Documentation

Mathlib.GroupTheory.GroupAction.Hom

Equivariant homomorphisms #

Main definitions #

The above types have corresponding classes:

Notation #

We introduce the following notation to code equivariant maps (the subscript index is for equivariant) :

When M = N and φ = MonoidHom.id M, we provide the backward compatible notation :

structure MulActionHom {M : Type u_2} {N : Type u_3} (φ : MN) (X : Type u_5) [SMul M X] (Y : Type u_6) [SMul N Y] :
Type (max u_5 u_6)

Equivariant functions : When φ : M → N is a function, and types X and Y are endowed with actions of M and N, a function f : X → Y is φ-equivariant if f (m • x) = (φ m) • (f x).

  • toFun : XY

    The underlying function.

  • map_smul' : ∀ (m : M) (x : X), self.toFun (m x) = φ m self.toFun x

    The proposition that the function commutes with the actions.

theorem MulActionHom.map_smul' {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] (self : X →ₑ[φ] Y) (m : M) (x : X) :
self.toFun (m x) = φ m self.toFun x

The proposition that the function commutes with the actions.

φ-equivariant functions X → Y, where φ : M → N, where M and N act on X and Y respectively

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

M-equivariant functions X → Y with respect to the action of M

This is the same as X →ₑ[@id M] Y

Equations
  • One or more equations did not get rendered due to their size.
class MulActionSemiHomClass (F : Type u_8) {M : outParam (Type u_9)} {N : outParam (Type u_10)} (φ : outParam (MN)) (X : outParam (Type u_11)) (Y : outParam (Type u_12)) [SMul M X] [SMul N Y] [FunLike F X Y] :

MulActionSemiHomClass F φ X Y states that F is a type of morphisms which are φ-equivariant.

You should extend this class when you extend MulActionHom.

  • map_smulₛₗ : ∀ (f : F) (c : M) (x : X), f (c x) = φ c f x

    The proposition that the function preserves the action.

Instances
    theorem MulActionSemiHomClass.map_smulₛₗ {F : Type u_8} {M : outParam (Type u_9)} {N : outParam (Type u_10)} {φ : outParam (MN)} {X : outParam (Type u_11)} {Y : outParam (Type u_12)} :
    ∀ {inst : SMul M X} {inst_1 : SMul N Y} {inst_2 : FunLike F X Y} [self : MulActionSemiHomClass F φ X Y] (f : F) (c : M) (x : X), f (c x) = φ c f x

    The proposition that the function preserves the action.

    @[reducible, inline]
    abbrev MulActionHomClass (F : Type u_8) (M : outParam (Type u_9)) (X : outParam (Type u_10)) (Y : outParam (Type u_11)) [SMul M X] [SMul M Y] [FunLike F X Y] :

    MulActionHomClass F M X Y states that F is a type of morphisms which are equivariant with respect to actions of M This is an abbreviation of MulActionSemiHomClass.

    Equations
    instance instFunLikeMulActionHom {M : Type u_2} {N : Type u_3} (φ : MN) (X : Type u_5) [SMul M X] (Y : Type u_6) [SMul N Y] :
    FunLike (X →ₑ[φ] Y) X Y
    Equations
    @[simp]
    theorem map_smul {F : Type u_8} {M : Type u_9} {X : Type u_10} {Y : Type u_11} [SMul M X] [SMul M Y] [FunLike F X Y] [MulActionHomClass F M X Y] (f : F) (c : M) (x : X) :
    f (c x) = c f x
    instance instMulActionSemiHomClassMulActionHom {M : Type u_2} {N : Type u_3} (φ : MN) (X : Type u_5) [SMul M X] (Y : Type u_6) [SMul N Y] :
    Equations
    • =
    def MulActionSemiHomClass.toMulActionHom {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {F : Type u_8} [FunLike F X Y] [MulActionSemiHomClass F φ X Y] (f : F) :
    X →ₑ[φ] Y

    Turn an element of a type F satisfying MulActionSemiHomClass F φ X Y into an actual MulActionHom. This is declared as the default coercion from F to MulActionSemiHom φ X Y.

