P. 270 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26901-27000   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-em 26901	Define the Euler-Mascheroni constant, γ = 0.57721.... This is the limit of the series Σ𝑘 ∈ (1...𝑚)(1 / 𝑘) − (log‘𝑚), with a proof that the limit exists in emcl 26911. (Contributed by Mario Carneiro, 11-Jul-2014.)
⊢ γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))
Theorem	logdifbnd 26902	Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)
⊢ (𝐴 ∈ ℝ+ → ((log‘(𝐴 + 1)) − (log‘𝐴)) ≤ (1 / 𝐴))
Theorem	logdiflbnd 26903	Lower bound on the difference of logs. (Contributed by Mario Carneiro, 3-Jul-2017.)
⊢ (𝐴 ∈ ℝ+ → (1 / (𝐴 + 1)) ≤ ((log‘(𝐴 + 1)) − (log‘𝐴)))
Theorem	emcllem1 26904*	Lemma for emcl 26911. 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)
Theorem	emcllem2 26905*	Lemma for emcl 26911. 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)) ≤ (𝐹‘𝑁) ∧ (𝐺‘𝑁) ≤ (𝐺‘(𝑁 + 1))))
Theorem	emcllem3 26906*	Lemma for emcl 26911. 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 → (𝐻‘𝑁) = ((𝐹‘𝑁) − (𝐺‘𝑁)))
Theorem	emcllem4 26907*	Lemma for emcl 26911. 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
Theorem	emcllem5 26908*	Lemma for emcl 26911. The partial sums of the series 𝑇, which is used in Definition df-em 26901, 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)
Theorem	emcllem6 26909*	Lemma for emcl 26911. 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)
Theorem	emcllem7 26910*	Lemma for emcl 26911 and harmonicbnd 26912. 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))
Theorem	emcl 26911	Closure and bounds for the Euler-Mascheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)
⊢ γ ∈ ((1 − (log‘2))[,]1)
Theorem	harmonicbnd 26912*	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))
Theorem	harmonicbnd2 26913*	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))
Theorem	emre 26914	The Euler-Mascheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)
⊢ γ ∈ ℝ
Theorem	emgt0 26915	The Euler-Mascheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)
⊢ 0 < γ
Theorem	harmonicbnd3 26916*	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))
Theorem	harmoniclbnd 26917*	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))
Theorem	harmonicubnd 26918*	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))
Theorem	harmonicbnd4 26919*	The asymptotic behavior of Σ𝑚 ≤ 𝐴, 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))
Theorem	fsumharmonic 26920*	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.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))    &   ⊢ (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐶)    &   ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛))    &   ⊢ (((𝜑 ∧ 𝑛 ∈ (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
Syntax	czeta 26921	The Riemann zeta function.
class ζ
Definition	df-zeta 26922*	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))) · (𝑓‘𝑠)) = Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1))))
Theorem	zetacvg 26923*	The zeta series is convergent. (Contributed by Mario Carneiro, 18-Jul-2014.)
⊢ (𝜑 → 𝑆 ∈ ℂ)    &   ⊢ (𝜑 → 1 < (ℜ‘𝑆))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝑘↑𝑐-𝑆))    ⇒   ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )
14.3.15  Gamma function
Syntax	clgam 26924	Logarithm of the Gamma function.
class log Γ
Syntax	cgam 26925	The Gamma function.
class Γ
Syntax	cigam 26926	The inverse Gamma function.
class 1/Γ
Definition	df-lgam 26927*	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)))
Definition	df-gam 26928	Define the Gamma function. See df-lgam 26927 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)
Definition	df-igam 26929	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))))
Theorem	eldmgm 26930	Elementhood in the set of non-nonpositive integers. (Contributed by Mario Carneiro, 12-Jul-2014.)
⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0))
Theorem	dmgmaddn0 26931	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)
Theorem	dmlogdmgm 26932	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)))
Theorem	rpdmgm 26933	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)))
Theorem	dmgmn0 26934	If 𝐴 is not a nonpositive integer, then 𝐴 is nonzero. (Contributed by Mario Carneiro, 3-Jul-2017.)
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	dmgmaddnn0 26935	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.)
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴 + 𝑁) ∈ (ℂ ∖ (ℤ ∖ ℕ)))
Theorem	dmgmdivn0 26936	Lemma for lgamf 26950. (Contributed by Mario Carneiro, 3-Jul-2017.)
