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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremftalem3 24601* Lemma for fta 24606. 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 24599; 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 24602* Lemma for fta 24606: Closure of the auxiliary variables for ftalem5 24603. (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 24603* Lemma for fta 24606: Main proof. We have already shifted the minimum found in ftalem3 24601 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)))

Theoremftalem6 24604* Lemma for fta 24606: Discharge the auxiliary variables in ftalem5 24603. (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 24605* Lemma for fta 24606. 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 24606* 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 24607 Lemma for basel 24616. 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 24608* Lemma for basel 24616. 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 24609* Lemma for basel 24616. 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 24610* Lemma for basel 24616. By basellem3 24609, 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 23867. (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 24611* Lemma for basel 24616. Using vieta1 23871, 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))

Theorembasellem6 24612 Lemma for basel 24616. 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

Theorembasellem7 24613 Lemma for basel 24616. 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

Theorembasellem8 24614* Lemma for basel 24616. 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 24611. (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}) ∘𝑓𝐺))    &   𝐽 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + ((ℕ × {-2}) ∘𝑓 · 𝐺)))    &   𝐾 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + 𝐺))    &   𝑁 = ((2 · 𝑀) + 1)       (𝑀 ∈ ℕ → ((𝐽𝑀) ≤ (𝐹𝑀) ∧ (𝐹𝑀) ≤ (𝐾𝑀)))

Theorembasellem9 24615* Lemma for basel 24616. Since by basellem8 24614 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 14219. 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}) ∘𝑓𝐺))    &   𝐽 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + ((ℕ × {-2}) ∘𝑓 · 𝐺)))    &   𝐾 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + 𝐺))       Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6)

Theorembasel 24616 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

Syntaxccht 24617 Extend class notation with the first Chebyshev function.
class θ

Syntaxcvma 24618 Extend class notation with the von Mangoldt function.
class Λ

Syntaxcchp 24619 Extend class notation with the second Chebyshev function.
class ψ

Syntaxcppi 24620 Extend class notation with the prime-counting function pi.
class π

Syntaxcmu 24621 Extend class notation with the Möbius function.
class μ

Syntaxcsgm 24622 Extend class notation with the divisor function.
class σ

Definitiondf-cht 24623* 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 24625. See https://en.wikipedia.org/wiki/Chebyshev_function for a discussion of the two functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝))

Definitiondf-vma 24624* 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 ∣ 𝑝𝑥} / 𝑠if((#‘𝑠) = 1, (log‘ 𝑠), 0))

Definitiondf-chp 24625* 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))

Definitiondf-ppi 24626 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)))

Definitiondf-mu 24627* 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 ∣ 𝑝𝑥}))))

Definitiondf-sgm 24628* 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 ∣ 𝑝𝑛} (𝑘𝑐𝑥))

Theoremefnnfsumcl 24629* Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → (exp‘𝐵) ∈ ℕ)       (𝜑 → (exp‘Σ𝑘𝐴 𝐵) ∈ ℕ)

Theoremppisval 24630 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))

Theoremppisval2 24631 The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ𝑀)) → ((0[,]𝐴) ∩ ℙ) = ((𝑀...(⌊‘𝐴)) ∩ ℙ))

Theoremppifi 24632 The set of primes less than 𝐴 is a finite set. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)

Theoremprmdvdsfi 24633* The set of prime divisors of a number is a finite set. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝𝐴} ∈ Fin)

Theoremchtf 24634 Domain and range of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
θ:ℝ⟶ℝ

Theoremchtcl 24635 Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘𝐴) ∈ ℝ)

Theoremchtval 24636* Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))

Theoremefchtcl 24637 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)

Theoremchtge0 24638 The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → 0 ≤ (θ‘𝐴))

Theoremvmaval 24639* Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
𝑆 = {𝑝 ∈ ℙ ∣ 𝑝𝐴}       (𝐴 ∈ ℕ → (Λ‘𝐴) = if((#‘𝑆) = 1, (log‘ 𝑆), 0))

Theoremisppw 24640* Two ways to say that 𝐴 is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃!𝑝 ∈ ℙ 𝑝𝐴))

Theoremisppw2 24641* 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 = (𝑝𝑘)))

Theoremvmappw 24642 Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (Λ‘(𝑃𝐾)) = (log‘𝑃))

Theoremvmaprm 24643 Value of the von Mangoldt function at a prime. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝑃 ∈ ℙ → (Λ‘𝑃) = (log‘𝑃))

Theoremvmacl 24644 Closure for the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (Λ‘𝐴) ∈ ℝ)

Theoremvmaf 24645 Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ:ℕ⟶ℝ

Theoremefvmacl 24646 The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (exp‘(Λ‘𝐴)) ∈ ℕ)

Theoremvmage0 24647 The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → 0 ≤ (Λ‘𝐴))

Theoremchpval 24648* 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))

Theoremchpf 24649 Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ:ℝ⟶ℝ

Theoremchpcl 24650 Closure for the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)

Theoremefchpcl 24651 The second Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (exp‘(ψ‘𝐴)) ∈ ℕ)

Theoremchpge0 24652 The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴))

Theoremppival 24653 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (π𝐴) = (#‘((0[,]𝐴) ∩ ℙ)))

Theoremppival2 24654 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℤ → (π𝐴) = (#‘((2...𝐴) ∩ ℙ)))

