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Theorem List for Metamath Proof Explorer - 25001-25100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremperfect 25001* 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 25002 Extend class notation with the group of Dirichlet characters.
class DChr

Definitiondf-dchr 25003* 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 25004* 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 25005* 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 25006 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 25007* 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 25008* 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 25009* 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 25010 Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)       (𝑋𝐷𝑁 ∈ ℕ)

Theoremdchrmhm 25011 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 25012 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 25013* 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 25014 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 25015 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 25016 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 25017 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 25018 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 25019 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 25020 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 25021* 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 25022* 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 25023* 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 25024 The set of Dirichlet characters is an Abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)       (𝑁 ∈ ℕ → 𝐺 ∈ Abel)

Theoremdchrfi 25025 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 25026 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 25027 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 25028* 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 25029 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 25030 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 25031 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 25032 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 25033 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 25034* Lemma for dchrpt 25037. (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 25035* Lemma for dchrpt 25037. (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 25036* Lemma for dchrpt 25037. (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 25037* 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 25038* 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 25039* 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 25040* Lemma for sumdchr 25042. (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 25041 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 25042* 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 25043* 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 25044* 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 25045 Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.)
(𝑁 ∈ ℕ0 → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘𝑁) · (!‘𝑁))))

Theorempcbcctr 25046* 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 25047 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 25048 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 25049 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 25050 A bound on the binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.)
(𝑁 ∈ (ℤ‘4) → ((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁))

Theoremefexple 25051 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 25052* Lemma for bpos1 25053. (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 25053* 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 25054 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 25055 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 25056* Lemma for bpos 25063. Since the binomial coefficient does not have any primes in the range (2𝑁 / 3, 𝑁] or (2𝑁, +∞) by bposlem2 25055 and prmfac1 15478, 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 25057* Lemma for bpos 25063. (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 25058* Lemma for bpos 25063. 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 25059* Lemma for bpos 25063. 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 25060* Lemma for bpos 25063. 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 25061 Lemma for bpos 25063. 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 25062* Lemma for bpos 25063. 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 25063* 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 25121.

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 25065 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 25064 Extend class notation with the Legendre symbol function.
class /L

Definitiondf-lgs 25065* 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 25066 {-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 25067 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 25068 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 25069* 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 25070* Lemma for lgsfcl2 25073. (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 25071* 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 25072* 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 25073* The function 𝐹 is closed in integers with absolute value less than 1 (namely {-1, 0, 1}, see zabsle1 25066). (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 25074* 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 25075* 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 25076* 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 25077* Lemma for lgsval2 25083. (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 25078* Lemma for lgsval4 25087. (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 25079* The Legendre symbol is an integer with absolute value less 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 25080 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 25081 The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℤ)

Theoremlgsle1 25082 The Legendre symbol has absolute value less or equal to 1. Together with lgscl 25081 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 25083 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)))

Theoremlgs2 25084 The Legendre symbol at 2. (Contributed by Mario Carneiro, 4-Feb-2015.)
(𝐴 ∈ ℤ → (𝐴 /L 2) = if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)))

Theoremlgsval3 25085 The Legendre symbol at an odd prime (this is the traditional domain of the Legendre symbol). (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝐴 /L 𝑃) = ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))

Theoremlgsvalmod 25086 The Legendre symbol is equivalent to 𝑎↑((𝑝 − 1) / 2), mod 𝑝. This theorem is also called "Euler's criterion", see theorem 9.2 in [ApostolNT] p. 180, or a representation of Euler's criterion using the Legendre symbol, see also lgsqr 25121. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) mod 𝑃) = ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃))

Theoremlgsval4 25087* Restate lgsval 25071 for nonzero 𝑁, where the function 𝐹 has been abbreviated into a self-referential expression taking the value of /L on the primes as given. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1))       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L 𝑁) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))))

Theoremlgsfcl3 25088* 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, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1))       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶ℤ)

Theoremlgsval4a 25089* Same as lgsval4 25087 for positive 𝑁. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1))       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 /L 𝑁) = (seq1( · , 𝐹)‘𝑁))

Theoremlgscl1 25090 The value of the Legendre symbol is either -1 or 0 or 1. (Contributed by AV, 13-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ {-1, 0, 1})

Theoremlgsneg 25091 The Legendre symbol is either even or odd under negation with respect to the second parameter according to the sign of the first. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L -𝑁) = (if(𝐴 < 0, -1, 1) · (𝐴 /L 𝑁)))

Theoremlgsneg1 25092 The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℕ0𝑁 ∈ ℤ) → (𝐴 /L -𝑁) = (𝐴 /L 𝑁))

Theoremlgsmod 25093 The Legendre (Jacobi) symbol is preserved under reduction mod 𝑛 when 𝑛 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → ((𝐴 mod 𝑁) /L 𝑁) = (𝐴 /L 𝑁))

Theoremlgsdilem 25094 Lemma for lgsdi 25104 and lgsdir 25102: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → if((𝑁 < 0 ∧ (𝐴 · 𝐵) < 0), -1, 1) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · if((𝑁 < 0 ∧ 𝐵 < 0), -1, 1)))

Theoremlgsdir2lem1 25095 Lemma for lgsdir2 25100. (Contributed by Mario Carneiro, 4-Feb-2015.)
(((1 mod 8) = 1 ∧ (-1 mod 8) = 7) ∧ ((3 mod 8) = 3 ∧ (-3 mod 8) = 5))

Theoremlgsdir2lem2 25096 Lemma for lgsdir2 25100. (Contributed by Mario Carneiro, 4-Feb-2015.)
(𝐾 ∈ ℤ ∧ 2 ∥ (𝐾 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝐾) → (𝐴 mod 8) ∈ 𝑆)))    &   𝑀 = (𝐾 + 1)    &   𝑁 = (𝑀 + 1)    &   𝑁𝑆       (𝑁 ∈ ℤ ∧ 2 ∥ (𝑁 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆)))

Theoremlgsdir2lem3 25097 Lemma for lgsdir2 25100. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → (𝐴 mod 8) ∈ ({1, 7} ∪ {3, 5}))

Theoremlgsdir2lem4 25098 Lemma for lgsdir2 25100. (Contributed by Mario Carneiro, 4-Feb-2015.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 mod 8) ∈ {1, 7}) → (((𝐴 · 𝐵) mod 8) ∈ {1, 7} ↔ (𝐵 mod 8) ∈ {1, 7}))

Theoremlgsdir2lem5 25099 Lemma for lgsdir2 25100. (Contributed by Mario Carneiro, 4-Feb-2015.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ {1, 7})

Theoremlgsdir2 25100 The Legendre symbol is completely multiplicative at 2. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 · 𝐵) /L 2) = ((𝐴 /L 2) · (𝐵 /L 2)))

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