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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvdsmulf1o 24901* If 𝑀 and 𝑁 are two coprime integers, multiplication forms a bijection from the set of pairs 𝑗, 𝑘 where 𝑗𝑀 and 𝑘𝑁, to the set of divisors of 𝑀 · 𝑁. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   𝑋 = {𝑥 ∈ ℕ ∣ 𝑥𝑀}    &   𝑌 = {𝑥 ∈ ℕ ∣ 𝑥𝑁}    &   𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}       (𝜑 → ( · ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto𝑍)

Theoremfsumdvdsmul 24902* Product of two divisor sums. (This is also the main part of the proof that "Σ𝑘𝑁𝐹(𝑘) is a multiplicative function if 𝐹 is".) (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   𝑋 = {𝑥 ∈ ℕ ∣ 𝑥𝑀}    &   𝑌 = {𝑥 ∈ ℕ ∣ 𝑥𝑁}    &   𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}    &   ((𝜑𝑗𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑌) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → (𝐴 · 𝐵) = 𝐷)    &   (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)       (𝜑 → (Σ𝑗𝑋 𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑖𝑍 𝐶)

Theoremsgmppw 24903* The value of the divisor function at a prime power. (Contributed by Mario Carneiro, 17-May-2016.)
((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝐴 σ (𝑃𝑁)) = Σ𝑘 ∈ (0...𝑁)((𝑃𝑐𝐴)↑𝑘))

Theorem0sgmppw 24904 A prime power 𝑃𝐾 has 𝐾 + 1 divisors. (Contributed by Mario Carneiro, 17-May-2016.)
((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ0) → (0 σ (𝑃𝐾)) = (𝐾 + 1))

Theorem1sgmprm 24905 The sum of divisors for a prime is 𝑃 + 1 because the only divisors are 1 and 𝑃. (Contributed by Mario Carneiro, 17-May-2016.)
(𝑃 ∈ ℙ → (1 σ 𝑃) = (𝑃 + 1))

Theorem1sgm2ppw 24906 The sum of the divisors of 2↑(𝑁 − 1). (Contributed by Mario Carneiro, 17-May-2016.)
(𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1))

Theoremsgmmul 24907 The divisor function for fixed parameter 𝐴 is a multiplicative function. (Contributed by Mario Carneiro, 2-Jul-2015.)
((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝐴 σ (𝑀 · 𝑁)) = ((𝐴 σ 𝑀) · (𝐴 σ 𝑁)))

Theoremppiublem1 24908 Lemma for ppiub 24910. (Contributed by Mario Carneiro, 12-Mar-2014.)
(𝑁 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑁...5) → (𝑃 mod 6) ∈ {1, 5})))    &   𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (2 ∥ 𝑀 ∨ 3 ∥ 𝑀𝑀 ∈ {1, 5})       (𝑀 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑀...5) → (𝑃 mod 6) ∈ {1, 5})))

Theoremppiublem2 24909 A prime greater than 3 does not divide 2 or 3, so its residue mod 6 is 1 or 5. (Contributed by Mario Carneiro, 12-Mar-2014.)
((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (𝑃 mod 6) ∈ {1, 5})

Theoremppiub 24910 An upper bound on the prime-counting function π, which counts the number of primes less than 𝑁. (Contributed by Mario Carneiro, 13-Mar-2014.)
((𝑁 ∈ ℝ ∧ 0 ≤ 𝑁) → (π𝑁) ≤ ((𝑁 / 3) + 2))

Theoremvmalelog 24911 The von Mangoldt function is less than the natural log. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (Λ‘𝐴) ≤ (log‘𝐴))

Theoremchtlepsi 24912 The first Chebyshev function is less than the second. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (θ‘𝐴) ≤ (ψ‘𝐴))

Theoremchprpcl 24913 Closure of the second Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 8-Apr-2016.)
((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (ψ‘𝐴) ∈ ℝ+)

Theoremchpeq0 24914 The second Chebyshev function is zero iff its argument is less than 2. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝐴 ∈ ℝ → ((ψ‘𝐴) = 0 ↔ 𝐴 < 2))

Theoremchteq0 24915 The first Chebyshev function is zero iff its argument is less than 2. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝐴 ∈ ℝ → ((θ‘𝐴) = 0 ↔ 𝐴 < 2))

Theoremchtleppi 24916 Upper bound on the θ function. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ+ → (θ‘𝐴) ≤ ((π𝐴) · (log‘𝐴)))