    Equations
    • f = { toFun := f, map_smul' := }
    instance MulActionHom.instCoeTCOfMulActionSemiHomClass {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {F : Type u_8} [FunLike F X Y] [MulActionSemiHomClass F φ X Y] :
    CoeTC F (X →ₑ[φ] Y)

    Any type satisfying MulActionSemiHomClass can be cast into MulActionHom via MulActionHomSemiClass.toMulActionHom.

    Equations
    • MulActionHom.instCoeTCOfMulActionSemiHomClass = { coe := MulActionSemiHomClass.toMulActionHom }
    theorem IsScalarTower.smulHomClass (M' : Type u_1) (X : Type u_5) [SMul M' X] (Y : Type u_6) [SMul M' Y] (F : Type u_8) [FunLike F X Y] [MulOneClass X] [SMul X Y] [IsScalarTower M' X Y] [MulActionHomClass F X X Y] :

    If Y/X/M forms a scalar tower, any map X → Y preserving X-action also preserves M-action.

    theorem MulActionHom.map_smul {M' : Type u_1} {X : Type u_5} [SMul M' X] {Y : Type u_6} [SMul M' Y] (f : X →ₑ[id] Y) (m : M') (x : X) :
    f (m x) = m f x
    theorem MulActionHom.ext_iff {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {f : X →ₑ[φ] Y} {g : X →ₑ[φ] Y} :
    f = g ∀ (x : X), f x = g x
    theorem MulActionHom.ext {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {f : X →ₑ[φ] Y} {g : X →ₑ[φ] Y} :
    (∀ (x : X), f x = g x)f = g
    theorem MulActionHom.congr_fun {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {f : X →ₑ[φ] Y} {g : X →ₑ[φ] Y} (h : f = g) (x : X) :
    f x = g x
    def MulActionHom.ofEq {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {φ' : MN} (h : φ = φ') (f : X →ₑ[φ] Y) :
    X →ₑ[φ'] Y

    Two equal maps on scalars give rise to an equivariant map for identity

    Equations
    @[simp]
    theorem MulActionHom.ofEq_coe {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {φ' : MN} (h : φ = φ') (f : X →ₑ[φ] Y) :
    (MulActionHom.ofEq h f).toFun = f.toFun
    @[simp]
    theorem MulActionHom.ofEq_apply {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {φ' : MN} (h : φ = φ') (f : X →ₑ[φ] Y) (a : X) :
    (MulActionHom.ofEq h f) a = f a
    def MulActionHom.id (M : Type u_2) {X : Type u_5} [SMul M X] :
    X →ₑ[id] X

    The identity map as an equivariant map.

    Equations
    @[simp]
    theorem MulActionHom.id_apply {M : Type u_2} {X : Type u_5} [SMul M X] (x : X) :
    def MulActionHom.comp {M : Type u_2} {N : Type u_3} {P : Type u_4} {φ : MN} {ψ : NP} {χ : MP} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {Z : Type u_7} [SMul P Z] (g : Y →ₑ[ψ] Z) (f : X →ₑ[φ] Y) [κ : CompTriple φ ψ χ] :
    X →ₑ[χ] Z

    Composition of two equivariant maps.

    Equations
    • g.comp f = { toFun := g f, map_smul' := }
    @[simp]
    theorem MulActionHom.comp_apply {M : Type u_2} {N : Type u_3} {P : Type u_4} {φ : MN} {ψ : NP} {χ : MP} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {Z : Type u_7} [SMul P Z] (g : Y →ₑ[ψ] Z) (f : X →ₑ[φ] Y) [CompTriple φ ψ χ] (x : X) :
    (g.comp f) x = g (f x)
    @[simp]
    theorem MulActionHom.id_comp {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] (f : X →ₑ[φ] Y) :
    (MulActionHom.id N).comp f = f
    @[simp]
    theorem MulActionHom.comp_id {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] (f : X →ₑ[φ] Y) :
    f.comp (MulActionHom.id M) = f
    @[simp]
    theorem MulActionHom.comp_assoc {M : Type u_2} {N : Type u_3} {P : Type u_4} {φ : MN} {ψ : NP} {χ : MP} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {Z : Type u_7} [SMul P Z] {Q : Type u_8} {T : Type u_9} [SMul Q T] {η : PQ} {θ : MQ} {ζ : NQ} (h : Z →ₑ[η] T) (g : Y →ₑ[ψ] Z) (f : X →ₑ[φ] Y) [CompTriple φ ψ χ] [CompTriple χ η θ] [CompTriple ψ η ζ] [CompTriple φ ζ θ] :
    h.comp (g.comp f) = (h.comp g).comp f
    @[simp]
    theorem MulActionHom.inverse_apply {M : Type u_2} {X : Type u_5} [SMul M X] {Y₁ : Type u_8} [SMul M Y₁] (f : X →ₑ[id] Y₁) (g : Y₁X) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
    ∀ (a : Y₁), (f.inverse g h₁ h₂) a = g a
    def MulActionHom.inverse {M : Type u_2} {X : Type u_5} [SMul M X] {Y₁ : Type u_8} [SMul M Y₁] (f : X →ₑ[id] Y₁) (g : Y₁X) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
    Y₁ →ₑ[id] X

