P. 279 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27801-27900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pntrsumo1 27801*	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 27802*	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 27803*	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 27804*	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 27805*	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 27806*	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 27807*	The sum of selberg3r 27805 and selberg4r 27806. (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 27808*	Define the "Selberg function", whose asymptotic behavior is the content of selberg 27784. (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 27809*	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 27810*	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 27811*	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 27812*	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 27813*	The sum of selberg3r 27805 and selberg4r 27806. (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 27814*	Lemma for pntrlog2bnd 27820. 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 27815*	Lemma for pntrlog2bnd 27820. 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 27816*	Lemma for pntrlog2bnd 27820. 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 27817*	Lemma for pntrlog2bnd 27820. 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 27818*	Lemma for pntrlog2bndlem6 27819. (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 27819*	Lemma for pntrlog2bnd 27820. 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 27820*	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 27821*	Lemma for pntpbnd 27824. (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 27822*	Lemma for pntpbnd 27824. (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 27823*	Lemma for pntpbnd 27824. (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 27824*	Lemma for pnt 27850. 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 27825	Lemma for pntibnd 27829. (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 27826*	Lemma for pntibndlem2 27827. (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 27827*	Lemma for pntibnd 27829. 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 27828*	Lemma for pntibnd 27829. Package up pntibndlem2 27827 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 27829*	Lemma for pnt 27850. 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 27830	Lemma for pnt 27850. 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 27831*	Lemma for pnt 27850. 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 27832*	Lemma for pnt 27850. 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 27833*	Lemma for pnt 27850. 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 27834*	Lemma for pnt 27850. 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 27835*	Lemma for pnt 27850. 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 27836*	Lemma for pnt 27850. 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 27837*	Lemma for pntlemj 27839. (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 27838*	Lemma for pntlemj 27839. (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 27839*	Lemma for pnt 27850. The induction step. Using pntibnd 27829, 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 27840*	Lemma for pnt 27850. Eliminate some assumptions from pntlemj 27839. (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 27841*	Lemma for pnt 27850. Add up the pieces in pntlemi 27840 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 27842*	Lemma for pnt 27850. 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 27843*	Lemma for pnt 27850. 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 27844*	Lemma for pnt 27850. Package up pntlemo 27843 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 27845*	Lemma for pnt 27850. 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 27846*	Lemma for pnt 27850. 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 27847*	Lemma for pnt 27850. 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 27848	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 27849	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 27850	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 27851*	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 27852*	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 27853*	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 27854*	Lemma for ostth2 27873, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 27873. 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 27855	The base set of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ ℚ = (Base‘𝑄)
Theorem	qdrng 27856	The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ 𝑄 ∈ DivRing
Theorem	qrng0 27857	The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ 0 = (0g‘𝑄)
Theorem	qrng1 27858	The unity element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ 1 = (1r‘𝑄)
Theorem	qrngneg 27859	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 27860	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 27861	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 27862	Induct the product rule abvmul 20987 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 27863*	Lemma for ostth 27875. 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 27864*	Lemma for ostth 27875. Refine ostthlem1 27863 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 27865	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 27866*	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 27867*	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 27868*	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 27869*	- Lemma for ostth 27875: 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 20987 of the absolute value, 𝐹 is equal to 1 on all the integers, and ostthlem1 27863 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 27870*	Lemma for ostth2 27873. (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 27871*	Lemma for ostth2 27873. (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 27872*	Lemma for ostth2 27873. (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 27873*	- Lemma for ostth 27875: 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 27874*	- Lemma for ostth 27875: 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 27875*	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  SURREAL NUMBERS
15.1  Sign sequence representation and Alling's axioms The surreal numbers can be represented in several equivalent ways. In [Alling], Norman Alling made this notion explicit by giving a set of axioms that all representations admit, then proving that there is an order and birthday preserving bijection between any systems that satisfy these axioms. In this section, we start with the definition of surreal numbers given in [Gonshor] and derive Alling's axioms. After deriving them we no longer refer to the explicit definition of surreals. In particular, we never take advantage of the fact that the empty set is a surreal number under our definition.
