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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremjensenlem2 24401* Lemma for jensen 24402. (Contributed by Mario Carneiro, 21-Jun-2015.)
(𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑 ∧ (𝑎𝐷𝑏𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑇:𝐴⟶(0[,)+∞))    &   (𝜑𝑋:𝐴𝐷)    &   (𝜑 → 0 < (ℂfld Σg 𝑇))    &   ((𝜑 ∧ (𝑥𝐷𝑦𝐷𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹𝑥)) + ((1 − 𝑡) · (𝐹𝑦))))    &   (𝜑 → ¬ 𝑧𝐵)    &   (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴)    &   𝑆 = (ℂfld Σg (𝑇𝐵))    &   𝐿 = (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧})))    &   (𝜑𝑆 ∈ ℝ+)    &   (𝜑 → ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷)    &   (𝜑 → (𝐹‘((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝐵)) / 𝑆)) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐵)) / 𝑆))       (𝜑 → (((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)))

Theoremjensen 24402* Jensen's inequality, a finite extension of the definition of convexity (the last hypothesis). (Contributed by Mario Carneiro, 21-Jun-2015.) (Proof shortened by AV, 27-Jul-2019.)
(𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑 ∧ (𝑎𝐷𝑏𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑇:𝐴⟶(0[,)+∞))    &   (𝜑𝑋:𝐴𝐷)    &   (𝜑 → 0 < (ℂfld Σg 𝑇))    &   ((𝜑 ∧ (𝑥𝐷𝑦𝐷𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹𝑥)) + ((1 − 𝑡) · (𝐹𝑦))))       (𝜑 → (((ℂfld Σg (𝑇𝑓 · 𝑋)) / (ℂfld Σg 𝑇)) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg (𝑇𝑓 · 𝑋)) / (ℂfld Σg 𝑇))) ≤ ((ℂfld Σg (𝑇𝑓 · (𝐹𝑋))) / (ℂfld Σg 𝑇))))

Theoremamgmlem 24403 Lemma for amgm 24404. (Contributed by Mario Carneiro, 21-Jun-2015.)
𝑀 = (mulGrp‘ℂfld)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐹:𝐴⟶ℝ+)       (𝜑 → ((𝑀 Σg 𝐹)↑𝑐(1 / (#‘𝐴))) ≤ ((ℂfld Σg 𝐹) / (#‘𝐴)))

Theoremamgm 24404 Inequality of arithmetic and geometric means. Here (𝑀 Σg 𝐹) calculates the group sum within the multiplicative monoid of the complex numbers (or in other words, it multiplies the elements 𝐹(𝑥), 𝑥𝐴 together), and (ℂfld Σg 𝐹) calculates the group sum in the additive group (i.e. the sum of the elements). This is Metamath 100 proof #38. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑀 = (mulGrp‘ℂfld)       ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(0[,)+∞)) → ((𝑀 Σg 𝐹)↑𝑐(1 / (#‘𝐴))) ≤ ((ℂfld Σg 𝐹) / (#‘𝐴)))

14.3.13  Euler-Mascheroni constant

Syntaxcem 24405 The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)
class γ

Definitiondf-em 24406 Define the Euler-Mascheroni constant, γ = 0.577... . This is the limit of the series Σ𝑘 ∈ (1...𝑚)(1 / 𝑘) − (log‘𝑚), with a proof that the limit exists in emcl 24416. (Contributed by Mario Carneiro, 11-Jul-2014.)
γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))

Theoremlogdifbnd 24407 Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)
(𝐴 ∈ ℝ+ → ((log‘(𝐴 + 1)) − (log‘𝐴)) ≤ (1 / 𝐴))

Theoremlogdiflbnd 24408 Lower bound on the difference of logs. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝐴 ∈ ℝ+ → (1 / (𝐴 + 1)) ≤ ((log‘(𝐴 + 1)) − (log‘𝐴)))