    The inverse of a bijective equivariant map is equivariant.

    Equations
    • f.inverse g h₁ h₂ = { toFun := g, map_smul' := }
    @[simp]
    theorem MulActionHom.inverse'_apply {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {φ' : NM} (f : X →ₑ[φ] Y) (g : YX) (k : Function.RightInverse φ' φ) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
    ∀ (a : Y), (f.inverse' g k h₁ h₂) a = g a
    def MulActionHom.inverse' {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {φ' : NM} (f : X →ₑ[φ] Y) (g : YX) (k : Function.RightInverse φ' φ) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
    Y →ₑ[φ'] X

    The inverse of a bijective equivariant map is equivariant.

    Equations
    • f.inverse' g k h₁ h₂ = { toFun := g, map_smul' := }
    theorem MulActionHom.inverse_eq_inverse' {M : Type u_2} {X : Type u_5} [SMul M X] {Y₁ : Type u_8} [SMul M Y₁] (f : X →ₑ[id] Y₁) (g : Y₁X) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
    f.inverse g h₁ h₂ = f.inverse' g h₁ h₂
    theorem MulActionHom.inverse'_inverse' {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {φ' : NM} {f : X →ₑ[φ] Y} {g : YX} {k₁ : Function.LeftInverse φ' φ} {k₂ : Function.RightInverse φ' φ} {h₁ : Function.LeftInverse g f} {h₂ : Function.RightInverse g f} :
    (f.inverse' g k₂ h₁ h₂).inverse' (⇑f) k₁ h₂ h₁ = f
    theorem MulActionHom.comp_inverse' {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {φ' : NM} {f : X →ₑ[φ] Y} {g : YX} {k₁ : Function.LeftInverse φ' φ} {k₂ : Function.RightInverse φ' φ} {h₁ : Function.LeftInverse g f} {h₂ : Function.RightInverse g f} :
    (f.inverse' g k₂ h₁ h₂).comp f = MulActionHom.id M
    theorem MulActionHom.inverse'_comp {M : Type u_2} {N : Type u_3} {φ : MN} {X : Type u_5} [SMul M X] {Y : Type u_6} [SMul N Y] {φ' : NM} {f : X →ₑ[φ] Y} {g : YX} {k₂ : Function.RightInverse φ' φ} {h₁ : Function.LeftInverse g f} {h₂ : Function.RightInverse g f} :
    f.comp (f.inverse' g k₂ h₁ h₂) = MulActionHom.id N
    @[simp]
    theorem SMulCommClass.toMulActionHom_apply {M : Type u_11} (N : Type u_9) (α : Type u_10) [SMul M α] [SMul N α] [SMulCommClass M N α] (c : M) :
    ∀ (x : α), (SMulCommClass.toMulActionHom N α c) x = c x
    def SMulCommClass.toMulActionHom {M : Type u_11} (N : Type u_9) (α : Type u_10) [SMul M α] [SMul N α] [SMulCommClass M N α] (c : M) :
    α →ₑ[id] α

    If actions of M and N on α commute, then for c : M, (c • · : α → α) is an N-action homomorphism.

    Equations
    structure DistribMulActionHom {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] (φ : M →* N) (A : Type u_4) [AddMonoid A] [DistribMulAction M A] (B : Type u_5) [AddMonoid B] [DistribMulAction N B] extends MulActionHom :
    Type (max u_4 u_5)

    Equivariant additive monoid homomorphisms.