15.1.1  Definitions and initial properties
Syntax	csur 27876	Declare the class of all surreal numbers (see df-no 27879).
class No
Syntax	clts 27877	Declare the less-than relation over surreal numbers (see df-lts 27880).
class <s
Syntax	cbday 27878	Declare the birthday function for surreal numbers (see df-bday 27881).
class bday
Definition	df-no 27879*	Define the class of surreal numbers. The surreal numbers are a proper class of numbers developed by John H. Conway and introduced by Donald Knuth in 1975. They form a proper class into which all ordered fields can be embedded. The approach we take to defining them was first introduced by Hary Gonshor, and is based on the conception of a "sign expansion" of a surreal number. We define the surreals as ordinal-indexed sequences of 1o and 2o, analogous to Gonshor's ( − ) and ( + ). After introducing this definition, we will abstract away from it using axioms that Norman Alling developed in "Foundations of Analysis over Surreal Number Fields." This is done in an effort to be agnostic towards the exact implementation of surreals. (Contributed by Scott Fenton, 9-Jun-2011.)
⊢ No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}
Definition	df-lts 27880*	Next, we introduce surreal less-than, a comparison relation over the surreals by lexicographically ordering them. (Contributed by Scott Fenton, 9-Jun-2011.)
⊢ <s = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ No ∧ 𝑔 ∈ No ) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)))}
Definition	df-bday 27881	Finally, we introduce the birthday function. This function maps each surreal to an ordinal. In our implementation, this is the domain of the sign function. The important properties of this function are established later. (Contributed by Scott Fenton, 11-Jun-2011.)
⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
Theorem	elno 27882*	Membership in the surreals. (Contributed by Scott Fenton, 11-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.) Avoid ax-rep 5232. (Revised by SN, 5-Jun-2025.)
⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
Theorem	ltsval 27883*	The value of the surreal less-than relation. (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))))
Theorem	bdayval 27884	The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)
Theorem	nofun 27885	A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (𝐴 ∈ No → Fun 𝐴)
Theorem	nodmon 27886	The domain of a surreal is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
Theorem	norn 27887	The range of a surreal is a subset of the surreal signs. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
Theorem	nofnbday 27888	A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (𝐴 ∈ No → 𝐴 Fn ( bday ‘𝐴))
Theorem	nodmord 27889	The domain of a surreal has the ordinal property. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (𝐴 ∈ No → Ord dom 𝐴)
Theorem	elno2 27890	An alternative condition for membership in No . (Contributed by Scott Fenton, 21-Mar-2012.)
⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
Theorem	elno3 27891	Another condition for membership in No . (Contributed by Scott Fenton, 14-Apr-2012.)
⊢ (𝐴 ∈ No ↔ (𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On))
Theorem	ltsval2 27892*	Alternate expression for surreal less-than. Two surreals obey surreal less-than iff they obey the sign ordering at the first place they differ. (Contributed by Scott Fenton, 17-Jun-2011.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})))
Theorem	nofv 27893	The function value of a surreal is either a sign or the empty set. (Contributed by Scott Fenton, 22-Jun-2011.)
⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))
Theorem	nosgnn0 27894	∅ is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ ¬ ∅ ∈ {1o, 2o}
Theorem	nosgnn0i 27895	If 𝑋 is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)
⊢ 𝑋 ∈ {1o, 2o}    ⇒   ⊢ ∅ ≠ 𝑋
Theorem	noreson 27896	The restriction of a surreal to an ordinal is still a surreal. (Contributed by Scott Fenton, 4-Sep-2011.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ↾ 𝐵) ∈ No )
Theorem	ltsintdifex 27897*	If 𝐴 <s 𝐵, then the intersection of all the ordinals that have differing signs in 𝐴 and 𝐵 exists. (Contributed by Scott Fenton, 22-Feb-2012.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ V))
Theorem	ltsres 27898	If the restrictions of two surreals to a given ordinal obey surreal less-than, then so do the two surreals themselves. (Contributed by Scott Fenton, 4-Sep-2011.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → 𝐴 <s 𝐵))
Theorem	noxp1o 27899	The Cartesian product of an ordinal and {1o} is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)
⊢ (𝐴 ∈ On → (𝐴 × {1o}) ∈ No )
Theorem	noseponlem 27900*	Lemma for nosepon 27901. Consider a case of proper subset domain. (Contributed by Scott Fenton, 21-Sep-2020.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))