Theorememcllem1 24409* Lemma for emcl 24416. The series 𝐹 and 𝐺 are sequences of real numbers that approach γ from above and below, respectively. (Contributed by Mario Carneiro, 11-Jul-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)))    &   𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))))       (𝐹:ℕ⟶ℝ ∧ 𝐺:ℕ⟶ℝ)

Theorememcllem2 24410* Lemma for emcl 24416. 𝐹 is increasing, and 𝐺 is decreasing. (Contributed by Mario Carneiro, 11-Jul-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)))    &   𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))))       (𝑁 ∈ ℕ → ((𝐹‘(𝑁 + 1)) ≤ (𝐹𝑁) ∧ (𝐺𝑁) ≤ (𝐺‘(𝑁 + 1))))

Theorememcllem3 24411* Lemma for emcl 24416. The function 𝐻 is the difference between 𝐹 and 𝐺. (Contributed by Mario Carneiro, 11-Jul-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)))    &   𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))))    &   𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛))))       (𝑁 ∈ ℕ → (𝐻𝑁) = ((𝐹𝑁) − (𝐺𝑁)))

Theorememcllem4 24412* Lemma for emcl 24416. The difference between series 𝐹 and 𝐺 tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)))    &   𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))))    &   𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛))))       𝐻 ⇝ 0

Theorememcllem5 24413* Lemma for emcl 24416. The partial sums of the series 𝑇, which is used in the definition df-em 24406, is in fact the same as 𝐺. (Contributed by Mario Carneiro, 11-Jul-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)))    &   𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))))    &   𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛))))    &   𝑇 = (𝑛 ∈ ℕ ↦ ((1 / 𝑛) − (log‘(1 + (1 / 𝑛)))))       𝐺 = seq1( + , 𝑇)

Theorememcllem6 24414* Lemma for emcl 24416. By the previous lemmas, 𝐹 and 𝐺 must approach a common limit, which is γ by definition. (Contributed by Mario Carneiro, 11-Jul-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)))    &   𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))))    &   𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛))))    &   𝑇 = (𝑛 ∈ ℕ ↦ ((1 / 𝑛) − (log‘(1 + (1 / 𝑛)))))       (𝐹 ⇝ γ ∧ 𝐺 ⇝ γ)

Theorememcllem7 24415* Lemma for emcl 24416 and harmonicbnd 24417. Derive bounds on γ as 𝐹(1) and 𝐺(1). (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)
𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)))    &   𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))))    &   𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛))))    &   𝑇 = (𝑛 ∈ ℕ ↦ ((1 / 𝑛) − (log‘(1 + (1 / 𝑛)))))       (γ ∈ ((1 − (log‘2))[,]1) ∧ 𝐹:ℕ⟶(γ[,]1) ∧ 𝐺:ℕ⟶((1 − (log‘2))[,]γ))

Theorememcl 24416 Closure and bounds for the Euler-Mascheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)
γ ∈ ((1 − (log‘2))[,]1)

Theoremharmonicbnd 24417* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ (γ[,]1))

Theoremharmonicbnd2 24418* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
(𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ((1 − (log‘2))[,]γ))

Theorememre 24419 The Euler-Mascheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)
γ ∈ ℝ

Theorememgt0 24420 The Euler-Mascheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)
0 < γ

Theoremharmonicbnd3 24421* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
(𝑁 ∈ ℕ0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))

Theoremharmoniclbnd 24422* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
(𝐴 ∈ ℝ+ → (log‘𝐴) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))

Theoremharmonicubnd 24423* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ≤ ((log‘𝐴) + 1))

Theoremharmonicbnd4 24424* The asymptotic behavior of Σ𝑚𝐴, 1 / 𝑚 = log𝐴 + γ + 𝑂(1 / 𝐴). (Contributed by Mario Carneiro, 14-May-2016.)
(𝐴 ∈ ℝ+ → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴))

Theoremfsumharmonic 24425* Bound a finite sum based on the harmonic series, where the "strong" bound 𝐶 only applies asymptotically, and there is a "weak" bound 𝑅 for the remaining values. (Contributed by Mario Carneiro, 18-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))    &   (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))    &   ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)    &   ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐶)    &   (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛))    &   (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝐴 / 𝑛) < 𝑇) → (abs‘𝐵) ≤ 𝑅)       (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))

