More operations on WithOne and WithZero

This file defines various bundled morphisms on WithOne and WithZero that were not available in Algebra/Group/WithOne/Defs.

Main definitions

• WithOne.lift, WithZero.lift
• WithOne.map, WithZero.map

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def WithZero.coeAddHom.proof_1 {α : Type u_1} [inst : Add α] : ∀ (x x_1 : α), ↑(x + x_1) = ↑(x + x_1)

Equations

• (_ : ↑(x + x) = ↑(x + x)) = (_ : ↑(x + x) = ↑(x + x))

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def WithZero.coeAddHom {α : Type u} [inst : Add α] : AddHom α (WithZero α)

WithZero.coe as a bundled morphism

Equations

• WithZero.coeAddHom = { toFun := WithZero.coe, map_add' := (_ : ∀ (x x_1 : α), ↑(x + x_1) = ↑(x + x_1)) }

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@[simp]

theorem WithOne.coeMulHom_apply {α : Type u} [inst : Mul α] : ∀ (a : α), ↑WithOne.coeMulHom a = ↑a

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@[simp]

theorem WithZero.coeAddHom_apply {α : Type u} [inst : Add α] : ∀ (a : α), ↑WithZero.coeAddHom a = ↑a

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def WithOne.coeMulHom {α : Type u} [inst : Mul α] : α →ₙ* WithOne α

WithOne.coe as a bundled morphism

Equations

• WithOne.coeMulHom = { toFun := WithOne.coe, map_mul' := (_ : ∀ (x x_1 : α), ↑(x * x_1) = ↑(x * x_1)) }

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def WithZero.lift {α : Type u} {β : Type v} [inst : Add α] [inst : AddZeroClass β] : AddHom α β ≃ (WithZero α →+ β)

Lift an add semigroup homomorphism f to a bundled add monoid homorphism.

Equations

• One or more equations did not get rendered due to their size.

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def WithZero.lift.proof_1 {α : Type u_2} {β : Type u_1} [inst : Add α] [inst : AddZeroClass β] (f : AddHom α β) : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0

Equations

• (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) = (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0)

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def WithZero.lift.proof_3 {α : Type u_1} {β : Type u_2} [inst : Add α] [inst : AddZeroClass β] (f : AddHom α β) : (fun F => AddHom.comp (↑F) WithZero.coeAddHom) ((fun f => { toZeroHom := { toFun := fun x => Option.casesOn x 0 ↑f, map_zero' := (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) }, map_add' := (_ : ∀ (x y : WithZero α), ZeroHom.toFun { toFun := fun x => Option.casesOn x 0 ↑f, map_zero' := (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) } (x + y) = ZeroHom.toFun { toFun := fun x => Option.casesOn x 0 ↑f, map_zero' := (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) } x + ZeroHom.toFun { toFun := fun x => Option.casesOn x 0 ↑f, map_zero' := (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) } y) }) f) = f

Equations

• One or more equations did not get rendered due to their size.

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def WithZero.lift.proof_4 {α : Type u_1} {β : Type u_2} [inst : Add α] [inst : AddZeroClass β] (F : WithZero α →+ β) : (fun f => { toZeroHom := { toFun := fun x => Option.casesOn x 0 ↑f, map_zero' := (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) }, map_add' := (_ : ∀ (x y : WithZero α), ZeroHom.toFun { toFun := fun x => Option.casesOn x 0 ↑f, map_zero' := (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) } (x + y) = ZeroHom.toFun { toFun := fun x => Option.casesOn x 0 ↑f, map_zero' := (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) } x + ZeroHom.toFun { toFun := fun x => Option.casesOn x 0 ↑f, map_zero' := (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) } y) }) ((fun F => AddHom.comp (↑F) WithZero.coeAddHom) F) = F

Equations

• One or more equations did not get rendered due to their size.

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def WithZero.lift.proof_2 {α : Type u_1} {β : Type u_2} [inst : Add α] [inst : AddZeroClass β] (f : AddHom α β) (x : WithZero α) (y : WithZero α) : ZeroHom.toFun { toFun := fun x => Option.casesOn x 0 ↑f, map_zero' := (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) } (x + y) = ZeroHom.toFun { toFun := fun x => Option.casesOn x 0 ↑f, map_zero' := (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) } x + ZeroHom.toFun { toFun := fun x => Option.casesOn x 0 ↑f, map_zero' := (_ : (fun x => Option.casesOn x 0 ↑f) 0 = (fun x => Option.casesOn x 0 ↑f) 0) } y

Equations

• One or more equations did not get rendered due to their size.