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((𝐴 / 𝑀) + 1) ≠ 0)
Theorem	lgamgulmlem1 26937*	Lemma for lgamgulm 26943. (Contributed by Mario Carneiro, 3-Jul-2017.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    ⇒   ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
Theorem	lgamgulmlem2 26938*	Lemma for lgamgulm 26943. (Contributed by Mario Carneiro, 3-Jul-2017.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → (2 · 𝑅) ≤ 𝑁)    ⇒   ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))
Theorem	lgamgulmlem3 26939*	Lemma for lgamgulm 26943. (Contributed by Mario Carneiro, 3-Jul-2017.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → (2 · 𝑅) ≤ 𝑁)    ⇒   ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))
Theorem	lgamgulmlem4 26940*	Lemma for lgamgulm 26943. (Contributed by Mario Carneiro, 3-Jul-2017.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((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 ⇝ )
Theorem	lgamgulmlem5 26941*	Lemma for lgamgulm 26943. (Contributed by Mario Carneiro, 3-Jul-2017.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((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 ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝐺‘𝑛)‘𝑦)) ≤ (𝑇‘𝑛))
Theorem	lgamgulmlem6 26942*	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.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((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 (⇝𝑢‘𝑈) ∧ (seq1( ∘f + , 𝐺)(⇝𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ 𝑂) → ∃𝑟 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘𝑂) ≤ 𝑟)))
Theorem	lgamgulm 26943*	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.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))    ⇒   ⊢ (𝜑 → seq1( ∘f + , 𝐺) ∈ dom (⇝𝑢‘𝑈))
Theorem	lgamgulm2 26944*	Rewrite the limit of the sequence 𝐺 in terms of the log-Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))    ⇒   ⊢ (𝜑 → (∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘f + , 𝐺)(⇝𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))
Theorem	lgambdd 26945*	The log-Gamma function is bounded on the region 𝑈. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))    ⇒   ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
Theorem	lgamucov 26946*	The 𝑈 regions used in the proof of lgamgulm 26943 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)))}    &   ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈))
Theorem	lgamucov2 26947*	The 𝑈 regions used in the proof of lgamgulm 26943 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)))}    &   ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    ⇒   ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ 𝑈)
Theorem	lgamcvglem 26948*	Lemma for lgamf 26950 and lgamcvg 26962. (Contributed by Mario Carneiro, 8-Jul-2017.)
⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}    &   ⊢ (𝜑 → 𝐴 ∈ (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))))
Theorem	lgamcl 26949	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)
Theorem	lgamf 26950	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
Theorem	gamf 26951	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
Theorem	gamcl 26952	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)
Theorem	eflgam 26953	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))
Theorem	gamne0 26954	The Gamma function is never zero. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ≠ 0)
Theorem	igamval 26955	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))))
Theorem	igamz 26956	Value of the inverse Gamma function on nonpositive integers. (Contributed by Mario Carneiro, 16-Jul-2017.)
⊢ (𝐴 ∈ (ℤ ∖ ℕ) → (1/Γ‘𝐴) = 0)
Theorem	igamgam 26957	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)))
Theorem	igamlgam 26958	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)))
Theorem	igamf 26959	Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
⊢ 1/Γ:ℂ⟶ℂ
Theorem	igamcl 26960	Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
⊢ (𝐴 ∈ ℂ → (1/Γ‘𝐴) ∈ ℂ)
Theorem	gamigam 26961	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)))
Theorem	lgamcvg 26962*	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))))    &   ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    ⇒   ⊢ (𝜑 → seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴)))
Theorem	lgamcvg2 26963*	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))))    &   ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    ⇒   ⊢ (𝜑 → seq1( + , 𝐺) ⇝ (log Γ‘(𝐴 + 1)))
Theorem	gamcvg 26964*	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))))    &   ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    ⇒   ⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ ((Γ‘𝐴) · 𝐴))
Theorem	lgamp1 26965	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)))
Theorem	gamp1 26966	The functional equation of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘(𝐴 + 1)) = ((Γ‘𝐴) · 𝐴))
Theorem	gamcvg2lem 26967*	Lemma for gamcvg2 26968. (Contributed by Mario Carneiro, 10-Jul-2017.)
⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1)))    &   ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))    ⇒   ⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) = seq1( · , 𝐹))
Theorem	gamcvg2 26968*	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)))    &   ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    ⇒   ⊢ (𝜑 → seq1( · , 𝐹) ⇝ ((Γ‘𝐴) · 𝐴))
Theorem	regamcl 26969	The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℝ)
Theorem	relgamcl 26970	The log-Gamma function is real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (𝐴 ∈ ℝ+ → (log Γ‘𝐴) ∈ ℝ)
Theorem	rpgamcl 26971	The log-Gamma function is positive real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (𝐴 ∈ ℝ+ → (Γ‘𝐴) ∈ ℝ+)
Theorem	lgam1 26972	The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (log Γ‘1) = 0
Theorem	gam1 26973	The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (Γ‘1) = 1
Theorem	facgam 26974	The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) = (Γ‘(𝑁 + 1)))
Theorem	gamfac 26975	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
Theorem	wilthlem1 26976	The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in ℤ / 𝑃ℤ are 1 and -1≡𝑃 − 1. (Note that from prmdiveq 16697, (𝑁↑(𝑃 − 2)) mod 𝑃 is the modular inverse of 𝑁 in ℤ / 𝑃ℤ. (Contributed by Mario Carneiro, 24-Jan-2015.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 = ((𝑁↑(𝑃 − 2)) mod 𝑃) ↔ (𝑁 = 1 ∨ 𝑁 = (𝑃 − 1))))
Theorem	wilthlem2 26977*	Lemma for wilth 26979: 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 𝑧 ≠ 𝑃 − 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 26976 gives that 𝑧 = 1 or 𝑃 − 1, and we've already excluded the second case, so the product gives 1; or 𝑧 ≠ 𝑧↑-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 ∧ ∀𝑦 ∈ 𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥)}    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐴)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝐴 (𝑠 ⊊ 𝑆 → ((𝑇 Σg ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))    ⇒   ⊢ (𝜑 → ((𝑇 Σg ( I ↾ 𝑆)) mod 𝑃) = (-1 mod 𝑃))
Theorem	wilthlem3 26978*	Lemma for wilth 26979. Here we round out the argument of wilthlem2 26977 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 ∧ ∀𝑦 ∈ 𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥)}    ⇒   ⊢ (𝑃 ∈ ℙ → 𝑃 ∥ ((!‘(𝑃 − 1)) + 1))
Theorem	wilth 26979	Wilson's theorem. A number is prime iff it is greater than 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 26978 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)))
Theorem	wilthimp 26980	The forward implication of Wilson's theorem wilth 26979 (see wilthlem3 26978), 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
Theorem	ftalem1 26981*	Lemma for fta 26988: "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)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ 𝑇 = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸)    ⇒   ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
Theorem	ftalem2 26982*	Lemma for fta 26988. 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)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑈 = if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1))    &   ⊢ 𝑇 = ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2))    ⇒   ⊢ (𝜑 → ∃𝑟 ∈ ℝ+ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))
Theorem	ftalem3 26983*	Lemma for fta 26988. 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 26981; 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)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅}    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ ℂ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))
Theorem	ftalem4 26984*	Lemma for fta 26988: Closure of the auxiliary variables for ftalem5 26985. (Contributed by Mario Carneiro, 20-Sep-2014.) (Revised by AV, 28-Sep-2020.)
⊢ 𝐴 = (coeff‘𝐹)    &   ⊢ 𝑁 = (deg‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐹‘0) ≠ 0)    &   ⊢ 𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < )    &   ⊢ 𝑇 = (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾))    &   ⊢ 𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1))    &   ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈)    ⇒   ⊢ (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+)))
Theorem	ftalem5 26985*	Lemma for fta 26988: Main proof. We have already shifted the minimum found in ftalem3 26983 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) · 𝑋↑𝐾 for the 𝐾-th term, and all higher terms are bounded because 𝑋 is small. Thus, abs(𝐹(𝑇𝑋)) ≤ abs(𝐹(0))(1 − 𝑋↑𝐾) < 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)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐹‘0) ≠ 0)    &   ⊢ 𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < )    &   ⊢ 𝑇 = (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾))    &   ⊢ 𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1))    &   ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹‘𝑥)) < (abs‘(𝐹‘0)))
Theorem	ftalem6 26986*	Lemma for fta 26988: Discharge the auxiliary variables in ftalem5 26985. (Contributed by Mario Carneiro, 20-Sep-2014.) (Proof shortened by AV, 28-Sep-2020.)
⊢ 𝐴 = (coeff‘𝐹)    &   ⊢ 𝑁 = (deg‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐹‘0) ≠ 0)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹‘𝑥)) < (abs‘(𝐹‘0)))
Theorem	ftalem7 26987*	Lemma for fta 26988. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.)
⊢ 𝐴 = (coeff‘𝐹)    &   ⊢ 𝑁 = (deg‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → (𝐹‘𝑋) ≠ 0)    ⇒   ⊢ (𝜑 → ¬ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑋)) ≤ (abs‘(𝐹‘𝑥)))
Theorem	fta 26988*	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 (𝐹‘𝑧) = 0)
14.4.3  The Basel problem (ζ(2) = π2/6)
Theorem	basellem1 26989	Lemma for basel 26998. Closure of the sequence of roots. (Contributed by Mario Carneiro, 30-Jul-2014.) Replace OLD theorem. (Revised by Wolf Lammen, 18-Sep-2020.)