14.3.14  Zeta function

Syntaxczeta 24426 The Riemann zeta function.
class ζ

Definitiondf-zeta 24427* Define the Riemann zeta function. This definition uses a series expansion of the alternating zeta function ~? zetaalt that is convergent everywhere except 1, but going from the alternating zeta function to the regular zeta function requires dividing by 1 − 2↑(1 − 𝑠), which has zeroes other than 1. To extract the correct value of the zeta function at these points, we extend the divided alternating zeta function by continuity. (Contributed by Mario Carneiro, 18-Jul-2014.)
ζ = (𝑓 ∈ ((ℂ ∖ {1})–cn→ℂ)∀𝑠 ∈ (ℂ ∖ {1})((1 − (2↑𝑐(1 − 𝑠))) · (𝑓𝑠)) = Σ𝑛 ∈ ℕ0𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1))))

Theoremzetacvg 24428* The zeta series is convergent. (Contributed by Mario Carneiro, 18-Jul-2014.)
(𝜑𝑆 ∈ ℂ)    &   (𝜑 → 1 < (ℜ‘𝑆))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = (𝑘𝑐-𝑆))       (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )

14.3.15  Gamma function

Syntaxclgam 24429 Logarithm of the Gamma function.
class log Γ

Syntaxcgam 24430 The Gamma function.
class Γ

Syntaxcigam 24431 The inverse Gamma function.
class 1/Γ

Definitiondf-lgam 24432* Define the log-Gamma function. We can work with this form of the gamma function a bit easier than the equivalent expression for the gamma function itself, and moreover this function is not actually equal to log(Γ(𝑥)) because the branch cuts are placed differently (we do have exp(log Γ(𝑥)) = Γ(𝑥), though). This definition is attributed to Euler, and unlike the usual integral definition is defined on the entire complex plane except the nonpositive integers ℤ ∖ ℕ, where the function has simple poles. (Contributed by Mario Carneiro, 12-Jul-2014.)
log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))

Definitiondf-gam 24433 Define the Gamma function. See df-lgam 24432 for more information about the reason for this definition in terms of the log-gamma function. (Contributed by Mario Carneiro, 12-Jul-2014.)
Γ = (exp ∘ log Γ)

Definitiondf-igam 24434 Define the inverse Gamma function, which is defined everywhere, unlike the Gamma function itself. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))

Theoremeldmgm 24435 Elementhood in the set of non-nonpositive integers. (Contributed by Mario Carneiro, 12-Jul-2014.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0))

Theoremdmgmaddn0 24436 If 𝐴 is not a nonpositive integer, then 𝐴 + 𝑁 is nonzero for any nonnegative integer 𝑁. (Contributed by Mario Carneiro, 12-Jul-2014.)
((𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ∧ 𝑁 ∈ ℕ0) → (𝐴 + 𝑁) ≠ 0)

Theoremdmlogdmgm 24437 If 𝐴 is in the continuous domain of the logarithm, then it is in the domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (-∞(,]0)) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))

Theoremrpdmgm 24438 A positive real number is in the domain of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ ℝ+𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))

Theoremdmgmn0 24439 If 𝐴 is not a nonpositive integer, then 𝐴 is nonzero. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑𝐴 ≠ 0)

Theoremdmgmaddnn0 24440 If 𝐴 is not a nonpositive integer and 𝑁 is a nonnegative integer, then 𝐴 + 𝑁 is also not a nonpositive integer. (Contributed by Mario Carneiro, 6-Jul-2017.)
(𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴 + 𝑁) ∈ (ℂ ∖ (ℤ ∖ ℕ)))

Theoremdmgmdivn0 24441 Lemma for lgamf 24455. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   (𝜑𝑀 ∈ ℕ)       (𝜑 → ((𝐴 / 𝑀) + 1) ≠ 0)