    • toFun : AB
    • map_smul' : ∀ (m : M) (x : A), self.toFun (m x) = φ m self.toFun x
    • map_zero' : self.toFun 0 = 0

      The proposition that the function preserves 0

    • map_add' : ∀ (x y : A), self.toFun (x + y) = self.toFun x + self.toFun y

      The proposition that the function preserves addition

    @[reducible]
    abbrev DistribMulActionHom.toAddMonoidHom {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (self : A →ₑ+[φ] B) :
    A →+ B

    Reinterpret an equivariant additive monoid homomorphism as an additive monoid homomorphism.

    Equations
    • self.toAddMonoidHom = { toFun := self.toFun, map_zero' := , map_add' := }

    Equivariant additive monoid homomorphisms.

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

    Equivariant additive monoid homomorphisms.

    Equations
    • One or more equations did not get rendered due to their size.
    class DistribMulActionSemiHomClass (F : Type u_10) {M : outParam (Type u_11)} {N : outParam (Type u_12)} (φ : outParam (MN)) (A : outParam (Type u_13)) (B : outParam (Type u_14)) [Monoid M] [Monoid N] [AddMonoid A] [AddMonoid B] [DistribMulAction M A] [DistribMulAction N B] [FunLike F A B] extends MulActionSemiHomClass , AddMonoidHomClass :

    DistribMulActionSemiHomClass F φ A B states that F is a type of morphisms preserving the additive monoid structure and equivariant with respect to φ. You should extend this class when you extend DistribMulActionSemiHom.

      Instances
        @[reducible, inline]
        abbrev DistribMulActionHomClass (F : Type u_10) (M : outParam (Type u_11)) (A : outParam (Type u_12)) (B : outParam (Type u_13)) [Monoid M] [AddMonoid A] [AddMonoid B] [DistribMulAction M A] [DistribMulAction M B] [FunLike F A B] :

        DistribMulActionHomClass F M A B states that F is a type of morphisms preserving the additive monoid structure and equivariant with respect to the action of M. It is an abbreviation to DistribMulActionHomClass F (MonoidHom.id M) A B You should extend this class when you extend DistribMulActionHom.

        Equations
        instance DistribMulActionHom.instFunLike {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] (φ : M →* N) (A : Type u_4) [AddMonoid A] [DistribMulAction M A] (B : Type u_5) [AddMonoid B] [DistribMulAction N B] :
        FunLike (A →ₑ+[φ] B) A B
        Equations
        instance DistribMulActionHom.instDistribMulActionSemiHomClassCoeMonoidHom {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] (φ : M →* N) (A : Type u_4) [AddMonoid A] [DistribMulAction M A] (B : Type u_5) [AddMonoid B] [DistribMulAction N B] :
        Equations
        • =
        def DistribMulActionSemiHomClass.toDistribMulActionHom {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {F : Type u_10} [FunLike F A B] [DistribMulActionSemiHomClass F (⇑φ) A B] (f : F) :
        A →ₑ+[φ] B

        Turn an element of a type F satisfying MulActionHomClass F M X Y into an actual MulActionHom. This is declared as the default coercion from F to MulActionHom M X Y.

        Equations
        • f = { toFun := (↑f).toFun, map_smul' := , map_zero' := , map_add' := }
        instance DistribMulActionHom.instCoeTCOfDistribMulActionSemiHomClassCoeMonoidHom {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {F : Type u_10} [FunLike F A B] [DistribMulActionSemiHomClass F (⇑φ) A B] :
        CoeTC F (A →ₑ+[φ] B)

        Any type satisfying MulActionHomClass can be cast into MulActionHom via MulActionHomClass.toMulActionHom.

        Equations
        • DistribMulActionHom.instCoeTCOfDistribMulActionSemiHomClassCoeMonoidHom = { coe := DistribMulActionSemiHomClass.toDistribMulActionHom }
        @[simp]
        theorem SMulCommClass.toDistribMulActionHom_toFun {M : Type u_13} (N : Type u_11) (A : Type u_12) [Monoid N] [AddMonoid A] [DistribSMul M A] [DistribMulAction N A] [SMulCommClass M N A] (c : M) :
        ∀ (x : A), (SMulCommClass.toDistribMulActionHom N A c) x = c x
        def SMulCommClass.toDistribMulActionHom {M : Type u_13} (N : Type u_11) (A : Type u_12) [Monoid N] [AddMonoid A] [DistribSMul M A] [DistribMulAction N A] [SMulCommClass M N A] (c : M) :
        A →+[N] A

        If DistribMulAction of M and N on A commute, then for each c : M, (c • ·) is an N-action additive homomorphism.

