P. 271 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lgamcvg 27001*	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 27002*	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 27003*	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 27004	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 27005	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 27006*	Lemma for gamcvg2 27007. (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 27007*	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 27008	The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℝ)
Theorem	relgamcl 27009	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 27010	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 27011	The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (log Γ‘1) = 0
Theorem	gam1 27012	The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (Γ‘1) = 1
Theorem	facgam 27013	The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) = (Γ‘(𝑁 + 1)))
Theorem	gamfac 27014	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 27015	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 16707, (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 27016*	Lemma for wilth 27018: 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 27015 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 27017*	Lemma for wilth 27018. Here we round out the argument of wilthlem2 27016 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 27018	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 27017 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 27019	The forward implication of Wilson's theorem wilth 27018 (see wilthlem3 27017), 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 27020*	Lemma for fta 27027: "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 27021*	Lemma for fta 27027. 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 27022*	Lemma for fta 27027. 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 27020; 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 27023*	Lemma for fta 27027: Closure of the auxiliary variables for ftalem5 27024. (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 27024*	Lemma for fta 27027: Main proof. We have already shifted the minimum found in ftalem3 27022 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 27025*	Lemma for fta 27027: Discharge the auxiliary variables in ftalem5 27024. (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 27026*	Lemma for fta 27027. 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 27027*	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 27028	Lemma for basel 27037. 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 27029*	Lemma for basel 27037. 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 27030*	Lemma for basel 27037. 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 27031*	Lemma for basel 27037. By basellem3 27030, 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 26253. (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 27032*	Lemma for basel 27037. Using vieta1 26257, 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 27033	Lemma for basel 27037. 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 27034	Lemma for basel 27037. 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 27035*	Lemma for basel 27037. 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 27032. (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 27036*	Lemma for basel 27037. Since by basellem8 27035 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 15558. 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 27037	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 27038	Extend class notation with the first Chebyshev function.
class θ
Syntax	cvma 27039	Extend class notation with the von Mangoldt function.
class Λ
Syntax	cchp 27040	Extend class notation with the second Chebyshev function.
class ψ
Syntax	cppi 27041	Extend class notation with the prime-counting function pi.
class π
Syntax	cmu 27042	Extend class notation with the Möbius function.
class μ
Syntax	csgm 27043	Extend class notation with the divisor function.
class σ
Definition	df-cht 27044*	Define the first Chebyshev function, which adds up the logarithms of all primes less than 𝑥, see definition in [ApostolNT] p. 75. The symbol used to represent this function is sometimes the variant greek letter theta shown here and sometimes the greek letter psi, ψ; however, this notation can also refer to the second Chebyshev function, which adds up the logarithms of prime powers instead, see df-chp 27046. See https://en.wikipedia.org/wiki/Chebyshev_function 27046 for a discussion of the two functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝))
Definition	df-vma 27045*	Define the von Mangoldt function, which gives the logarithm of the prime at a prime power, and is zero elsewhere, see definition in [ApostolNT] p. 32. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))
Definition	df-chp 27046*	Define the second Chebyshev function, which adds up the logarithms of the primes corresponding to the prime powers less than 𝑥, see definition in [ApostolNT] p. 75. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
Definition	df-ppi 27047	Define the prime π function, which counts the number of primes less than or equal to 𝑥, see definition in [ApostolNT] p. 8. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
Definition	df-mu 27048*	Define the Möbius function, which is zero for non-squarefree numbers and is -1 or 1 for squarefree numbers according as to the number of prime divisors of the number is even or odd, see definition in [ApostolNT] p. 24. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}))))
Definition	df-sgm 27049*	Define the sum of positive divisors function (𝑥 σ 𝑛), which is the sum of the xth powers of the positive integer divisors of n, see definition in [ApostolNT] p. 38. For 𝑥 = 0, (𝑥 σ 𝑛) counts the number of divisors of 𝑛, i.e. (0 σ 𝑛) is the divisor function, see remark in [ApostolNT] p. 38. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥))
Theorem	efnnfsumcl 27050*	Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (exp‘𝐵) ∈ ℕ)    ⇒   ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ)