Theoremlgamgulmlem1 24442* Lemma for lgamgulm 24448. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}       (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))

Theoremlgamgulmlem2 24443* Lemma for lgamgulm 24448. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝑈)    &   (𝜑 → (2 · 𝑅) ≤ 𝑁)       (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((1 / (𝑁𝑅)) − (1 / 𝑁))))

Theoremlgamgulmlem3 24444* Lemma for lgamgulm 24448. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝑈)    &   (𝜑 → (2 · 𝑅) ≤ 𝑁)       (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))

Theoremlgamgulmlem4 24445* Lemma for lgamgulm 24448. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))    &   𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))       (𝜑 → seq1( + , 𝑇) ∈ dom ⇝ )

Theoremlgamgulmlem5 24446* Lemma for lgamgulm 24448. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))    &   𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))       ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) ≤ (𝑇𝑛))

Theoremlgamgulmlem6 24447* The series 𝐺 is uniformly convergent on the compact region 𝑈, which describes a circle of radius 𝑅 with holes of size 1 / 𝑅 around the poles of the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))    &   𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))       (𝜑 → (seq1( ∘𝑓 + , 𝐺) ∈ dom (⇝𝑢𝑈) ∧ (seq1( ∘𝑓 + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈𝑂) → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘𝑂) ≤ 𝑟)))

Theoremlgamgulm 24448* The series 𝐺 is uniformly convergent on the compact region 𝑈, which describes a circle of radius 𝑅 with holes of size 1 / 𝑅 around the poles of the gamma function. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))       (𝜑 → seq1( ∘𝑓 + , 𝐺) ∈ dom (⇝𝑢𝑈))

Theoremlgamgulm2 24449* Rewrite the limit of the sequence 𝐺 in terms of the log-Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))       (𝜑 → (∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘𝑓 + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))

Theoremlgambdd 24450* The log-Gamma function is bounded on the region 𝑈. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))       (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)

Theoremlgamucov 24451* The 𝑈 regions used in the proof of lgamgulm 24448 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   𝐽 = (TopOpen‘ℂfld)       (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈))

Theoremlgamucov2 24452* The 𝑈 regions used in the proof of lgamgulm 24448 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → ∃𝑟 ∈ ℕ 𝐴𝑈)

Theoremlgamcvglem 24453* Lemma for lgamf 24455 and lgamcvg 24467. (Contributed by Mario Carneiro, 8-Jul-2017.)
𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))       (𝜑 → ((log Γ‘𝐴) ∈ ℂ ∧ seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴))))

Theoremlgamcl 24454 The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 8-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘𝐴) ∈ ℂ)

Theoremlgamf 24455 The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)
log Γ:(ℂ ∖ (ℤ ∖ ℕ))⟶ℂ

Theoremgamf 24456 The Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)
Γ:(ℂ ∖ (ℤ ∖ ℕ))⟶ℂ

Theoremgamcl 24457 The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℂ)

Theoremeflgam 24458 The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (exp‘(log Γ‘𝐴)) = (Γ‘𝐴))

Theoremgamne0 24459 The Gamma function is never zero. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ≠ 0)

Theoremigamval 24460 Value of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ ℂ → (1/Γ‘𝐴) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))))

Theoremigamz 24461 Value of the inverse Gamma function on nonpositive integers. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ (ℤ ∖ ℕ) → (1/Γ‘𝐴) = 0)

Theoremigamgam 24462 Value of the inverse Gamma function in terms of the Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (1/Γ‘𝐴) = (1 / (Γ‘𝐴)))

Theoremigamlgam 24463 Value of the inverse Gamma function in terms of the log-Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (1/Γ‘𝐴) = (exp‘-(log Γ‘𝐴)))

Theoremigamf 24464 Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/Γ:ℂ⟶ℂ

Theoremigamcl 24465 Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ ℂ → (1/Γ‘𝐴) ∈ ℂ)

Theoremgamigam 24466 The Gamma function is the inverse of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) = (1 / (1/Γ‘𝐴)))