Theoremchtublem 24917 Lemma for chtub 24918. (Contributed by Mario Carneiro, 13-Mar-2014.)
(𝑁 ∈ ℕ → (θ‘((2 · 𝑁) − 1)) ≤ ((θ‘𝑁) + ((log‘4) · (𝑁 − 1))))

Theoremchtub 24918 An upper bound on the Chebyshev function. (Contributed by Mario Carneiro, 13-Mar-2014.) (Revised 22-Sep-2014.)
((𝑁 ∈ ℝ ∧ 2 < 𝑁) → (θ‘𝑁) < ((log‘2) · ((2 · 𝑁) − 3)))

Theoremfsumvma 24919* Rewrite a sum over the von Mangoldt function as a sum over prime powers. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝑥 = (𝑝𝑘) → 𝐵 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑 → ((𝑝𝑃𝑘𝐾) ↔ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝𝑘) ∈ 𝐴)))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ (𝑥𝐴 ∧ (Λ‘𝑥) = 0)) → 𝐵 = 0)       (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑝𝑃 Σ𝑘𝐾 𝐶)

Theoremfsumvma2 24920* Apply fsumvma 24919 for the common case of all numbers less than a real number 𝐴. (Contributed by Mario Carneiro, 30-Apr-2016.)
(𝑥 = (𝑝𝑘) → 𝐵 = 𝐶)    &   (𝜑𝐴 ∈ ℝ)    &   ((𝜑𝑥 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ (𝑥 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑥) = 0)) → 𝐵 = 0)       (𝜑 → Σ𝑥 ∈ (1...(⌊‘𝐴))𝐵 = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))𝐶)

Theorempclogsum 24921* The logarithmic analogue of pcprod 15580. The sum of the logarithms of the primes dividing 𝐴 multiplied by their powers yields the logarithm of 𝐴. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝐴 ∈ ℕ → Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)((𝑝 pCnt 𝐴) · (log‘𝑝)) = (log‘𝐴))

Theoremvmasum 24922* The sum of the von Mangoldt function over the divisors of 𝑛. Equation 9.2.4 of [Shapiro], p. 328 and theorem 2.10 in [ApostolNT] p. 32. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝐴 ∈ ℕ → Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝐴} (Λ‘𝑛) = (log‘𝐴))

Theoremlogfac2 24923* Another expression for the logarithm of a factorial, in terms of the von Mangoldt function. Equation 9.2.7 of [Shapiro], p. 329. (Contributed by Mario Carneiro, 15-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) = Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))

Theoremchpval2 24924* Express the second Chebyshev function directly as a sum over the primes less than 𝐴 (instead of indirectly through the von Mangoldt function). (Contributed by Mario Carneiro, 8-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))

Theoremchpchtsum 24925* The second Chebyshev function is the sum of the theta function at arguments quickly approaching zero. (This is usually stated as an infinite sum, but after a certain point, the terms are all zero, and it is easier for us to use an explicit finite sum.) (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))))

Theoremchpub 24926 An upper bound on the second Chebyshev function. (Contributed by Mario Carneiro, 8-Apr-2016.)
((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (ψ‘𝐴) ≤ ((θ‘𝐴) + ((√‘𝐴) · (log‘𝐴))))

Theoremlogfacubnd 24927 A simple upper bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)
((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴)))

Theoremlogfaclbnd 24928 A lower bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)
(𝐴 ∈ ℝ+ → (𝐴 · ((log‘𝐴) − 2)) ≤ (log‘(!‘(⌊‘𝐴))))

Theoremlogfacbnd3 24929 Show the stronger statement log(𝑥!) = 𝑥log𝑥𝑥 + 𝑂(log𝑥) alluded to in logfacrlim 24930. (Contributed by Mario Carneiro, 20-May-2016.)
((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ≤ ((log‘𝐴) + 1))

Theoremlogfacrlim 24930 Combine the estimates logfacubnd 24927 and logfaclbnd 24928, to get log(𝑥!) = 𝑥log𝑥 + 𝑂(𝑥). Equation 9.2.9 of [Shapiro], p. 329. This is a weak form of the even stronger statement, log(𝑥!) = 𝑥log𝑥𝑥 + 𝑂(log𝑥). (Contributed by Mario Carneiro, 16-Apr-2016.) (Revised by Mario Carneiro, 21-May-2016.)
(𝑥 ∈ ℝ+ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ⇝𝑟 1