Theorem	ppisval 27051	The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ))
Theorem	ppisval2 27052	The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) → ((0[,]𝐴) ∩ ℙ) = ((𝑀...(⌊‘𝐴)) ∩ ℙ))
Theorem	ppifi 27053	The set of primes less than 𝐴 is a finite set. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
Theorem	prmdvdsfi 27054*	The set of prime divisors of a number is a finite set. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin)
Theorem	chtf 27055	Domain and codoamin of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ θ:ℝ⟶ℝ
Theorem	chtcl 27056	Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℝ → (θ‘𝐴) ∈ ℝ)
Theorem	chtval 27057*	Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
Theorem	efchtcl 27058	The Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℝ → (exp‘(θ‘𝐴)) ∈ ℕ)
Theorem	chtge0 27059	The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℝ → 0 ≤ (θ‘𝐴))
Theorem	vmaval 27060*	Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}    ⇒   ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0))
Theorem	isppw 27061*	Two ways to say that 𝐴 is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃!𝑝 ∈ ℙ 𝑝 ∥ 𝐴))
Theorem	isppw2 27062*	Two ways to say that 𝐴 is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝐴 = (𝑝↑𝑘)))
Theorem	vmappw 27063	Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (Λ‘(𝑃↑𝐾)) = (log‘𝑃))
Theorem	vmaprm 27064	Value of the von Mangoldt function at a prime. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝑃 ∈ ℙ → (Λ‘𝑃) = (log‘𝑃))
Theorem	vmacl 27065	Closure for the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) ∈ ℝ)
Theorem	vmaf 27066	Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ Λ:ℕ⟶ℝ
Theorem	efvmacl 27067	The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℕ → (exp‘(Λ‘𝐴)) ∈ ℕ)
Theorem	vmage0 27068	The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℕ → 0 ≤ (Λ‘𝐴))
Theorem	chpval 27069*	Value of the second Chebyshev function, or summary von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))
Theorem	chpf 27070	Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ ψ:ℝ⟶ℝ
Theorem	chpcl 27071	Closure for the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)
Theorem	efchpcl 27072	The second Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℝ → (exp‘(ψ‘𝐴)) ∈ ℕ)
Theorem	chpge0 27073	The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴))
Theorem	ppival 27074	Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℝ → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))
Theorem	ppival2 27075	Value of the prime-counting function pi. (Contributed by Mario Carneiro, 18-Sep-2014.)
⊢ (𝐴 ∈ ℤ → (π‘𝐴) = (♯‘((2...𝐴) ∩ ℙ)))
Theorem	ppival2g 27076	Value of the prime-counting function pi. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 2 ∈ (ℤ≥‘𝑀)) → (π‘𝐴) = (♯‘((𝑀...𝐴) ∩ ℙ)))
Theorem	ppif 27077	Domain and codomain of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ π:ℝ⟶ℕ0
Theorem	ppicl 27078	Real closure of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ0)
Theorem	muval 27079*	The value of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (𝐴 ∈ ℕ → (μ‘𝐴) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))))
Theorem	muval1 27080	The value of the Möbius function at a non-squarefree number. (Contributed by Mario Carneiro, 21-Sep-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝑃 ∈ (ℤ≥‘2) ∧ (𝑃↑2) ∥ 𝐴) → (μ‘𝐴) = 0)
Theorem	muval2 27081*	The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014.)
⊢ ((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) → (μ‘𝐴) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))
Theorem	isnsqf 27082*	Two ways to say that a number is not squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) = 0 ↔ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴))
Theorem	issqf 27083*	Two ways to say that a number is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1))
Theorem	sqfpc 27084	The prime count of a squarefree number is at most 1. (Contributed by Mario Carneiro, 1-Jul-2015.)
⊢ ((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0 ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt 𝐴) ≤ 1)
Theorem	dvdssqf 27085	A divisor of a squarefree number is squarefree. (Contributed by Mario Carneiro, 1-Jul-2015.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴) → ((μ‘𝐴) ≠ 0 → (μ‘𝐵) ≠ 0))
Theorem	sqf11 27086*	A squarefree number is completely determined by the set of its prime divisors. (Contributed by Mario Carneiro, 1-Jul-2015.)
⊢ (((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝐴 ↔ 𝑝 ∥ 𝐵)))
Theorem	muf 27087	The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ μ:ℕ⟶ℤ
Theorem	mucl 27088	Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (𝐴 ∈ ℕ → (μ‘𝐴) ∈ ℤ)
Theorem	sgmval 27089*	The value of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝑐𝐴))
Theorem	sgmval2 27090*	The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴))
Theorem	0sgm 27091*	The value of the sum-of-divisors function, usually denoted σ<SUB>0</SUB>(<i>n</i>). (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))
Theorem	sgmf 27092	The divisor function is a function into the complex numbers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
⊢ σ :(ℂ × ℕ)⟶ℂ
Theorem	sgmcl 27093	Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℂ)
Theorem	sgmnncl 27094	Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℕ)
Theorem	mule1 27095	The Möbius function takes on values in magnitude at most 1. (Together with mucl 27088, this implies that it takes a value in {-1, 0, 1} for every positive integer.) (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (𝐴 ∈ ℕ → (abs‘(μ‘𝐴)) ≤ 1)
Theorem	chtfl 27096	The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (𝐴 ∈ ℝ → (θ‘(⌊‘𝐴)) = (θ‘𝐴))
Theorem	chpfl 27097	The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016.)
⊢ (𝐴 ∈ ℝ → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))
Theorem	ppiprm 27098	The prime-counting function π at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = ((π‘𝐴) + 1))
Theorem	ppinprm 27099	The prime-counting function π at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ ¬ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = (π‘𝐴))
Theorem	chtprm 27100	The Chebyshev function at a prime. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = ((θ‘𝐴) + (log‘(𝐴 + 1))))