Theoremlgamcvg 24467* The series 𝐺 converges to log Γ(𝐴) + log(𝐴). (Contributed by Mario Carneiro, 6-Jul-2017.)
𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴)))

Theoremlgamcvg2 24468* The series 𝐺 converges to log Γ(𝐴 + 1). (Contributed by Mario Carneiro, 9-Jul-2017.)
𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → seq1( + , 𝐺) ⇝ (log Γ‘(𝐴 + 1)))

Theoremgamcvg 24469* The pointwise exponential of the series 𝐺 converges to Γ(𝐴) · 𝐴. (Contributed by Mario Carneiro, 6-Jul-2017.)
𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ ((Γ‘𝐴) · 𝐴))

Theoremlgamp1 24470 The functional equation of the (log) Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘(𝐴 + 1)) = ((log Γ‘𝐴) + (log‘𝐴)))

Theoremgamp1 24471 The functional equation of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘(𝐴 + 1)) = ((Γ‘𝐴) · 𝐴))

Theoremgamcvg2lem 24472* Lemma for gamcvg2 24473. (Contributed by Mario Carneiro, 10-Jul-2017.)
𝐹 = (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1)))    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))       (𝜑 → (exp ∘ seq1( + , 𝐺)) = seq1( · , 𝐹))

Theoremgamcvg2 24473* An infinite product expression for the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
𝐹 = (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1)))    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → seq1( · , 𝐹) ⇝ ((Γ‘𝐴) · 𝐴))

Theoremregamcl 24474 The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℝ)

Theoremrelgamcl 24475 The log-Gamma function is real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ ℝ+ → (log Γ‘𝐴) ∈ ℝ)

Theoremrpgamcl 24476 The log-Gamma function is positive real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ ℝ+ → (Γ‘𝐴) ∈ ℝ+)

Theoremlgam1 24477 The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
(log Γ‘1) = 0

Theoremgam1 24478 The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
(Γ‘1) = 1

Theoremfacgam 24479 The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝑁 ∈ ℕ0 → (!‘𝑁) = (Γ‘(𝑁 + 1)))

Theoremgamfac 24480 The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝑁 ∈ ℕ → (Γ‘𝑁) = (!‘(𝑁 − 1)))

14.4  Basic number theory

14.4.1  Wilson's theorem

Theoremwilthlem1 24481 The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in ℤ / 𝑃 are 1 and -1≡𝑃 − 1. (Note that from prmdiveq 15207, (𝑁↑(𝑃 − 2)) mod 𝑃 is the modular inverse of 𝑁 in ℤ / 𝑃. (Contributed by Mario Carneiro, 24-Jan-2015.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 = ((𝑁↑(𝑃 − 2)) mod 𝑃) ↔ (𝑁 = 1 ∨ 𝑁 = (𝑃 − 1))))

Theoremwilthlem2 24482* Lemma for wilth 24484: induction step. The "hand proof" version of this theorem works by writing out the list of all numbers from 1 to 𝑃 − 1 in pairs such that a number is paired with its inverse. Every number has a unique inverse different from itself except 1 and 𝑃 − 1, and so each pair multiplies to 1, and 1 and 𝑃 − 1≡-1 multiply to -1, so the full product is equal to -1. Here we make this precise by doing the product pair by pair.

The induction hypothesis says that every subset 𝑆 of 1...(𝑃 − 1) that is closed under inverse (i.e. all pairs are matched up) and contains 𝑃 − 1 multiplies to -1 mod 𝑃. Given such a set, we take out one element 𝑧𝑃 − 1. If there are no such elements, then 𝑆 = {𝑃 − 1} which forms the base case. Otherwise, 𝑆 ∖ {𝑧, 𝑧↑-1} is also closed under inverse and contains 𝑃 − 1, so the induction hypothesis says that this equals -1; and the remaining two elements are either equal to each other, in which case wilthlem1 24481 gives that 𝑧 = 1 or 𝑃 − 1, and we've already excluded the second case, so the product gives 1; or 𝑧𝑧↑-1 and their product is 1. In either case the accumulated product is unaffected. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by AV, 27-Jul-2019.)