Theoremlogexprlim 24931* The sum Σ𝑛𝑥, log↑𝑁(𝑥 / 𝑛) has the asymptotic expansion (𝑁!)𝑥 + 𝑜(𝑥). (More precisely, the omitted term has order 𝑂(log↑𝑁(𝑥) / 𝑥).) (Contributed by Mario Carneiro, 22-May-2016.)
(𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥)) ⇝𝑟 (!‘𝑁))

Theoremlogfacrlim2 24932* Write out logfacrlim 24930 as a sum of logs. (Contributed by Mario Carneiro, 18-May-2016.) (Revised by Mario Carneiro, 22-May-2016.)
(𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) ⇝𝑟 1

14.4.5  Perfect Number Theorem

Theoremmersenne 24933 A Mersenne prime is a prime number of the form 2↑𝑃 − 1. This theorem shows that the 𝑃 in this expression is necessarily also prime. (Contributed by Mario Carneiro, 17-May-2016.)
((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → 𝑃 ∈ ℙ)

Theoremperfect1 24934 Euclid's contribution to the Euclid-Euler theorem. A number of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1) is a perfect number. (Contributed by Mario Carneiro, 17-May-2016.)
((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (1 σ ((2↑(𝑃 − 1)) · ((2↑𝑃) − 1))) = ((2↑𝑃) · ((2↑𝑃) − 1)))

Theoremperfectlem1 24935 Lemma for perfect 24937. (Contributed by Mario Carneiro, 7-Jun-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐵)    &   (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))       (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))

Theoremperfectlem2 24936 Lemma for perfect 24937. (Contributed by Mario Carneiro, 17-May-2016.) Replace OLD theorem. (Revised by Wolf Lammen, 17-Sep-2020.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐵)    &   (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))       (𝜑 → (𝐵 ∈ ℙ ∧ 𝐵 = ((2↑(𝐴 + 1)) − 1)))

Theoremperfect 24937* The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer 𝑁 is a perfect number (that is, its divisor sum is 2𝑁) if and only if it is of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1), where 2↑𝑝 − 1 is prime (a Mersenne prime). (It follows from this that 𝑝 is also prime.) This is Metamath 100 proof #70. (Contributed by Mario Carneiro, 17-May-2016.)
((𝑁 ∈ ℕ ∧ 2 ∥ 𝑁) → ((1 σ 𝑁) = (2 · 𝑁) ↔ ∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)))))

14.4.6  Characters of Z/nZ

Syntaxcdchr 24938 Extend class notation with the group of Dirichlet characters.
class DChr

Definitiondf-dchr 24939* The group of Dirichlet characters mod 𝑛 is the set of monoid homomorphisms from ℤ / 𝑛 to the multiplicative monoid of the complex numbers, equipped with the group operation of pointwise multiplication. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr = (𝑛 ∈ ℕ ↦ (ℤ/nℤ‘𝑛) / 𝑧{𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 · ↾ (𝑏 × 𝑏))⟩})

Theoremdchrval 24940* Value of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵𝑈) × {0}) ⊆ 𝑥})       (𝜑𝐺 = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))⟩})

Theoremdchrbas 24941* Base set of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = (Base‘𝐺)       (𝜑𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵𝑈) × {0}) ⊆ 𝑥})

Theoremdchrelbas 24942 A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/n to the multiplicative monoid of , which is zero off the group of units of ℤ/n. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = (Base‘𝐺)       (𝜑 → (𝑋𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ((𝐵𝑈) × {0}) ⊆ 𝑋)))

Theoremdchrelbas2 24943* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/n to the multiplicative monoid of , which is zero off the group of units of ℤ/n. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = (Base‘𝐺)       (𝜑 → (𝑋𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥𝐵 ((𝑋𝑥) ≠ 0 → 𝑥𝑈))))

Theoremdchrelbas3 24944* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/n to the multiplicative monoid of , which is zero off the group of units of ℤ/n. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = (Base‘𝐺)       (𝜑 → (𝑋𝐷 ↔ (𝑋:𝐵⟶ℂ ∧ (∀𝑥𝑈𝑦𝑈 (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)) ∧ (𝑋‘(1r𝑍)) = 1 ∧ ∀𝑥𝐵 ((𝑋𝑥) ≠ 0 → 𝑥𝑈)))))

