⇒   ⊢ (𝐺 ∈ TarskiGB ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))
Theorem	istrkgcb 25400*	Property of being a Tarski geometry - congruence and betweenness part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    ⇒   ⊢ (𝐺 ∈ TarskiGCB ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∀𝑐 ∈ 𝑃 ∀𝑣 ∈ 𝑃 (((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 − 𝑦) = (𝑎 − 𝑏) ∧ (𝑦 − 𝑧) = (𝑏 − 𝑐)) ∧ ((𝑥 − 𝑢) = (𝑎 − 𝑣) ∧ (𝑦 − 𝑢) = (𝑏 − 𝑣)))) → (𝑧 − 𝑢) = (𝑐 − 𝑣)) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)))))