Semirings and rings #

This file gives lemmas about semirings, rings and domains. This is analogous to Algebra.Group.Basic, the difference being that the former is about + and * separately, while the present file is about their interaction.

For the definitions of semirings and rings see Algebra.Ring.Defs.

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@[simp]

theorem AddHom.mulLeft_apply {R : Type u_1} [Distrib R] (r : R) :

↑(AddHom.mulLeft r) = (fun x x_1 => x * x_1) r

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def AddHom.mulLeft {R : Type u_1} [Distrib R] (r : R) :

AddHom R R

Left multiplication by an element of a type with distributive multiplication is an AddHom.

Equations

AddHom.mulLeft r = { toFun := (fun x x_1 => x * x_1) r, map_add' := (_ : ∀ (b c : R), r * (b + c) = r * b + r * c) }

Instances For

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@[simp]

theorem AddHom.mulRight_apply {R : Type u_1} [Distrib R] (r : R) :

↑(AddHom.mulRight r) = fun a => a * r

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def AddHom.mulRight {R : Type u_1} [Distrib R] (r : R) :

AddHom R R

Left multiplication by an element of a type with distributive multiplication is an AddHom.

Equations

AddHom.mulRight r = { toFun := fun a => a * r, map_add' := (_ : ∀ (x x_1 : R), (x + x_1) * r = x * r + x_1 * r) }

Instances For

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@[simp, deprecated]

theorem map_bit0 {α : Type u_3} {β : Type u_2} {F : Type u_1} [NonAssocSemiring α] [NonAssocSemiring β] [AddHomClass F α β] (f : F) (a : α) :

↑f (bit0 a) = bit0 (↑f a)

Additive homomorphisms preserve bit0.

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def AddMonoidHom.mulLeft {R : Type u_1} [NonUnitalNonAssocSemiring R] (r : R) :

R →+ R

Left multiplication by an element of a (semi)ring is an AddMonoidHom

Equations

AddMonoidHom.mulLeft r = { toZeroHom := { toFun := fun x => r * x, map_zero' := (_ : r * 0 = 0) }, map_add' := (_ : ∀ (b c : R), r * (b + c) = r * b + r * c) }

Instances For

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@[simp]

theorem AddMonoidHom.coe_mulLeft {R : Type u_1} [NonUnitalNonAssocSemiring R] (r : R) :

↑(AddMonoidHom.mulLeft r) = HMul.hMul r

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def AddMonoidHom.mulRight {R : Type u_1} [NonUnitalNonAssocSemiring R] (r : R) :

R →+ R

Right multiplication by an element of a (semi)ring is an AddMonoidHom

Equations

AddMonoidHom.mulRight r = { toZeroHom := { toFun := fun a => a * r, map_zero' := (_ : 0 * r = 0) }, map_add' := (_ : ∀ (x x_1 : R), (x + x_1) * r = x * r + x_1 * r) }

Instances For

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@[simp]

theorem AddMonoidHom.coe_mulRight {R : Type u_1} [NonUnitalNonAssocSemiring R] (r : R) :

↑(AddMonoidHom.mulRight r) = fun x => x * r

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theorem AddMonoidHom.mulRight_apply {R : Type u_1} [NonUnitalNonAssocSemiring R] (a : R) (r : R) :

↑(AddMonoidHom.mulRight r) a = a * r

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instance hasDistribNeg {α : Type u_1} [Mul α] [HasDistribNeg α] :

HasDistribNeg αᵐᵒᵖ

Equations

hasDistribNeg = let src := MulOpposite.involutiveNeg α; HasDistribNeg.mk (_ : ∀ (x x_1 : αᵐᵒᵖ), -x * x_1 = -(x * x_1)) (_ : ∀ (x x_1 : αᵐᵒᵖ), x * -x_1 = -(x * x_1))

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@[simp]

theorem inv_neg' {α : Type u_1} [Group α] [HasDistribNeg α] (a : α) :

(-a)⁻¹ = -a⁻¹

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theorem vieta_formula_quadratic {α : Type u_1} [NonUnitalCommRing α] {b : α} {c : α} {x : α} (h : x * x - b * x + c = 0) :

∃ y, y * y - b * y + c = 0 ∧ x + y = b ∧ x * y = c

Vieta's formula for a quadratic equation, relating the coefficients of the polynomial with its roots. This particular version states that if we have a root x of a monic quadratic polynomial, then there is another root y such that x + y is negative the a_1 coefficient and x * y is the a_0 coefficient.

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