Theoremdchrelbasd 24945* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/n to the multiplicative monoid of , which is zero off the group of units of ℤ/n. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = (Base‘𝐺)    &   (𝑘 = 𝑥𝑋 = 𝐴)    &   (𝑘 = 𝑦𝑋 = 𝐶)    &   (𝑘 = (𝑥(.r𝑍)𝑦) → 𝑋 = 𝐸)    &   (𝑘 = (1r𝑍) → 𝑋 = 𝑌)    &   ((𝜑𝑘𝑈) → 𝑋 ∈ ℂ)    &   ((𝜑 ∧ (𝑥𝑈𝑦𝑈)) → 𝐸 = (𝐴 · 𝐶))    &   (𝜑𝑌 = 1)       (𝜑 → (𝑘𝐵 ↦ if(𝑘𝑈, 𝑋, 0)) ∈ 𝐷)

Theoremdchrrcl 24946 Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)       (𝑋𝐷𝑁 ∈ ℕ)

Theoremdchrmhm 24947 A Dirichlet character is a monoid homomorphism. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)       𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))

Theoremdchrf 24948 A Dirichlet character is a function. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑋𝐷)       (𝜑𝑋:𝐵⟶ℂ)

Theoremdchrelbas4 24949* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/n to the multiplicative monoid of , which is zero off the group of units of ℤ/n. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐿 = (ℤRHom‘𝑍)       (𝑋𝐷 ↔ (𝑁 ∈ ℕ ∧ 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ ℤ (1 < (𝑥 gcd 𝑁) → (𝑋‘(𝐿𝑥)) = 0)))

Theoremdchrzrh1 24950 Value of a Dirichlet character at one. (Contributed by Mario Carneiro, 4-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑋𝐷)       (𝜑 → (𝑋‘(𝐿‘1)) = 1)

Theoremdchrzrhcl 24951 A Dirichlet character takes values in the complex numbers. (Contributed by Mario Carneiro, 12-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝐴 ∈ ℤ)       (𝜑 → (𝑋‘(𝐿𝐴)) ∈ ℂ)

Theoremdchrzrhmul 24952 A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)       (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿𝐴)) · (𝑋‘(𝐿𝐶))))

Theoremdchrplusg 24953 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &    · = (+g𝐺)    &   (𝜑𝑁 ∈ ℕ)       (𝜑· = ( ∘𝑓 · ↾ (𝐷 × 𝐷)))

Theoremdchrmul 24954 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &    · = (+g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)       (𝜑 → (𝑋 · 𝑌) = (𝑋𝑓 · 𝑌))

Theoremdchrmulcl 24955 Closure of the group operation on Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &    · = (+g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)       (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)

Theoremdchrn0 24956 A Dirichlet character is nonzero on the units of ℤ/n. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝑋𝐴) ≠ 0 ↔ 𝐴𝑈))

Theoremdchr1cl 24957* Closure of the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &    1 = (𝑘𝐵 ↦ if(𝑘𝑈, 1, 0))    &   (𝜑𝑁 ∈ ℕ)       (𝜑1𝐷)

Theoremdchrmulid2 24958* Left identity for the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &    1 = (𝑘𝐵 ↦ if(𝑘𝑈, 1, 0))    &    · = (+g𝐺)    &   (𝜑𝑋𝐷)       (𝜑 → ( 1 · 𝑋) = 𝑋)

Theoremdchrinvcl 24959* Closure of the group inverse operation on Dirichlet characters. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &    1 = (𝑘𝐵 ↦ if(𝑘𝑈, 1, 0))    &    · = (+g𝐺)    &   (𝜑𝑋𝐷)    &   𝐾 = (𝑘𝐵 ↦ if(𝑘𝑈, (1 / (𝑋𝑘)), 0))       (𝜑 → (𝐾𝐷 ∧ (𝐾 · 𝑋) = 1 ))

Theoremdchrabl 24960 The set of Dirichlet characters is an Abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)       (𝑁 ∈ ℕ → 𝐺 ∈ Abel)

Theoremdchrfi 24961 The group of Dirichlet characters is a finite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)       (𝑁 ∈ ℕ → 𝐷 ∈ Fin)

Theoremdchrghm 24962 A Dirichlet character restricted to the unit group of ℤ/n is a group homomorphism into the multiplicative group of nonzero complex numbers. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑈 = (Unit‘𝑍)    &   𝐻 = ((mulGrp‘𝑍) ↾s 𝑈)    &   𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    &   (𝜑𝑋𝐷)       (𝜑 → (𝑋𝑈) ∈ (𝐻 GrpHom 𝑀))

