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Theorem List for Metamath Proof Explorer - 25801-25900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfsumvma2 25801* Apply fsumvma 25800 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 25802* The logarithmic analogue of pcprod 16229. 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 25803* 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 25804* 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 25805* 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 25806* 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 25807 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 25808 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 25809 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 25810 Show the stronger statement log(𝑥!) = 𝑥log𝑥𝑥 + 𝑂(log𝑥) alluded to in logfacrlim 25811. (Contributed by Mario Carneiro, 20-May-2016.)
((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ≤ ((log‘𝐴) + 1))

Theoremlogfacrlim 25811 Combine the estimates logfacubnd 25808 and logfaclbnd 25809, 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 25812* 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 25813* Write out logfacrlim 25811 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 25814 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 25815 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 25816 Lemma for perfect 25818. (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 25817 Lemma for perfect 25818. (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 25818* 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 25819 Extend class notation with the group of Dirichlet characters.
class DChr

Definitiondf-dchr 25820* 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 25821* 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 25822* 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 25823 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 25824* 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 25825* 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 25826* 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 25827 Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)       (𝑋𝐷𝑁 ∈ ℕ)

Theoremdchrmhm 25828 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 25829 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 25830* 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 25831 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 25832 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 25833 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 25834 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 25835 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 25836 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 25837 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 25838* 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 25839* 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 25840* 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 25841 The set of Dirichlet characters is an Abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)       (𝑁 ∈ ℕ → 𝐺 ∈ Abel)

Theoremdchrfi 25842 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 25843 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 25844 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 25845* 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 25846 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 25847 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 25848 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 25849 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 25850 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 25851* Lemma for dchrpt 25854. (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 25852* Lemma for dchrpt 25854. (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 25853* Lemma for dchrpt 25854. (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 25854* 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 25855* 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 25856* 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 25857* Lemma for sumdchr 25859. (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 25858 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 25859* 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 25860* 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 25861* 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 25862 Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.)
(𝑁 ∈ ℕ0 → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘𝑁) · (!‘𝑁))))

Theorempcbcctr 25863* 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 25864 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 25865 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 25866 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 25867 A bound on the binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.)
(𝑁 ∈ (ℤ‘4) → ((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁))

Theoremefexple 25868 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 25869* Lemma for bpos1 25870. (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 25870* 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 25871 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 25872 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 25873* Lemma for bpos 25880. Since the binomial coefficient does not have any primes in the range (2𝑁 / 3, 𝑁] or (2𝑁, +∞) by bposlem2 25872 and prmfac1 16061, 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 25874* Lemma for bpos 25880. (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 25875* Lemma for bpos 25880. 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 25876* Lemma for bpos 25880. 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 25877* Lemma for bpos 25880. 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 25878 Lemma for bpos 25880. 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 25879* Lemma for bpos 25880. 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 25880* 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 25938.

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 25882 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 25881 Extend class notation with the Legendre symbol function.
class /L

Definitiondf-lgs 25882* Define the Legendre symbol (actually the Kronecker symbol, which extends the Legendre symbol to all integers, and also the Jacobi symbol, which restricts the Kronecker symbol to positive odd integers). See definition in [ApostolNT] p. 179 resp. definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 4-Feb-2015.)
/L = (𝑎 ∈ ℤ, 𝑛 ∈ ℤ ↦ if(𝑛 = 0, if((𝑎↑2) = 1, 1, 0), (if((𝑛 < 0 ∧ 𝑎 < 0), -1, 1) · (seq1( · , (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, (if(𝑚 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑚 − 1) / 2)) + 1) mod 𝑚) − 1))↑(𝑚 pCnt 𝑛)), 1)))‘(abs‘𝑛)))))

Theoremzabsle1 25883 {-1, 0, 1} is the set of all integers with absolute value at most 1. (Contributed by AV, 13-Jul-2021.)
(𝑍 ∈ ℤ → (𝑍 ∈ {-1, 0, 1} ↔ (abs‘𝑍) ≤ 1))

Theoremlgslem1 25884 When 𝑎 is coprime to the prime 𝑝, 𝑎↑((𝑝 − 1) / 2) is equivalent mod 𝑝 to 1 or -1, and so adding 1 makes it equivalent to 0 or 2. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2})

Theoremlgslem2 25885 The set 𝑍 of all integers with absolute value at most 1 contains {-1, 0, 1}. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}       (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍)

Theoremlgslem3 25886* The set 𝑍 of all integers with absolute value at most 1 is closed under multiplication. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}       ((𝐴𝑍𝐵𝑍) → (𝐴 · 𝐵) ∈ 𝑍)

Theoremlgslem4 25887* Lemma for lgsfcl2 25890. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 19-Mar-2022.)
𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}       ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍)

Theoremlgsval 25888* Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) = if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))))

Theoremlgsfval 25889* Value of the function 𝐹 which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))       (𝑀 ∈ ℕ → (𝐹𝑀) = if(𝑀 ∈ ℙ, (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)), 1))

Theoremlgsfcl2 25890* The function 𝐹 is closed in integers with absolute value less than 1 (namely {-1, 0, 1}, see zabsle1 25883). (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))    &   𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶𝑍)

Theoremlgscllem 25891* The Legendre symbol is an element of 𝑍. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))    &   𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ 𝑍)

Theoremlgsfcl 25892* Closure of the function 𝐹 which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶ℤ)

Theoremlgsfle1 25893* The function 𝐹 has magnitude less or equal to 1. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))       (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑀 ∈ ℕ) → (abs‘(𝐹𝑀)) ≤ 1)

Theoremlgsval2lem 25894* Lemma for lgsval2 25900. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℙ) → (𝐴 /L 𝑁) = if(𝑁 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑁 − 1) / 2)) + 1) mod 𝑁) − 1)))

Theoremlgsval4lem 25895* Lemma for lgsval4 25904. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))

Theoremlgscl2 25896* The Legendre symbol is an integer with absolute value less than or equal to 1. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ 𝑍)

Theoremlgs0 25897 The Legendre symbol when the second argument is zero. (Contributed by Mario Carneiro, 4-Feb-2015.)
(𝐴 ∈ ℤ → (𝐴 /L 0) = if((𝐴↑2) = 1, 1, 0))

Theoremlgscl 25898 The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℤ)

Theoremlgsle1 25899 The Legendre symbol has absolute value less than or equal to 1. Together with lgscl 25898 this implies that it takes values in {-1, 0, 1}. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴 /L 𝑁)) ≤ 1)

Theoremlgsval2 25900 The Legendre symbol at a prime (this is the traditional domain of the Legendre symbol, except for the addition of prime 2). (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ 𝑃 ∈ ℙ) → (𝐴 /L 𝑃) = if(𝑃 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1)))

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