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def WithOne.lift {α : Type u} {β : Type v} [inst : Mul α] [inst : MulOneClass β] : (α →ₙ* β) ≃ (WithOne α →* β)

Lift a semigroup homomorphism f to a bundled monoid homorphism.

Equations

• One or more equations did not get rendered due to their size.

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@[simp]

theorem WithZero.lift_coe {α : Type u} {β : Type v} [inst : Add α] [inst : AddZeroClass β] (f : AddHom α β) (x : α) : ↑(↑WithZero.lift f) ↑x = ↑f x

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@[simp]

theorem WithOne.lift_coe {α : Type u} {β : Type v} [inst : Mul α] [inst : MulOneClass β] (f : α →ₙ* β) (x : α) : ↑(↑WithOne.lift f) ↑x = ↑f x

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theorem WithZero.lift_zero {α : Type u} {β : Type v} [inst : Add α] [inst : AddZeroClass β] (f : AddHom α β) : ↑(↑WithZero.lift f) 0 = 0

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theorem WithOne.lift_one {α : Type u} {β : Type v} [inst : Mul α] [inst : MulOneClass β] (f : α →ₙ* β) : ↑(↑WithOne.lift f) 1 = 1

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theorem WithZero.lift_unique {α : Type u} {β : Type v} [inst : Add α] [inst : AddZeroClass β] (f : WithZero α →+ β) : f = ↑WithZero.lift (AddHom.comp (↑f) WithZero.coeAddHom)

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theorem WithOne.lift_unique {α : Type u} {β : Type v} [inst : Mul α] [inst : MulOneClass β] (f : WithOne α →* β) : f = ↑WithOne.lift (MulHom.comp (↑f) WithOne.coeMulHom)

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def WithZero.map {α : Type u} {β : Type v} [inst : Add α] [inst : Add β] (f : AddHom α β) : WithZero α →+ WithZero β

Given an additive map from α → β returns an add monoid homomorphism from WithZero α to WithZero β

Equations

• WithZero.map f = ↑WithZero.lift (AddHom.comp WithZero.coeAddHom f)

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def WithOne.map {α : Type u} {β : Type v} [inst : Mul α] [inst : Mul β] (f : α →ₙ* β) : WithOne α →* WithOne β

Given a multiplicative map from α → β returns a monoid homomorphism from WithOne α to WithOne β

Equations

• WithOne.map f = ↑WithOne.lift (MulHom.comp WithOne.coeMulHom f)

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@[simp]

theorem WithZero.map_coe {α : Type u} {β : Type v} [inst : Add α] [inst : Add β] (f : AddHom α β) (a : α) : ↑(WithZero.map f) ↑a = ↑(↑f a)

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@[simp]

theorem WithOne.map_coe {α : Type u} {β : Type v} [inst : Mul α] [inst : Mul β] (f : α →ₙ* β) (a : α) : ↑(WithOne.map f) ↑a = ↑(↑f a)

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@[simp]

theorem WithZero.map_id {α : Type u} [inst : Add α] : WithZero.map (AddHom.id α) = AddMonoidHom.id (WithZero α)

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@[simp]

theorem WithOne.map_id {α : Type u} [inst : Mul α] : WithOne.map (MulHom.id α) = MonoidHom.id (WithOne α)

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theorem WithZero.map_map {α : Type u} {β : Type v} {γ : Type w} [inst : Add α] [inst : Add β] [inst : Add γ] (f : AddHom α β) (g : AddHom β γ) (x : WithZero α) : ↑(WithZero.map g) (↑(WithZero.map f) x) = ↑(WithZero.map (AddHom.comp g f)) x

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theorem WithOne.map_map {α : Type u} {β : Type v} {γ : Type w} [inst : Mul α] [inst : Mul β] [inst : Mul γ] (f : α →ₙ* β) (g : β →ₙ* γ) (x : WithOne α) : ↑(WithOne.map g) (↑(WithOne.map f) x) = ↑(WithOne.map (MulHom.comp g f)) x

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@[simp]

theorem WithZero.map_comp {α : Type u} {β : Type v} {γ : Type w} [inst : Add α] [inst : Add β] [inst : Add γ] (f : AddHom α β) (g : AddHom β γ) : WithZero.map (AddHom.comp g f) = AddMonoidHom.comp (WithZero.map g) (WithZero.map f)

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@[simp]

theorem WithOne.map_comp {α : Type u} {β : Type v} {γ : Type w} [inst : Mul α] [inst : Mul β] [inst : Mul γ] (f : α →ₙ* β) (g : β →ₙ* γ) : WithOne.map (MulHom.comp g f) = MonoidHom.comp (WithOne.map g) (WithOne.map f)

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def AddEquiv.withZeroCongr.proof_1 {α : Type u_1} {β : Type u_2} [inst : Add α] [inst : Add β] (e : α ≃+ β) : ∀ (x : WithZero α), ↑(WithZero.map (AddEquiv.toAddHom (AddEquiv.symm e))) (↑(WithZero.map (AddEquiv.toAddHom e)) x) = x

Equations

• One or more equations did not get rendered due to their size.