Theoremdchr1 24963 Value of the principal Dirichlet character. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &    1 = (0g𝐺)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝑈)       (𝜑 → ( 1𝐴) = 1)

Theoremdchreq 24964* A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)       (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘𝑈 (𝑋𝑘) = (𝑌𝑘)))

Theoremdchrresb 24965 A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)       (𝜑 → ((𝑋𝑈) = (𝑌𝑈) ↔ 𝑋 = 𝑌))

Theoremdchrabs 24966 A Dirichlet character takes values on the unit circle. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   (𝜑𝑋𝐷)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝐴𝑈)       (𝜑 → (abs‘(𝑋𝐴)) = 1)

Theoremdchrinv 24967 The inverse of a Dirichlet character is the conjugate (which is also the multiplicative inverse, because the values of 𝑋 are unimodular). (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   (𝜑𝑋𝐷)    &   𝐼 = (invg𝐺)       (𝜑 → (𝐼𝑋) = (∗ ∘ 𝑋))

Theoremdchrabs2 24968 A Dirichlet character takes values inside the unit circle. (Contributed by Mario Carneiro, 3-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝐴𝐵)       (𝜑 → (abs‘(𝑋𝐴)) ≤ 1)

Theoremdchr1re 24969 The principal Dirichlet character is a real character. (Contributed by Mario Carneiro, 2-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &    1 = (0g𝐺)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑁 ∈ ℕ)       (𝜑1 :𝐵⟶ℝ)

Theoremdchrptlem1 24970* Lemma for dchrpt 24973. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &    1 = (1r𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴1 )    &   𝑈 = (Unit‘𝑍)    &   𝐻 = ((mulGrp‘𝑍) ↾s 𝑈)    &    · = (.g𝐻)    &   𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊𝑘))))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ Word 𝑈)    &   (𝜑𝐻dom DProd 𝑆)    &   (𝜑 → (𝐻 DProd 𝑆) = 𝑈)    &   𝑃 = (𝐻dProj𝑆)    &   𝑂 = (od‘𝐻)    &   𝑇 = (-1↑𝑐(2 / (𝑂‘(𝑊𝐼))))    &   (𝜑𝐼 ∈ dom 𝑊)    &   (𝜑 → ((𝑃𝐼)‘𝐴) ≠ 1 )    &   𝑋 = (𝑢𝑈 ↦ (℩𝑚 ∈ ℤ (((𝑃𝐼)‘𝑢) = (𝑚 · (𝑊𝐼)) ∧ = (𝑇𝑚))))       (((𝜑𝐶𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃𝐼)‘𝐶) = (𝑀 · (𝑊𝐼)))) → (𝑋𝐶) = (𝑇𝑀))

Theoremdchrptlem2 24971* Lemma for dchrpt 24973. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &    1 = (1r𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴1 )    &   𝑈 = (Unit‘𝑍)    &   𝐻 = ((mulGrp‘𝑍) ↾s 𝑈)    &    · = (.g𝐻)    &   𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊𝑘))))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ Word 𝑈)    &   (𝜑𝐻dom DProd 𝑆)    &   (𝜑 → (𝐻 DProd 𝑆) = 𝑈)    &   𝑃 = (𝐻dProj𝑆)    &   𝑂 = (od‘𝐻)    &   𝑇 = (-1↑𝑐(2 / (𝑂‘(𝑊𝐼))))    &   (𝜑𝐼 ∈ dom 𝑊)    &   (𝜑 → ((𝑃𝐼)‘𝐴) ≠ 1 )    &   𝑋 = (𝑢𝑈 ↦ (℩𝑚 ∈ ℤ (((𝑃𝐼)‘𝑢) = (𝑚 · (𝑊𝐼)) ∧ = (𝑇𝑚))))       (𝜑 → ∃𝑥𝐷 (𝑥𝐴) ≠ 1)

Theoremdchrptlem3 24972* Lemma for dchrpt 24973. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &    1 = (1r𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴1 )    &   𝑈 = (Unit‘𝑍)    &   𝐻 = ((mulGrp‘𝑍) ↾s 𝑈)    &    · = (.g𝐻)    &   𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊𝑘))))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ Word 𝑈)    &   (𝜑𝐻dom DProd 𝑆)    &   (𝜑 → (𝐻 DProd 𝑆) = 𝑈)       (𝜑 → ∃𝑥𝐷 (𝑥𝐴) ≠ 1)