Theoremppival2g 24655 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℤ ∧ 2 ∈ (ℤ𝑀)) → (π𝐴) = (#‘((𝑀...𝐴) ∩ ℙ)))

Theoremppif 24656 Domain and range of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
π:ℝ⟶ℕ0

Theoremppicl 24657 Real closure of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (π𝐴) ∈ ℕ0)

Theoremmuval 24658* 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 ∣ 𝑝𝐴}))))

Theoremmuval1 24659 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)

Theoremmuval2 24660* 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 ∣ 𝑝𝐴})))

Theoremisnsqf 24661* 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))

Theoremissqf 24662* 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))

Theoremsqfpc 24663 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)

Theoremdvdssqf 24664 A divisor of a squarefree number is squarefree. (Contributed by Mario Carneiro, 1-Jul-2015.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵𝐴) → ((μ‘𝐴) ≠ 0 → (μ‘𝐵) ≠ 0))

Theoremsqf11 24665* 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 (𝑝𝐴𝑝𝐵)))

Theoremmuf 24666 The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
μ:ℕ⟶ℤ

Theoremmucl 24667 Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℕ → (μ‘𝐴) ∈ ℤ)

Theoremsgmval 24668* 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 ∣ 𝑝𝐵} (𝑘𝑐𝐴))

Theoremsgmval2 24669* The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝𝐵} (𝑘𝐴))

Theorem0sgm 24670* 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 ∣ 𝑝𝐴}))

Theoremsgmf 24671 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

Theoremsgmcl 24672 Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℂ)

Theoremsgmnncl 24673 Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℕ)

Theoremmule1 24674 The Möbius function takes on values in magnitude at most 1. (Together with mucl 24667, 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)

Theoremchtfl 24675 The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘(⌊‘𝐴)) = (θ‘𝐴))

Theoremchpfl 24676 The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))

Theoremppiprm 24677 The prime-counting function π at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = ((π𝐴) + 1))

Theoremppinprm 24678 The prime-counting function π at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℤ ∧ ¬ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = (π𝐴))

Theoremchtprm 24679 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))))

Theoremchtnprm 24680 The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℤ ∧ ¬ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = (θ‘𝐴))

Theoremchpp1 24681 The second Chebyshev function at a successor. (Contributed by Mario Carneiro, 11-Apr-2016.)
(𝐴 ∈ ℕ0 → (ψ‘(𝐴 + 1)) = ((ψ‘𝐴) + (Λ‘(𝐴 + 1))))

Theoremchtwordi 24682 The Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (θ‘𝐴) ≤ (θ‘𝐵))

Theoremchpwordi 24683 The second Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 9-Apr-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (ψ‘𝐴) ≤ (ψ‘𝐵))

Theoremchtdif 24684* The difference of the Chebyshev function at two points sums the logarithms of the primes in an interval. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝑁 ∈ (ℤ𝑀) → ((θ‘𝑁) − (θ‘𝑀)) = Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝))

Theoremefchtdvds 24685 The exponentiated Chebyshev function forms a divisibility chain between any two points. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (exp‘(θ‘𝐴)) ∥ (exp‘(θ‘𝐵)))

Theoremppifl 24686 The prime-counting function π does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π𝐴))

Theoremppip1le 24687 The prime-counting function π cannot locally increase faster than the identity function. (Contributed by Mario Carneiro, 21-Sep-2014.)
(𝐴 ∈ ℝ → (π‘(𝐴 + 1)) ≤ ((π𝐴) + 1))

Theoremppiwordi 24688 The prime-counting function π is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (π𝐴) ≤ (π𝐵))

Theoremppidif 24689 The difference of the prime-counting function π at two points counts the number of primes in an interval. (Contributed by Mario Carneiro, 21-Sep-2014.)
(𝑁 ∈ (ℤ𝑀) → ((π𝑁) − (π𝑀)) = (#‘(((𝑀 + 1)...𝑁) ∩ ℙ)))

Theoremppi1 24690 The prime-counting function π at 1. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘1) = 0

Theoremcht1 24691 The Chebyshev function at 1. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘1) = 0

Theoremvma1 24692 The von Mangoldt function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.)
(Λ‘1) = 0

Theoremchp1 24693 The second Chebyshev function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.)
(ψ‘1) = 0

Theoremppi1i 24694 Inference form of ppiprm 24677. (Contributed by Mario Carneiro, 21-Sep-2014.)
𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (π𝑀) = 𝐾    &   𝑁 ∈ ℙ       (π𝑁) = (𝐾 + 1)

Theoremppi2i 24695 Inference form of ppinprm 24678. (Contributed by Mario Carneiro, 21-Sep-2014.)
𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (π𝑀) = 𝐾    &    ¬ 𝑁 ∈ ℙ       (π𝑁) = 𝐾

Theoremppi2 24696 The prime-counting function π at 2. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘2) = 1

Theoremppi3 24697 The prime-counting function π at 3. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘3) = 2

Theoremcht2 24698 The Chebyshev function at 2. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘2) = (log‘2)

Theoremcht3 24699 The Chebyshev function at 3. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘3) = (log‘6)

Theoremppinncl 24700 Closure of the prime-counting function π in the positive integers. (Contributed by Mario Carneiro, 21-Sep-2014.)
((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π𝐴) ∈ ℕ)

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