P. 272 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27101-27200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dchrfi 27101	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 27102	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 27103	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 27104*	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 27105	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 27106	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 27107	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 27108	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 27109	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 27110*	Lemma for dchrpt 27113. (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 27111*	Lemma for dchrpt 27113. (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 27112*	Lemma for dchrpt 27113. (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 27113*	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 27114*	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 27115*	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 27116*	Lemma for sumdchr 27118. (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 27117	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 27118*	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 27119*	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 27120*	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))
14.4.7  Bertrand's postulate
Theorem	bcctr 27121	Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.)
⊢ (𝑁 ∈ ℕ0 → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘𝑁) · (!‘𝑁))))
Theorem	pcbcctr 27122*	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 27123	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 27124	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 27125	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 27126	A bound on the binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.)
⊢ (𝑁 ∈ (ℤ≥‘4) → ((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁))
Theorem	efexple 27127	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 27128*	Lemma for bpos1 27129. (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 27129*	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 27130	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 27131	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 27132*	Lemma for bpos 27139. Since the binomial coefficient does not have any primes in the range (2𝑁 / 3, 𝑁] or (2𝑁, +∞) by bposlem2 27131 and prmfac1 16665, 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 27133*	Lemma for bpos 27139. (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 27134*	Lemma for bpos 27139. 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 27135*	Lemma for bpos 27139. 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 27136*	Lemma for bpos 27139. 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 27137	Lemma for bpos 27139. 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 27138*	Lemma for bpos 27139. 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 27139*	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)))
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 27197. 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 27141 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 27140	Extend class notation with the Legendre symbol function.
class /L
Definition	df-lgs 27141*	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 27142	{-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 27143	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 27144	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 27145*	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 27146*	Lemma for lgsfcl2 27149. (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 27147*	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 27148*	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 27149*	The function 𝐹 is closed in integers with absolute value less than 1 (namely {-1, 0, 1}, see zabsle1 27142). (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 27150*	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 27151*	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 27152*	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 27153*	Lemma for lgsval2 27159. (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 27154*	Lemma for lgsval4 27163. (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 27155*	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 27156	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 27157	The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℤ)
Theorem	lgsle1 27158	The Legendre symbol has absolute value less than or equal to 1. Together with lgscl 27157 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 27159	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 27160	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 27161	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 27162	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 27197. (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 27163*	Restate lgsval 27147 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 27164*	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 27165*	Same as lgsval4 27163 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 27166	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 27167	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 27168	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 27169	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 27170	Lemma for lgsdi 27180 and lgsdir 27178: 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 27171	Lemma for lgsdir2 27176. (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 27172	Lemma for lgsdir2 27176. (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 27173	Lemma for lgsdir2 27176. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → (𝐴 mod 8) ∈ ({1, 7} ∪ {3, 5}))