Theoremdchrpt 24973* For any element other than 1, there is a Dirichlet character that is not one at the given element. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &    1 = (1r𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴1 )    &   (𝜑𝐴𝐵)       (𝜑 → ∃𝑥𝐷 (𝑥𝐴) ≠ 1)

Theoremdchrsum2 24974* An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character 𝑋 is 0 if 𝑋 is non-principal and ϕ(𝑛) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   𝑈 = (Unit‘𝑍)       (𝜑 → Σ𝑎𝑈 (𝑋𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0))

Theoremdchrsum 24975* An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character 𝑋 is 0 if 𝑋 is non-principal and ϕ(𝑛) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   𝐵 = (Base‘𝑍)       (𝜑 → Σ𝑎𝐵 (𝑋𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0))

Theoremsumdchr2 24976* Lemma for sumdchr 24978. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑍 = (ℤ/nℤ‘𝑁)    &    1 = (1r𝑍)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝐵)       (𝜑 → Σ𝑥𝐷 (𝑥𝐴) = if(𝐴 = 1 , (#‘𝐷), 0))

Theoremdchrhash 24977 There are exactly ϕ(𝑁) Dirichlet characters modulo 𝑁. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)       (𝑁 ∈ ℕ → (#‘𝐷) = (ϕ‘𝑁))

Theoremsumdchr 24978* An orthogonality relation for Dirichlet characters: the sum of 𝑥(𝐴) for fixed 𝐴 and all 𝑥 is 0 if 𝐴 = 1 and ϕ(𝑛) otherwise. Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑍 = (ℤ/nℤ‘𝑁)    &    1 = (1r𝑍)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝐵)       (𝜑 → Σ𝑥𝐷 (𝑥𝐴) = if(𝐴 = 1 , (ϕ‘𝑁), 0))

Theoremdchr2sum 24979* An orthogonality relation for Dirichlet characters: the sum of 𝑋(𝑎) · ∗𝑌(𝑎) over all 𝑎 is nonzero only when 𝑋 = 𝑌. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)       (𝜑 → Σ𝑎𝐵 ((𝑋𝑎) · (∗‘(𝑌𝑎))) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0))

Theoremsum2dchr 24980* An orthogonality relation for Dirichlet characters: the sum of 𝑥(𝐴) for fixed 𝐴 and all 𝑥 is 0 if 𝐴 = 1 and ϕ(𝑛) otherwise. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝑈)       (𝜑 → Σ𝑥𝐷 ((𝑥𝐴) · (∗‘(𝑥𝐶))) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0))

14.4.7  Bertrand's postulate

Theorembcctr 24981 Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.)
(𝑁 ∈ ℕ0 → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘𝑁) · (!‘𝑁))))

Theorempcbcctr 24982* Prime count of a central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.)
((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) = Σ𝑘 ∈ (1...(2 · 𝑁))((⌊‘((2 · 𝑁) / (𝑃𝑘))) − (2 · (⌊‘(𝑁 / (𝑃𝑘))))))

Theorembcmono 24983 The binomial coefficient is monotone in its second argument, up to the midway point. (Contributed by Mario Carneiro, 5-Mar-2014.)
((𝑁 ∈ ℕ0𝐵 ∈ (ℤ𝐴) ∧ 𝐵 ≤ (𝑁 / 2)) → (𝑁C𝐴) ≤ (𝑁C𝐵))

Theorembcmax 24984 The binomial coefficient takes its maximum value at the center. (Contributed by Mario Carneiro, 5-Mar-2014.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → ((2 · 𝑁)C𝐾) ≤ ((2 · 𝑁)C𝑁))

Theorembcp1ctr 24985 Ratio of two central binomial coefficients. (Contributed by Mario Carneiro, 10-Mar-2014.)
(𝑁 ∈ ℕ0 → ((2 · (𝑁 + 1))C(𝑁 + 1)) = (((2 · 𝑁)C𝑁) · (2 · (((2 · 𝑁) + 1) / (𝑁 + 1)))))

Theorembclbnd 24986 A bound on the binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.)
(𝑁 ∈ (ℤ‘4) → ((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁))