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def AddEquiv.withZeroCongr.proof_2 {α : Type u_2} {β : Type u_1} [inst : Add α] [inst : Add β] (e : α ≃+ β) : ∀ (x : WithZero β), ↑(WithZero.map (AddEquiv.toAddHom e)) (↑(WithZero.map (AddEquiv.toAddHom (AddEquiv.symm e))) x) = x

Equations

• One or more equations did not get rendered due to their size.

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def AddEquiv.withZeroCongr.proof_3 {α : Type u_1} {β : Type u_2} [inst : Add α] [inst : Add β] (e : α ≃+ β) (x : WithZero α) (y : WithZero α) : ZeroHom.toFun (↑(WithZero.map (AddEquiv.toAddHom e))) (x + y) = ZeroHom.toFun (↑(WithZero.map (AddEquiv.toAddHom e))) x + ZeroHom.toFun (↑(WithZero.map (AddEquiv.toAddHom e))) y

Equations

• One or more equations did not get rendered due to their size.

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def AddEquiv.withZeroCongr {α : Type u} {β : Type v} [inst : Add α] [inst : Add β] (e : α ≃+ β) : WithZero α ≃+ WithZero β

A version of Equiv.optionCongr for WithZero.

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• One or more equations did not get rendered due to their size.

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@[simp]

theorem AddEquiv.withZeroCongr_apply {α : Type u} {β : Type v} [inst : Add α] [inst : Add β] (e : α ≃+ β) (a : WithZero α) : ↑(AddEquiv.withZeroCongr e) a = ↑(WithZero.map (AddEquiv.toAddHom e)) a

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@[simp]

theorem MulEquiv.withOneCongr_apply {α : Type u} {β : Type v} [inst : Mul α] [inst : Mul β] (e : α ≃* β) (a : WithOne α) : ↑(MulEquiv.withOneCongr e) a = ↑(WithOne.map (MulEquiv.toMulHom e)) a

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def MulEquiv.withOneCongr {α : Type u} {β : Type v} [inst : Mul α] [inst : Mul β] (e : α ≃* β) : WithOne α ≃* WithOne β

A version of Equiv.optionCongr for WithOne.

Equations

• One or more equations did not get rendered due to their size.

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@[simp]

theorem AddEquiv.withZeroCongr_refl {α : Type u} [inst : Add α] : AddEquiv.withZeroCongr (AddEquiv.refl α) = AddEquiv.refl (WithZero α)

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@[simp]

theorem MulEquiv.withOneCongr_refl {α : Type u} [inst : Mul α] : MulEquiv.withOneCongr (MulEquiv.refl α) = MulEquiv.refl (WithOne α)

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@[simp]

theorem AddEquiv.withZeroCongr_symm {α : Type u} {β : Type v} [inst : Add α] [inst : Add β] (e : α ≃+ β) : AddEquiv.symm (AddEquiv.withZeroCongr e) = AddEquiv.withZeroCongr (AddEquiv.symm e)

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@[simp]

theorem MulEquiv.withOneCongr_symm {α : Type u} {β : Type v} [inst : Mul α] [inst : Mul β] (e : α ≃* β) : MulEquiv.symm (MulEquiv.withOneCongr e) = MulEquiv.withOneCongr (MulEquiv.symm e)

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@[simp]

theorem AddEquiv.withZeroCongr_trans {α : Type u} {β : Type v} {γ : Type w} [inst : Add α] [inst : Add β] [inst : Add γ] (e₁ : α ≃+ β) (e₂ : β ≃+ γ) : AddEquiv.trans (AddEquiv.withZeroCongr e₁) (AddEquiv.withZeroCongr e₂) = AddEquiv.withZeroCongr (AddEquiv.trans e₁ e₂)

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@[simp]

theorem MulEquiv.withOneCongr_trans {α : Type u} {β : Type v} {γ : Type w} [inst : Mul α] [inst : Mul β] [inst : Mul γ] (e₁ : α ≃* β) (e₂ : β ≃* γ) : MulEquiv.trans (MulEquiv.withOneCongr e₁) (MulEquiv.withOneCongr e₂) = MulEquiv.withOneCongr (MulEquiv.trans e₁ e₂)