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Type | Label | Description |
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Statement | ||
Theorem | dchrelbas4 27301* | A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of ℂ, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐿 = (ℤRHom‘𝑍) ⇒ ⊢ (𝑋 ∈ 𝐷 ↔ (𝑁 ∈ ℕ ∧ 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ ℤ (1 < (𝑥 gcd 𝑁) → (𝑋‘(𝐿‘𝑥)) = 0))) | ||
Theorem | dchrzrh1 27302 | 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) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1) | ||
Theorem | dchrzrhcl 27303 | 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) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑋‘(𝐿‘𝐴)) ∈ ℂ) | ||
Theorem | dchrzrhmul 27304 | 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) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) | ||
Theorem | dchrplusg 27305 | 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) & ⊢ · = (+g‘𝐺) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → · = ( ∘f · ↾ (𝐷 × 𝐷))) | ||
Theorem | dchrmul 27306 | 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) & ⊢ · = (+g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 ∘f · 𝑌)) | ||
Theorem | dchrmulcl 27307 | 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) & ⊢ · = (+g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐷) | ||
Theorem | dchrn0 27308 | A Dirichlet character is nonzero on the units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 ↔ 𝐴 ∈ 𝑈)) | ||
Theorem | dchr1cl 27309* | 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 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → 1 ∈ 𝐷) | ||
Theorem | dchrmullid 27310* | 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 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) & ⊢ · = (+g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) | ||
Theorem | dchrinvcl 27311* | 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 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) & ⊢ · = (+g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ 𝐾 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0)) ⇒ ⊢ (𝜑 → (𝐾 ∈ 𝐷 ∧ (𝐾 · 𝑋) = 1 )) | ||
Theorem | dchrabl 27312 | The set of Dirichlet characters is an Abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) | ||
Theorem | dchrfi 27313 | 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) | ||
Theorem | dchrghm 27314 | A Dirichlet character restricted to the unit group of ℤ/nℤ 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})) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀)) | ||
Theorem | dchr1 27315 | Value of the principal Dirichlet character. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 1 = (0g‘𝐺) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → ( 1 ‘𝐴) = 1) | ||
Theorem | dchreq 27316* | 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) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘))) | ||
Theorem | dchrresb 27317 | 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) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → ((𝑋 ↾ 𝑈) = (𝑌 ↾ 𝑈) ↔ 𝑋 = 𝑌)) | ||
Theorem | dchrabs 27318 | A Dirichlet character takes values on the unit circle. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → (abs‘(𝑋‘𝐴)) = 1) | ||
Theorem | dchrinv 27319 | 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) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) = (∗ ∘ 𝑋)) | ||
Theorem | dchrabs2 27320 | 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) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → (abs‘(𝑋‘𝐴)) ≤ 1) | ||
Theorem | dchr1re 27321 | The principal Dirichlet character is a real character. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 1 = (0g‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → 1 :𝐵⟶ℝ) | ||
Theorem | dchrptlem1 27322* | Lemma for dchrpt 27325. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 1 = (1r‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ≠ 1 ) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ 𝐻 = ((mulGrp‘𝑍) ↾s 𝑈) & ⊢ · = (.g‘𝐻) & ⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘)))) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑈) & ⊢ (𝜑 → 𝐻dom DProd 𝑆) & ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈) & ⊢ 𝑃 = (𝐻dProj𝑆) & ⊢ 𝑂 = (od‘𝐻) & ⊢ 𝑇 = (-1↑𝑐(2 / (𝑂‘(𝑊‘𝐼)))) & ⊢ (𝜑 → 𝐼 ∈ dom 𝑊) & ⊢ (𝜑 → ((𝑃‘𝐼)‘𝐴) ≠ 1 ) & ⊢ 𝑋 = (𝑢 ∈ 𝑈 ↦ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)))) ⇒ ⊢ (((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) → (𝑋‘𝐶) = (𝑇↑𝑀)) | ||