Theoremefexple 24987 Convert a bound on a power to a bound on the exponent. (Contributed by Mario Carneiro, 11-Mar-2014.)
(((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℝ+) → ((𝐴𝑁) ≤ 𝐵𝑁 ≤ (⌊‘((log‘𝐵) / (log‘𝐴)))))

Theorembpos1lem 24988* Lemma for bpos1 24989. (Contributed by Mario Carneiro, 12-Mar-2014.)
(∃𝑝 ∈ ℙ (𝑁 < 𝑝𝑝 ≤ (2 · 𝑁)) → 𝜑)    &   (𝑁 ∈ (ℤ𝑃) → 𝜑)    &   𝑃 ∈ ℙ    &   𝐴 ∈ ℕ0    &   (𝐴 · 2) = 𝐵    &   𝐴 < 𝑃    &   (𝑃 < 𝐵𝑃 = 𝐵)       (𝑁 ∈ (ℤ𝐴) → 𝜑)

Theorembpos1 24989* Bertrand's postulate, checked numerically for 𝑁 ≤ 64, using the prime sequence 2, 3, 5, 7, 13, 23, 43, 83. (Contributed by Mario Carneiro, 12-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)
((𝑁 ∈ ℕ ∧ 𝑁64) → ∃𝑝 ∈ ℙ (𝑁 < 𝑝𝑝 ≤ (2 · 𝑁)))

Theorembposlem1 24990 An upper bound on the prime powers dividing a central binomial coefficient. (Contributed by Mario Carneiro, 9-Mar-2014.)
((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(𝑃 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁))

Theorembposlem2 24991 There are no odd primes in the range (2𝑁 / 3, 𝑁] dividing the 𝑁-th central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → 2 < 𝑃)    &   (𝜑 → ((2 · 𝑁) / 3) < 𝑃)    &   (𝜑𝑃𝑁)       (𝜑 → (𝑃 pCnt ((2 · 𝑁)C𝑁)) = 0)

Theorembposlem3 24992* Lemma for bpos 24999. Since the binomial coefficient does not have any primes in the range (2𝑁 / 3, 𝑁] or (2𝑁, +∞) by bposlem2 24991 and prmfac1 15412, respectively, and it does not have any in the range (𝑁, 2𝑁] by hypothesis, the product of the primes up through 2𝑁 / 3 must be sufficient to compose the whole binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.)
(𝜑𝑁 ∈ (ℤ‘5))    &   (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝𝑝 ≤ (2 · 𝑁)))    &   𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt ((2 · 𝑁)C𝑁))), 1))    &   𝐾 = (⌊‘((2 · 𝑁) / 3))       (𝜑 → (seq1( · , 𝐹)‘𝐾) = ((2 · 𝑁)C𝑁))

Theorembposlem4 24993* Lemma for bpos 24999. (Contributed by Mario Carneiro, 13-Mar-2014.)
(𝜑𝑁 ∈ (ℤ‘5))    &   (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝𝑝 ≤ (2 · 𝑁)))    &   𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt ((2 · 𝑁)C𝑁))), 1))    &   𝐾 = (⌊‘((2 · 𝑁) / 3))    &   𝑀 = (⌊‘(√‘(2 · 𝑁)))       (𝜑𝑀 ∈ (3...𝐾))

Theorembposlem5 24994* Lemma for bpos 24999. Bound the product of all small primes in the binomial coefficient. (Contributed by Mario Carneiro, 15-Mar-2014.) (Proof shortened by AV, 15-Sep-2021.)
(𝜑𝑁 ∈ (ℤ‘5))    &   (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝𝑝 ≤ (2 · 𝑁)))    &   𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt ((2 · 𝑁)C𝑁))), 1))    &   𝐾 = (⌊‘((2 · 𝑁) / 3))    &   𝑀 = (⌊‘(√‘(2 · 𝑁)))       (𝜑 → (seq1( · , 𝐹)‘𝑀) ≤ ((2 · 𝑁)↑𝑐(((√‘(2 · 𝑁)) / 3) + 2)))

Theorembposlem6 24995* Lemma for bpos 24999. By using the various bounds at our disposal, arrive at an inequality that is false for 𝑁 large enough. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Wolf Lammen, 12-Sep-2020.)
(𝜑𝑁 ∈ (ℤ‘5))    &   (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝𝑝 ≤ (2 · 𝑁)))    &   𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt ((2 · 𝑁)C𝑁))), 1))    &   𝐾 = (⌊‘((2 · 𝑁) / 3))    &   𝑀 = (⌊‘(√‘(2 · 𝑁)))       (𝜑 → ((4↑𝑁) / 𝑁) < (((2 · 𝑁)↑𝑐(((√‘(2 · 𝑁)) / 3) + 2)) · (2↑𝑐(((4 · 𝑁) / 3) − 5))))