𝑇 = (mulGrp‘ℂfld)    &   𝐴 = {𝑥 ∈ 𝒫 (1...(𝑃 − 1)) ∣ ((𝑃 − 1) ∈ 𝑥 ∧ ∀𝑦𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥)}    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑆𝐴)    &   (𝜑 → ∀𝑠𝐴 (𝑠𝑆 → ((𝑇 Σg ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))       (𝜑 → ((𝑇 Σg ( I ↾ 𝑆)) mod 𝑃) = (-1 mod 𝑃))

Theoremwilthlem3 24483* Lemma for wilth 24484. Here we round out the argument of wilthlem2 24482 with the final step of the induction. The induction argument shows that every subset of 1...(𝑃 − 1) that is closed under inverse and contains 𝑃 − 1 multiplies to -1 mod 𝑃, and clearly 1...(𝑃 − 1) itself is such a set. Thus, the product of all the elements is -1, and all that is left is to translate the group sum notation (which we used for its unordered summing capabilities) into an ordered sequence to match the definition of the factorial. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by AV, 27-Jul-2019.)
𝑇 = (mulGrp‘ℂfld)    &   𝐴 = {𝑥 ∈ 𝒫 (1...(𝑃 − 1)) ∣ ((𝑃 − 1) ∈ 𝑥 ∧ ∀𝑦𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥)}       (𝑃 ∈ ℙ → 𝑃 ∥ ((!‘(𝑃 − 1)) + 1))

Theoremwilth 24484 Wilson's theorem. A number is prime iff it is greater or equal to 2 and (𝑁 − 1)! is congruent to -1, mod 𝑁, or alternatively if 𝑁 divides (𝑁 − 1)! + 1. In this part of the proof we show the relatively simple reverse implication; see wilthlem3 24483 for the forward implication. This is Metamath 100 proof #51. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
(𝑁 ∈ ℙ ↔ (𝑁 ∈ (ℤ‘2) ∧ 𝑁 ∥ ((!‘(𝑁 − 1)) + 1)))

Theoremwilthimp 24485 The forward implication of Wilson's theorem wilth 24484 (see wilthlem3 24483), expressed using the modulo operation: For any prime 𝑝 we have (𝑝 − 1)!≡ − 1 (mod 𝑝), see theorem 5.24 in [ApostolNT] p. 116. (Contributed by AV, 21-Jul-2021.)
(𝑃 ∈ ℙ → ((!‘(𝑃 − 1)) mod 𝑃) = (-1 mod 𝑃))

14.4.2  The Fundamental Theorem of Algebra

Theoremftalem1 24486* Lemma for fta 24495: "growth lemma". There exists some 𝑟 such that 𝐹 is arbitrarily close in proportion to its dominant term. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐸 ∈ ℝ+)    &   𝑇 = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸)       (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))

Theoremftalem2 24487* Lemma for fta 24495. There exists some 𝑟 such that 𝐹 has magnitude greater than 𝐹(0) outside the closed ball B(0,r). (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   𝑈 = if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1))    &   𝑇 = ((abs‘(𝐹‘0)) / ((abs‘(𝐴𝑁)) / 2))       (𝜑 → ∃𝑟 ∈ ℝ+𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹𝑥))))

Theoremftalem3 24488* Lemma for fta 24495. There exists a global minimum of the function abs ∘ 𝐹. The proof uses a circle of radius 𝑟 where 𝑟 is the value coming from ftalem1 24486; since this is a compact set, the minimum on this disk is achieved, and this must then be the global minimum. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅}    &   𝐽 = (TopOpen‘ℂfld)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹𝑥))))       (𝜑 → ∃𝑧 ∈ ℂ ∀𝑥 ∈ ℂ (abs‘(𝐹𝑧)) ≤ (abs‘(𝐹𝑥)))