Theorem | dchrptlem2 27323* | Lemma for dchrpt 27325. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 1 = (1r‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ≠ 1 ) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ 𝐻 = ((mulGrp‘𝑍) ↾s 𝑈) & ⊢ · = (.g‘𝐻) & ⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘)))) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑈) & ⊢ (𝜑 → 𝐻dom DProd 𝑆) & ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈) & ⊢ 𝑃 = (𝐻dProj𝑆) & ⊢ 𝑂 = (od‘𝐻) & ⊢ 𝑇 = (-1↑𝑐(2 / (𝑂‘(𝑊‘𝐼)))) & ⊢ (𝜑 → 𝐼 ∈ dom 𝑊) & ⊢ (𝜑 → ((𝑃‘𝐼)‘𝐴) ≠ 1 ) & ⊢ 𝑋 = (𝑢 ∈ 𝑈 ↦ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1) | ||
Theorem | dchrptlem3 27324* | Lemma for dchrpt 27325. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 1 = (1r‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ≠ 1 ) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ 𝐻 = ((mulGrp‘𝑍) ↾s 𝑈) & ⊢ · = (.g‘𝐻) & ⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘)))) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑈) & ⊢ (𝜑 → 𝐻dom DProd 𝑆) & ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1) | ||
Theorem | dchrpt 27325* | 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 = (1r‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ≠ 1 ) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1) | ||
Theorem | dchrsum2 27326* | 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 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ 𝑈 = (Unit‘𝑍) ⇒ ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0)) | ||
Theorem | dchrsum 27327* | 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 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ 𝐵 = (Base‘𝑍) ⇒ ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0)) | ||
Theorem | sumdchr2 27328* | Lemma for sumdchr 27330. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 1 = (1r‘𝑍) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)) | ||
Theorem | dchrhash 27329 | 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)) | ||
Theorem | sumdchr 27330* | 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 = (1r‘𝑍) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (ϕ‘𝑁), 0)) | ||
Theorem | dchr2sum 27331* | 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) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎))) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0)) | ||
Theorem | sum2dchr 27332* | 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) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0)) | ||
Theorem | bcctr 27333 | Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ (𝑁 ∈ ℕ0 → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘𝑁) · (!‘𝑁)))) | ||
Theorem | pcbcctr 27334* | 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) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))))) | ||
Theorem | bcmono 27335 | The binomial coefficient is monotone in its second argument, up to the midway point. (Contributed by Mario Carneiro, 5-Mar-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐵 ≤ (𝑁 / 2)) → (𝑁C𝐴) ≤ (𝑁C𝐵)) | ||
Theorem | bcmax 27336 | The binomial coefficient takes its maximum value at the center. (Contributed by Mario Carneiro, 5-Mar-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → ((2 · 𝑁)C𝐾) ≤ ((2 · 𝑁)C𝑁)) | ||
Theorem | bcp1ctr 27337 | 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))))) | ||
Theorem | bclbnd 27338 | A bound on the binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) |
⊢ (𝑁 ∈ (ℤ≥‘4) → ((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁)) | ||
Theorem | efexple 27339 | 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+) → ((𝐴↑𝑁) ≤ 𝐵 ↔ 𝑁 ≤ (⌊‘((log‘𝐵) / (log‘𝐴))))) | ||
Theorem | bpos1lem 27340* | Lemma for bpos1 27341. (Contributed by Mario Carneiro, 12-Mar-2014.) |
⊢ (∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)) → 𝜑) & ⊢ (𝑁 ∈ (ℤ≥‘𝑃) → 𝜑) & ⊢ 𝑃 ∈ ℙ & ⊢ 𝐴 ∈ ℕ0 & ⊢ (𝐴 · 2) = 𝐵 & ⊢ 𝐴 < 𝑃 & ⊢ (𝑃 < 𝐵 ∨ 𝑃 = 𝐵) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝐴) → 𝜑) | ||
Theorem | bpos1 27341* | 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 ∧ 𝑁 ≤ ;64) → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) | ||
Theorem | bposlem1 27342 | 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)) | ||
Theorem | bposlem2 27343 | 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.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 2 < 𝑃) & ⊢ (𝜑 → ((2 · 𝑁) / 3) < 𝑃) & ⊢ (𝜑 → 𝑃 ≤ 𝑁) ⇒ ⊢ (𝜑 → (𝑃 pCnt ((2 · 𝑁)C𝑁)) = 0) | ||