Theorembposlem7 24996* Lemma for bpos 24999. The function 𝐹 is decreasing. (Contributed by Mario Carneiro, 13-Mar-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ ((((√‘2) · (𝐺‘(√‘𝑛))) + ((9 / 4) · (𝐺‘(𝑛 / 2)))) + ((log‘2) / (√‘(2 · 𝑛)))))    &   𝐺 = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) / 𝑥))    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → (e↑2) ≤ 𝐴)    &   (𝜑 → (e↑2) ≤ 𝐵)       (𝜑 → (𝐴 < 𝐵 → (𝐹𝐵) < (𝐹𝐴)))

Theorembposlem8 24997 Lemma for bpos 24999. Evaluate 𝐹(64) and show it is less than log2. (Contributed by Mario Carneiro, 14-Mar-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ ((((√‘2) · (𝐺‘(√‘𝑛))) + ((9 / 4) · (𝐺‘(𝑛 / 2)))) + ((log‘2) / (√‘(2 · 𝑛)))))    &   𝐺 = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) / 𝑥))       ((𝐹64) ∈ ℝ ∧ (𝐹64) < (log‘2))

Theorembposlem9 24998* Lemma for bpos 24999. Derive a contradiction. (Contributed by Mario Carneiro, 14-Mar-2014.) (Proof shortened by AV, 15-Sep-2021.)
𝐹 = (𝑛 ∈ ℕ ↦ ((((√‘2) · (𝐺‘(√‘𝑛))) + ((9 / 4) · (𝐺‘(𝑛 / 2)))) + ((log‘2) / (√‘(2 · 𝑛)))))    &   𝐺 = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) / 𝑥))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑64 < 𝑁)    &   (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝𝑝 ≤ (2 · 𝑁)))       (𝜑𝜓)

Theorembpos 24999* Bertrand's postulate: there is a prime between 𝑁 and 2𝑁 for every positive integer 𝑁. This proof follows Erdős's method, for the most part, but with some refinements due to Shigenori Tochiori to save us some calculations of large primes. See http://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate for an overview of the proof strategy. This is Metamath 100 proof #98. (Contributed by Mario Carneiro, 14-Mar-2014.)
(𝑁 ∈ ℕ → ∃𝑝 ∈ ℙ (𝑁 < 𝑝𝑝 ≤ (2 · 𝑁)))

14.4.8  Quadratic residues and the Legendre symbol

If the congruence ((𝑥↑2) mod 𝑝) = (𝑛 mod 𝑝) has a solution we say that 𝑛 is a quadratic residue mod 𝑝. If the congruence has no solution we say that 𝑛 is a quadratic nonresidue mod 𝑝, see definition in [ApostolNT] p. 178. The Legendre symbol (𝑛 /L 𝑝) is defined in a way that its value is 1 if 𝑛 is a quadratic residue mod 𝑝 and -1 if 𝑛 is a quadratic nonresidue mod 𝑝 (and 0 if 𝑝 divides 𝑛), see lgsqr 25057.

Originally, the Legendre symbol (𝑁 /L 𝑃) was defined for odd primes 𝑃 only (and arbitrary integers 𝑁) by Adrien-Marie Legendre in 1798, see definition in [ApostolNT] p. 179. It was generalized to be defined for any positive odd integer by Carl Gustav Jacob Jacobi in 1837 (therefore called "Jacobi symbol" since then), see definition in [ApostolNT] p. 188. Finally, it was generalized to be defined for any integer by Leopold Kronecker in 1885 (therefore called "Kronecker symbol" since then). The definition df-lgs 25001 for the "Legendre symbol" /L is actually the definition of the "Kronecker symbol". Since only one definition (and one class symbol) are provided in set.mm, the names "Legendre symbol", "Jacobi symbol" and "Kronecker symbol" are used synonymously for /L, but mostly it is called "Legendre symbol", even if it is used in the context of a "Jacobi symbol" or "Kronecker symbol".

Syntaxclgs 25000 Extend class notation with the Legendre symbol function.
class /L

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