Theoremftalem4 24489* Lemma for fta 24495: Closure of the auxiliary variables for ftalem5 24490. (Contributed by Mario Carneiro, 20-Sep-2014.) (Revised by AV, 28-Sep-2020.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐹‘0) ≠ 0)    &   𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴𝑛) ≠ 0}, ℝ, < )    &   𝑇 = (-((𝐹‘0) / (𝐴𝐾))↑𝑐(1 / 𝐾))    &   𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴𝑘) · (𝑇𝑘))) + 1))    &   𝑋 = if(1 ≤ 𝑈, 1, 𝑈)       (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+𝑋 ∈ ℝ+)))

Theoremftalem5 24490* Lemma for fta 24495: Main proof. We have already shifted the minimum found in ftalem3 24488 to zero by a change of variables, and now we show that the minimum value is zero. Expanding in a series about the minimum value, let 𝐾 be the lowest term in the polynomial that is nonzero, and let 𝑇 be a 𝐾-th root of -𝐹(0) / 𝐴(𝐾). Then an evaluation of 𝐹(𝑇𝑋) where 𝑋 is a sufficiently small positive number yields 𝐹(0) for the first term and -𝐹(0) · 𝑋𝐾 for the 𝐾-th term, and all higher terms are bounded because 𝑋 is small. Thus, abs(𝐹(𝑇𝑋)) ≤ abs(𝐹(0))(1 − 𝑋𝐾) < abs(𝐹(0)), in contradiction to our choice of 𝐹(0) as the minimum. (Contributed by Mario Carneiro, 14-Sep-2014.) (Revised by AV, 28-Sep-2020.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐹‘0) ≠ 0)    &   𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴𝑛) ≠ 0}, ℝ, < )    &   𝑇 = (-((𝐹‘0) / (𝐴𝐾))↑𝑐(1 / 𝐾))    &   𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴𝑘) · (𝑇𝑘))) + 1))    &   𝑋 = if(1 ≤ 𝑈, 1, 𝑈)       (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹𝑥)) < (abs‘(𝐹‘0)))

Theoremftalem4OLD 24491* Lemma for fta 24495: Closure of the auxiliary variables for ftalem5 24490. (Contributed by Mario Carneiro, 20-Sep-2014.) Obsolete version of ftalem4 24489 as of 1-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐹‘0) ≠ 0)    &   𝐾 = sup({𝑛 ∈ ℕ ∣ (𝐴𝑛) ≠ 0}, ℝ, < )    &   𝑇 = (-((𝐹‘0) / (𝐴𝐾))↑𝑐(1 / 𝐾))    &   𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴𝑘) · (𝑇𝑘))) + 1))    &   𝑋 = if(1 ≤ 𝑈, 1, 𝑈)       (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+𝑋 ∈ ℝ+)))

Theoremftalem5OLD 24492* Lemma for fta 24495: Main proof. We have already shifted the minimum found in ftalem3 24488 to zero by a change of variables, and now we show that the minimum value is zero. Expanding in a series about the minimum value, let 𝐾 be the lowest term in the polynomial that is nonzero, and let 𝑇 be a 𝐾-th root of -𝐹(0) / 𝐴(𝐾). Then an evaluation of 𝐹(𝑇𝑋) where 𝑋 is a sufficiently small positive number yields 𝐹(0) for the first term and -𝐹(0) · 𝑋𝐾 for the 𝐾-th term, and all higher terms are bounded because 𝑋 is small. Thus, abs(𝐹(𝑇𝑋)) ≤ abs(𝐹(0))(1 − 𝑋𝐾) < abs(𝐹(0)), in contradiction to our choice of 𝐹(0) as the minimum. (Contributed by Mario Carneiro, 14-Sep-2014.) Obsolete version of ftalem5 24490 as of 1-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐹‘0) ≠ 0)    &   𝐾 = sup({𝑛 ∈ ℕ ∣ (𝐴𝑛) ≠ 0}, ℝ, < )    &   𝑇 = (-((𝐹‘0) / (𝐴𝐾))↑𝑐(1 / 𝐾))    &   𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴𝑘) · (𝑇𝑘))) + 1))    &   𝑋 = if(1 ≤ 𝑈, 1, 𝑈)       (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹𝑥)) < (abs‘(𝐹‘0)))