⊢ 𝑁 = ((2 · 𝑀) + 1)    ⇒   ⊢ ((𝑀 ∈ ℕ ∧ 𝐾 ∈ (1...𝑀)) → ((𝐾 · π) / 𝑁) ∈ (0(,)(π / 2)))
Theorem	basellem2 26990*	Lemma for basel 26998. 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↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))    ⇒   ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))))
Theorem	basellem3 26991*	Lemma for basel 26998. Using the binomial theorem and de Moivre's formula, we have the identity e↑i𝑁𝑥 / (sin𝑥)↑𝑛 = Σ𝑚 ∈ (0...𝑁)(𝑁C𝑚)(i↑𝑚)(cot𝑥)↑(𝑁 − 𝑚), so taking imaginary parts yields sin(𝑁𝑥) / (sin𝑥)↑𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑁C2𝑗)(-1)↑(𝑀 − 𝑗) (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↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))    ⇒   ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑃‘((tan‘𝐴)↑-2)) = ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)))
Theorem	basellem4 26992*	Lemma for basel 26998. By basellem3 26991, 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 26214. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ 𝑁 = ((2 · 𝑀) + 1)    &   ⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))    &   ⊢ 𝑇 = (𝑛 ∈ (1...𝑀) ↦ ((tan‘((𝑛 · π) / 𝑁))↑-2))    ⇒   ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1-onto→(◡𝑃 “ {0}))
Theorem	basellem5 26993*	Lemma for basel 26998. Using vieta1 26218, 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↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))    &   ⊢ 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))
Theorem	basellem6 26994	Lemma for basel 26998. The function 𝐺 goes to zero because it is bounded by 1 / 𝑛. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))    ⇒   ⊢ 𝐺 ⇝ 0
Theorem	basellem7 26995	Lemma for basel 26998. The function 1 + 𝐴 · 𝐺 for any fixed 𝐴 goes to 1. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))    &   ⊢ 𝐴 ∈ ℂ    ⇒   ⊢ ((ℕ × {1}) ∘f + ((ℕ × {𝐴}) ∘f · 𝐺)) ⇝ 1
Theorem	basellem8 26996*	Lemma for basel 26998. The function 𝐹 of partial sums of the inverse squares is bounded below by 𝐽 and above by 𝐾, obtained by summing the inequality cot↑2𝑥 ≤ 1 / 𝑥↑2 ≤ csc↑2𝑥 = cot↑2𝑥 + 1 over the 𝑀 roots of the polynomial 𝑃, and applying the identity basellem5 26993. (Contributed by Mario Carneiro, 29-Jul-2014.)
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))    &   ⊢ 𝐹 = seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2)))    &   ⊢ 𝐻 = ((ℕ × {((π↑2) / 6)}) ∘f · ((ℕ × {1}) ∘f − 𝐺))    &   ⊢ 𝐽 = (𝐻 ∘f · ((ℕ × {1}) ∘f + ((ℕ × {-2}) ∘f · 𝐺)))    &   ⊢ 𝐾 = (𝐻 ∘f · ((ℕ × {1}) ∘f + 𝐺))    &   ⊢ 𝑁 = ((2 · 𝑀) + 1)    ⇒   ⊢ (𝑀 ∈ ℕ → ((𝐽‘𝑀) ≤ (𝐹‘𝑀) ∧ (𝐹‘𝑀) ≤ (𝐾‘𝑀)))
Theorem	basellem9 26997*	Lemma for basel 26998. Since by basellem8 26996 𝐹 is bounded by two expressions that tend to π↑2 / 6, 𝐹 must also go to π↑2 / 6 by the squeeze theorem climsqz 15548. But the series 𝐹 is exactly the partial sums of 𝑘↑-2, so it follows that this is also the value of the infinite sum Σ𝑘 ∈ ℕ(𝑘↑-2). (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))    &   ⊢ 𝐹 = seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2)))    &   ⊢ 𝐻 = ((ℕ × {((π↑2) / 6)}) ∘f · ((ℕ × {1}) ∘f − 𝐺))    &   ⊢ 𝐽 = (𝐻 ∘f · ((ℕ × {1}) ∘f + ((ℕ × {-2}) ∘f · 𝐺)))    &   ⊢ 𝐾 = (𝐻 ∘f · ((ℕ × {1}) ∘f + 𝐺))    ⇒   ⊢ Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6)
Theorem	basel 26998	The sum of the inverse squares is π↑2 / 6. This is commonly known as the Basel problem, with the first known proof attributed to Euler. See http://en.wikipedia.org/wiki/Basel_problem. This particular proof approach is due to Cauchy (1821). This is Metamath 100 proof #14. (Contributed by Mario Carneiro, 30-Jul-2014.)
⊢ Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6)
14.4.4  Number-theoretical functions
Syntax	ccht 26999	Extend class notation with the first Chebyshev function.
class θ
Syntax	cvma 27000	Extend class notation with the von Mangoldt function.
class Λ