Theorem	lgsdir2lem4 27174	Lemma for lgsdir2 27176. (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 27175	Lemma for lgsdir2 27176. (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 27176	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 27177	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 27178	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 27197 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 27179*	Lemma for lgsdi 27180. (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 27180	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 27181	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 27182	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 27183	The Legendre symbol at a square is equal to 1. Together with lgsmod 27169 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 27184	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 27185	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 27186	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 27187	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 27188	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)))
Theorem	lgsmulsqcoprm 27189	The Legendre (Jacobi) symbol is preserved under multiplication with a square of an integer coprime to the second argument. Theorem 9.9(d) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) → (((𝐴↑2) · 𝐵) /L 𝑁) = (𝐵 /L 𝑁))
Theorem	lgsdirnn0 27190	Variation on lgsdir 27178 valid for all 𝐴, 𝐵 but only for positive 𝑁. (The exact location of the failure of this law is for 𝐴 = 0, 𝐵 < 0, 𝑁 = -1 in which case (0 /L -1) = 1 but (𝐵 /L -1) = -1.) (Contributed by Mario Carneiro, 28-Apr-2016.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝐴 · 𝐵) /L 𝑁) = ((𝐴 /L 𝑁) · (𝐵 /L 𝑁)))
Theorem	lgsdinn0 27191	Variation on lgsdi 27180 valid for all 𝑀, 𝑁 but only for positive 𝐴. (The exact location of the failure of this law is for 𝐴 = -1, 𝑀 = 0, and some 𝑁 in which case (-1 /L 0) = 1 but (-1 /L 𝑁) = -1 when -1 is not a quadratic residue mod 𝑁.) (Contributed by Mario Carneiro, 28-Apr-2016.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L (𝑀 · 𝑁)) = ((𝐴 /L 𝑀) · (𝐴 /L 𝑁)))
Theorem	lgsqrlem1 27192	Lemma for lgsqr 27197. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑃)    &   ⊢ 𝑆 = (Poly1‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐷 = ( deg1 ‘𝑌)    &   ⊢ 𝑂 = (eval1‘𝑌)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ 𝑋 = (var1‘𝑌)    &   ⊢ − = (-g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    &   ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 )    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃) = (1 mod 𝑃))    ⇒   ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0g‘𝑌))
Theorem	lgsqrlem2 27193*	Lemma for lgsqr 27197. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑃)    &   ⊢ 𝑆 = (Poly1‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐷 = ( deg1 ‘𝑌)    &   ⊢ 𝑂 = (eval1‘𝑌)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ 𝑋 = (var1‘𝑌)    &   ⊢ − = (-g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    &   ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 )    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐺 = (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(𝑦↑2)))    ⇒   ⊢ (𝜑 → 𝐺:(1...((𝑃 − 1) / 2))–1-1→(◡(𝑂‘𝑇) “ {(0g‘𝑌)}))
Theorem	lgsqrlem3 27194*	Lemma for lgsqr 27197. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑃)    &   ⊢ 𝑆 = (Poly1‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐷 = ( deg1 ‘𝑌)    &   ⊢ 𝑂 = (eval1‘𝑌)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ 𝑋 = (var1‘𝑌)    &   ⊢ − = (-g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    &   ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 )    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐺 = (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(𝑦↑2)))    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → (𝐴 /L 𝑃) = 1)    ⇒   ⊢ (𝜑 → (𝐿‘𝐴) ∈ (◡(𝑂‘𝑇) “ {(0g‘𝑌)}))
Theorem	lgsqrlem4 27195*	Lemma for lgsqr 27197. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑃)    &   ⊢ 𝑆 = (Poly1‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐷 = ( deg1 ‘𝑌)    &   ⊢ 𝑂 = (eval1‘𝑌)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ 𝑋 = (var1‘𝑌)    &   ⊢ − = (-g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    &   ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 )    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐺 = (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(𝑦↑2)))    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → (𝐴 /L 𝑃) = 1)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℤ 𝑃 ∥ ((𝑥↑2) − 𝐴))
Theorem	lgsqrlem5 27196*	Lemma for lgsqr 27197. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ (𝐴 /L 𝑃) = 1) → ∃𝑥 ∈ ℤ 𝑃 ∥ ((𝑥↑2) − 𝐴))
Theorem	lgsqr 27197*	The Legendre symbol for odd primes is 1 iff the number is not a multiple of the prime (in which case it is 0, see lgsne0 27181) and the number is a quadratic residue mod 𝑃 (it is -1 for nonresidues by the process of elimination from lgsabs1 27182). Given our definition of the Legendre symbol, this theorem is equivalent to Euler's criterion. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) = 1 ↔ (¬ 𝑃 ∥ 𝐴 ∧ ∃𝑥 ∈ ℤ 𝑃 ∥ ((𝑥↑2) − 𝐴))))
Theorem	lgsqrmod 27198*	If the Legendre symbol of an integer for an odd prime is 1, then the number is a quadratic residue mod 𝑃. (Contributed by AV, 20-Aug-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) = 1 → ∃𝑥 ∈ ℤ ((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃)))
Theorem	lgsqrmodndvds 27199*	If the Legendre symbol of an integer 𝐴 for an odd prime is 1, then the number is a quadratic residue mod 𝑃 with a solution 𝑥 of the congruence (𝑥↑2)≡𝐴 (mod 𝑃) which is not divisible by the prime. (Contributed by AV, 20-Aug-2021.) (Proof shortened by AV, 18-Mar-2022.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) = 1 → ∃𝑥 ∈ ℤ (((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃) ∧ ¬ 𝑃 ∥ 𝑥)))
Theorem	lgsdchrval 27200*	The Legendre symbol function 𝑋(𝑚) = (𝑚 /L 𝑁), where 𝑁 is an odd positive number, is a Dirichlet character modulo 𝑁. (Contributed by Mario Carneiro, 28-Apr-2016.)
⊢ 𝐺 = (DChr‘𝑁)    &   ⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐷 = (Base‘𝐺)    &   ⊢ 𝐵 = (Base‘𝑍)    &   ⊢ 𝐿 = (ℤRHom‘𝑍)    &   ⊢ 𝑋 = (𝑦 ∈ 𝐵 ↦ (℩ℎ∃𝑚 ∈ ℤ (𝑦 = (𝐿‘𝑚) ∧ ℎ = (𝑚 /L 𝑁))))    ⇒   ⊢ (((𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝐴 ∈ ℤ) → (𝑋‘(𝐿‘𝐴)) = (𝐴 /L 𝑁))