P. 277 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27601-27700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pntrlog2bndlem4 27601*	Lemma for pntrlog2bnd 27605. 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 27602*	Lemma for pntrlog2bnd 27605. 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 27603*	Lemma for pntrlog2bndlem6 27604. (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 27604*	Lemma for pntrlog2bnd 27605. 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 27605*	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 27606*	Lemma for pntpbnd 27609. (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 27607*	Lemma for pntpbnd 27609. (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 27608*	Lemma for pntpbnd 27609. (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 27609*	Lemma for pnt 27635. 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 27610	Lemma for pntibnd 27614. (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 27611*	Lemma for pntibndlem2 27612. (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 27612*	Lemma for pntibnd 27614. 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 27613*	Lemma for pntibnd 27614. Package up pntibndlem2 27612 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 27614*	Lemma for pnt 27635. 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 27615	Lemma for pnt 27635. 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 27616*	Lemma for pnt 27635. 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 27617*	Lemma for pnt 27635. 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 27618*	Lemma for pnt 27635. 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 27619*	Lemma for pnt 27635. 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 27620*	Lemma for pnt 27635. 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 27621*	Lemma for pnt 27635. 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 27622*	Lemma for pntlemj 27624. (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 27623*	Lemma for pntlemj 27624. (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 27624*	Lemma for pnt 27635. The induction step. Using pntibnd 27614, 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 27625*	Lemma for pnt 27635. Eliminate some assumptions from pntlemj 27624. (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 27626*	Lemma for pnt 27635. Add up the pieces in pntlemi 27625 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 27627*	Lemma for pnt 27635. 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 27628*	Lemma for pnt 27635. 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 27629*	Lemma for pnt 27635. Package up pntlemo 27628 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 27630*	Lemma for pnt 27635. 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 27631*	Lemma for pnt 27635. 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 27632*	Lemma for pnt 27635. 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 27633	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 27634	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 27635	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 27636*	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 27637*	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 27638*	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 27639*	Lemma for ostth2 27658, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 27658. 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 27640	The base set of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ ℚ = (Base‘𝑄)
Theorem	qdrng 27641	The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ 𝑄 ∈ DivRing
Theorem	qrng0 27642	The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ 0 = (0g‘𝑄)
Theorem	qrng1 27643	The unity element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ 1 = (1r‘𝑄)
Theorem	qrngneg 27644	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 27645	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 27646	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 27647	Induct the product rule abvmul 20749 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 27648*	Lemma for ostth 27660. 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 27649*	Lemma for ostth 27660. Refine ostthlem1 27648 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 27650	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 27651*	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 27652*	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 27653*	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 27654*	- Lemma for ostth 27660: 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 20749 of the absolute value, 𝐹 is equal to 1 on all the integers, and ostthlem1 27648 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 27655*	Lemma for ostth2 27658. (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 27656*	Lemma for ostth2 27658. (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 27657*	Lemma for ostth2 27658. (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 27658*	- Lemma for ostth 27660: 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 27659*	- Lemma for ostth 27660: 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 27660*	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 27661	Declare the class of all surreal numbers (see df-no 27664).
class No
Syntax	cslt 27662	Declare the less-than relation over surreal numbers (see df-slt 27665).
class <s
Syntax	cbday 27663	Declare the birthday function for surreal numbers (see df-bday 27666).
class bday
Definition	df-no 27664*	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-slt 27665*	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 27666	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 27667*	Membership in the surreals. (Contributed by Scott Fenton, 11-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.) Avoid ax-rep 5289. (Revised by SN, 5-Jun-2025.)
⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
Theorem	elnoOLD 27668*	Obsolete version of elno 27667 as of 5-Jun-2025. (Contributed by Scott Fenton, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
Theorem	sltval 27669*	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 27670	The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)
Theorem	nofun 27671	A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (𝐴 ∈ No → Fun 𝐴)
Theorem	nodmon 27672	The domain of a surreal is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
Theorem	norn 27673	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 27674	A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (𝐴 ∈ No → 𝐴 Fn ( bday ‘𝐴))
Theorem	nodmord 27675	The domain of a surreal has the ordinal property. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (𝐴 ∈ No → Ord dom 𝐴)
Theorem	elno2 27676	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 27677	Another condition for membership in No . (Contributed by Scott Fenton, 14-Apr-2012.)
⊢ (𝐴 ∈ No ↔ (𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On))
Theorem	sltval2 27678*	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 27679	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 27680	∅ is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ ¬ ∅ ∈ {1o, 2o}
Theorem	nosgnn0i 27681	If 𝑋 is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)
⊢ 𝑋 ∈ {1o, 2o}    ⇒   ⊢ ∅ ≠ 𝑋
Theorem	noreson 27682	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	sltintdifex 27683*	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	sltres 27684	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 27685	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 27686*	Lemma for nosepon 27687. 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))
Theorem	nosepon 27687*	Given two unequal surreals, the minimal ordinal at which they differ is an ordinal. (Contributed by Scott Fenton, 21-Sep-2020.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On)
Theorem	noextend 27688	Extending a surreal by one sign value results in a new surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
⊢ 𝑋 ∈ {1o, 2o}    ⇒   ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No )
Theorem	noextendseq 27689	Extend a surreal by a sequence of ordinals. (Contributed by Scott Fenton, 30-Nov-2021.)
⊢ 𝑋 ∈ {1o, 2o}    ⇒   ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No )
Theorem	noextenddif 27690*	Calculate the place where a surreal and its extension differ. (Contributed by Scott Fenton, 22-Nov-2021.)
⊢ 𝑋 ∈ {1o, 2o}    ⇒   ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} = dom 𝐴)
Theorem	noextendlt 27691	Extending a surreal with a negative sign results in a smaller surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) <s 𝐴)
Theorem	noextendgt 27692	Extending a surreal with a positive sign results in a bigger surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
⊢ (𝐴 ∈ No → 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}))
Theorem	nolesgn2o 27693	Given 𝐴 less-than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2o, then 𝐵(𝑋) = 2o. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐵‘𝑋) = 2o)
Theorem	nolesgn2ores 27694	Given 𝐴 less-than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2o, then (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋). (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋))
Theorem	nogesgn1o 27695	Given 𝐴 greater than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 1o, then 𝐵(𝑋) = 1o. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐵‘𝑋) = 1o)
Theorem	nogesgn1ores 27696	Given 𝐴 greater than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 1o, then (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋). (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋))
15.1.2  Ordering
Theorem	sltsolem1 27697	Lemma for sltso 27698. The "sign expansion" binary relation totally orders the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.)
⊢ {⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or ({1o, 2o} ∪ {∅})
Theorem	sltso 27698	Less-than totally orders the surreals. Axiom O of [Alling] p. 184. (Contributed by Scott Fenton, 9-Jun-2011.)
⊢ <s Or No
15.1.3  Birthday Function
Theorem	bdayfo 27699	The birthday function maps the surreals onto the ordinals. Axiom B of [Alling] p. 184. (Proof shortened on 14-Apr-2012 by SF). (Contributed by Scott Fenton, 11-Jun-2011.)
⊢ bday : No –onto→On
15.1.4  Density
Theorem	fvnobday 27700	The value of a surreal at its birthday is ∅. (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.)
⊢ (𝐴 ∈ No → (𝐴‘( bday ‘𝐴)) = ∅)