Theorem | bposlem3 27344* | Lemma for bpos 27351. Since the binomial coefficient does not have any primes in the range (2𝑁 / 3, 𝑁] or (2𝑁, +∞) by bposlem2 27343 and prmfac1 16753, 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.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘5)) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (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)) | ||
Theorem | bposlem4 27345* | Lemma for bpos 27351. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘5)) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt ((2 · 𝑁)C𝑁))), 1)) & ⊢ 𝐾 = (⌊‘((2 · 𝑁) / 3)) & ⊢ 𝑀 = (⌊‘(√‘(2 · 𝑁))) ⇒ ⊢ (𝜑 → 𝑀 ∈ (3...𝐾)) | ||
Theorem | bposlem5 27346* | Lemma for bpos 27351. 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.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘5)) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (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))) | ||
Theorem | bposlem6 27347* | Lemma for bpos 27351. 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.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘5)) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (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)))) | ||
Theorem | bposlem7 27348* | Lemma for bpos 27351. 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)) & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → (e↑2) ≤ 𝐴) & ⊢ (𝜑 → (e↑2) ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 → (𝐹‘𝐵) < (𝐹‘𝐴))) | ||
Theorem | bposlem8 27349 | Lemma for bpos 27351. 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)) ⇒ ⊢ ((𝐹‘;64) ∈ ℝ ∧ (𝐹‘;64) < (log‘2)) | ||
Theorem | bposlem9 27350* | Lemma for bpos 27351. 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)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ;64 < 𝑁) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | bpos 27351* | 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 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) | ||
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 27409. 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 27353 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". | ||
Syntax | clgs 27352 | Extend class notation with the Legendre symbol function. |
class /L | ||
Definition | df-lgs 27353* | 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))))) | ||
Theorem | zabsle1 27354 | {-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)) | ||
Theorem | lgslem1 27355 | 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}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2}) | ||
Theorem | lgslem2 27356 | 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) | ||
Theorem | lgslem3 27357* | 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} ⇒ ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑍) → (𝐴 · 𝐵) ∈ 𝑍) | ||
Theorem | lgslem4 27358* | Lemma for lgsfcl2 27361. (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) | ||
Theorem | lgsval 27359* | 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))))) | ||
Theorem | lgsfval 27360* | 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 → (𝐹‘𝑀) = 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)) | ||
Theorem | lgsfcl2 27361* | The function 𝐹 is closed in integers with absolute value less than 1 (namely {-1, 0, 1}, see zabsle1 27354). (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) | ||
Theorem | lgscllem 27362* | 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) | ||
Theorem | lgsfcl 27363* | 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) | ||
Theorem | lgsfle1 27364* | 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‘(𝐹‘𝑀)) ≤ 1) | ||
Theorem | lgsval2lem 27365* | Lemma for lgsval2 27371. (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))) | ||
Theorem | lgsval4lem 27366* | Lemma for lgsval4 27375. (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))) | ||
Theorem | lgscl2 27367* | 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) | ||
Theorem | lgs0 27368 | 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)) | ||
Theorem | lgscl 27369 | The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℤ) | ||
Theorem | lgsle1 27370 | The Legendre symbol has absolute value less than or equal to 1. Together with lgscl 27369 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) | ||
Theorem | lgsval2 27371 | 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))) | ||
Theorem | lgs2 27372 | 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))) | ||
Theorem | lgsval3 27373 | 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)) | ||
Theorem | lgsvalmod 27374 | 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 27409. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) mod 𝑃) = ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃)) | ||