Theoremftalem6 24493* Lemma for fta 24495: Discharge the auxiliary variables in ftalem5 24490. (Contributed by Mario Carneiro, 20-Sep-2014.) (Proof shortened by AV, 28-Sep-2020.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐹‘0) ≠ 0)       (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹𝑥)) < (abs‘(𝐹‘0)))

Theoremftalem7 24494* Lemma for fta 24495. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑 → (𝐹𝑋) ≠ 0)       (𝜑 → ¬ ∀𝑥 ∈ ℂ (abs‘(𝐹𝑋)) ≤ (abs‘(𝐹𝑥)))

Theoremfta 24495* The Fundamental Theorem of Algebra. Any polynomial with positive degree (i.e. non-constant) has a root. This is Metamath 100 proof #2. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) ∈ ℕ) → ∃𝑧 ∈ ℂ (𝐹𝑧) = 0)

14.4.3  The Basel problem (ζ(2) = π2/6)

Theorembasellem1 24496 Lemma for basel 24505. Closure of the sequence of roots. (Contributed by Mario Carneiro, 30-Jul-2014.) Replace OLD theorem. (Revised ba Wolf Lammen, 18-Sep-2020.)
𝑁 = ((2 · 𝑀) + 1)       ((𝑀 ∈ ℕ ∧ 𝐾 ∈ (1...𝑀)) → ((𝐾 · π) / 𝑁) ∈ (0(,)(π / 2)))

Theorembasellem2 24497* Lemma for basel 24505. Show that 𝑃 is a polynomial of degree 𝑀, and compute its coefficient function. (Contributed by Mario Carneiro, 30-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))       (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀𝑛))))))

Theorembasellem3 24498* Lemma for basel 24505. Using the binomial theorem and de Moivre's formula, we have the identity e↑i𝑁𝑥 / (sin𝑥)↑𝑛 = Σ𝑚 ∈ (0...𝑁)(𝑁C𝑚)(i↑𝑚)(cot𝑥)↑(𝑁𝑚), so taking imaginary parts yields sin(𝑁𝑥) / (sin𝑥)↑𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑁C2𝑗)(-1)↑(𝑀𝑗) (cot𝑥)↑(-2𝑗) = 𝑃((cot𝑥)↑2), where 𝑁 = 2𝑀 + 1. (Contributed by Mario Carneiro, 30-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))       ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑃‘((tan‘𝐴)↑-2)) = ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)))

Theorembasellem4 24499* Lemma for basel 24505. By basellem3 24498, the expression 𝑃((cot𝑥)↑2) = sin(𝑁𝑥) / (sin𝑥)↑𝑁 goes to zero whenever 𝑥 = 𝑛π / 𝑁 for some 𝑛 ∈ (1...𝑀), so this function enumerates 𝑀 distinct roots of a degree- 𝑀 polynomial, which must therefore be all the roots by fta1 23751. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))    &   𝑇 = (𝑛 ∈ (1...𝑀) ↦ ((tan‘((𝑛 · π) / 𝑁))↑-2))       (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1-onto→(𝑃 “ {0}))

Theorembasellem5 24500* Lemma for basel 24505. Using vieta1 23755, we can calculate the sum of the roots of 𝑃 as the quotient of the top two coefficients, and since the function 𝑇 enumerates the roots, we are left with an equation that sums the cot↑2 function at the 𝑀 different roots. (Contributed by Mario Carneiro, 29-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))    &   𝑇 = (𝑛 ∈ (1...𝑀) ↦ ((tan‘((𝑛 · π) / 𝑁))↑-2))       (𝑀 ∈ ℕ → Σ𝑘 ∈ (1...𝑀)((tan‘((𝑘 · π) / 𝑁))↑-2) = (((2 · 𝑀) · ((2 · 𝑀) − 1)) / 6))

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