Theorem | lgsval4 27375* | Restate lgsval 27359 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)))) | ||
Theorem | lgsfcl3 27376* | 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) | ||
Theorem | lgsval4a 27377* | Same as lgsval4 27375 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)) | ||
Theorem | lgscl1 27378 | 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}) | ||
Theorem | lgsneg 27379 | 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))) | ||
Theorem | lgsneg1 27380 | The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → (𝐴 /L -𝑁) = (𝐴 /L 𝑁)) | ||
Theorem | lgsmod 27381 | 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)) | ||
Theorem | lgsdilem 27382 | Lemma for lgsdi 27392 and lgsdir 27390: 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))) | ||
Theorem | lgsdir2lem1 27383 | Lemma for lgsdir2 27388. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (((1 mod 8) = 1 ∧ (-1 mod 8) = 7) ∧ ((3 mod 8) = 3 ∧ (-3 mod 8) = 5)) | ||
Theorem | lgsdir2lem2 27384 | Lemma for lgsdir2 27388. (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) & ⊢ 𝑁 ∈ 𝑆 ⇒ ⊢ (𝑁 ∈ ℤ ∧ 2 ∥ (𝑁 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆))) | ||
Theorem | lgsdir2lem3 27385 | Lemma for lgsdir2 27388. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → (𝐴 mod 8) ∈ ({1, 7} ∪ {3, 5})) | ||
Theorem | lgsdir2lem4 27386 | Lemma for lgsdir2 27388. (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})) | ||
Theorem | lgsdir2lem5 27387 | Lemma for lgsdir2 27388. (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}) | ||
Theorem | lgsdir2 27388 | 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))) | ||
Theorem | lgsdirprm 27389 | The Legendre symbol is completely multiplicative at the primes. See theorem 9.3 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 18-Mar-2022.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑃 ∈ ℙ) → ((𝐴 · 𝐵) /L 𝑃) = ((𝐴 /L 𝑃) · (𝐵 /L 𝑃))) | ||
Theorem | lgsdir 27390 | The Legendre symbol is completely multiplicative in its left argument. Generalization of theorem 9.9(a) in [ApostolNT] p. 188 (which assumes that 𝐴 and 𝐵 are odd positive integers). Together with lgsqr 27409 this implies that the product of two quadratic residues or nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → ((𝐴 · 𝐵) /L 𝑁) = ((𝐴 /L 𝑁) · (𝐵 /L 𝑁))) | ||
Theorem | lgsdilem2 27391* | Lemma for lgsdi 27392. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ≠ 0) & ⊢ (𝜑 → 𝑁 ≠ 0) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑀)), 1)) ⇒ ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) = (seq1( · , 𝐹)‘(abs‘(𝑀 · 𝑁)))) | ||
Theorem | lgsdi 27392 | The Legendre symbol is completely multiplicative in its right argument. Generalization of theorem 9.9(b) in [ApostolNT] p. 188 (which assumes that 𝑀 and 𝑁 are odd positive integers). (Contributed by Mario Carneiro, 5-Feb-2015.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝐴 /L (𝑀 · 𝑁)) = ((𝐴 /L 𝑀) · (𝐴 /L 𝑁))) | ||
Theorem | lgsne0 27393 | The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 /L 𝑁) ≠ 0 ↔ (𝐴 gcd 𝑁) = 1)) | ||
Theorem | lgsabs1 27394 | The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ (𝐴 gcd 𝑁) = 1)) | ||
Theorem | lgssq 27395 | The Legendre symbol at a square is equal to 1. Together with lgsmod 27381 this implies that the Legendre symbol takes value 1 at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.) (Revised by AV, 20-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑2) /L 𝑁) = 1) | ||
Theorem | lgssq2 27396 | The Legendre symbol at a square is equal to 1. (Contributed by Mario Carneiro, 5-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝑁) = 1) → (𝐴 /L (𝑁↑2)) = 1) | ||
Theorem | lgsprme0 27397 | The Legendre symbol at any prime (even at 2) is 0 iff the prime does not divide the first argument. See definition in [ApostolNT] p. 179. (Contributed by AV, 20-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ ℙ) → ((𝐴 /L 𝑃) = 0 ↔ (𝐴 mod 𝑃) = 0)) | ||
Theorem | 1lgs 27398 | The Legendre symbol at 1. See example 1 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ (𝑁 ∈ ℤ → (1 /L 𝑁) = 1) | ||
Theorem | lgs1 27399 | The Legendre symbol at 1. See definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ (𝐴 ∈ ℤ → (𝐴 /L 1) = 1) | ||
Theorem | lgsmodeq 27400 | The Legendre (Jacobi) symbol is preserved under reduction mod 𝑛 when 𝑛 is odd. Theorem 9.9(c) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → ((𝐴 mod 𝑁) = (𝐵 mod 𝑁) → (𝐴 /L 𝑁) = (𝐵 /L 𝑁))) |
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