        Equations
        @[simp]
        theorem DistribMulActionHom.toFun_eq_coe {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B) :
        f.toFun = f
        theorem DistribMulActionHom.coe_fn_coe {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B) :
        f = f
        theorem DistribMulActionHom.coe_fn_coe' {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B) :
        f = f
        theorem DistribMulActionHom.ext_iff {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {f : A →ₑ+[φ] B} {g : A →ₑ+[φ] B} :
        f = g ∀ (x : A), f x = g x
        theorem DistribMulActionHom.ext {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {f : A →ₑ+[φ] B} {g : A →ₑ+[φ] B} :
        (∀ (x : A), f x = g x)f = g
        theorem DistribMulActionHom.congr_fun {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {f : A →ₑ+[φ] B} {g : A →ₑ+[φ] B} (h : f = g) (x : A) :
        f x = g x
        theorem DistribMulActionHom.toMulActionHom_injective {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {f : A →ₑ+[φ] B} {g : A →ₑ+[φ] B} (h : f = g) :
        f = g
        theorem DistribMulActionHom.toAddMonoidHom_injective {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {f : A →ₑ+[φ] B} {g : A →ₑ+[φ] B} (h : f = g) :
        f = g
        theorem DistribMulActionHom.map_zero {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B) :
        f 0 = 0
        theorem DistribMulActionHom.map_add {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B) (x : A) (y : A) :
        f (x + y) = f x + f y
        theorem DistribMulActionHom.map_neg {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} (A' : Type u_8) [AddGroup A'] [DistribMulAction M A'] (B' : Type u_9) [AddGroup B'] [DistribMulAction N B'] (f : A' →ₑ+[φ] B') (x : A') :
        f (-x) = -f x
        theorem DistribMulActionHom.map_sub {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} (A' : Type u_8) [AddGroup A'] [DistribMulAction M A'] (B' : Type u_9) [AddGroup B'] [DistribMulAction N B'] (f : A' →ₑ+[φ] B') (x : A') (y : A') :
        f (x - y) = f x - f y
        theorem DistribMulActionHom.map_smulₑ {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B) (m : M) (x : A) :
        f (m x) = φ m f x
        def DistribMulActionHom.id (M : Type u_1) [Monoid M] {A : Type u_4} [AddMonoid A] [DistribMulAction M A] :
        A →+[M] A

        The identity map as an equivariant additive monoid homomorphism.

        Equations
        @[simp]
        theorem DistribMulActionHom.id_apply (M : Type u_1) [Monoid M] {A : Type u_4} [AddMonoid A] [DistribMulAction M A] (x : A) :
        instance DistribMulActionHom.instZero {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] :
        Zero (A →ₑ+[φ] B)
        Equations
        • DistribMulActionHom.instZero = { zero := let __src := 0; { toFun := (↑__src).toFun, map_smul' := , map_zero' := , map_add' := } }
        instance DistribMulActionHom.instOneId {M : Type u_1} [Monoid M] {A : Type u_4} [AddMonoid A] [DistribMulAction M A] :
        One (A →+[M] A)
        Equations
        @[simp]
        theorem DistribMulActionHom.coe_zero {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] :
        0 = 0
        @[simp]
        theorem DistribMulActionHom.coe_one {M : Type u_1} [Monoid M] {A : Type u_4} [AddMonoid A] [DistribMulAction M A] :
        1 = id
        theorem DistribMulActionHom.zero_apply {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (a : A) :
        0 a = 0
        theorem DistribMulActionHom.one_apply {M : Type u_1} [Monoid M] {A : Type u_4} [AddMonoid A] [DistribMulAction M A] (a : A) :
        1 a = a
        instance DistribMulActionHom.instInhabited {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] :
        Equations
        • DistribMulActionHom.instInhabited = { default := 0 }
        def DistribMulActionHom.comp {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {P : Type u_3} [Monoid P] {φ : M →* N} {ψ : N →* P} {χ : M →* P} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {C : Type u_7} [AddMonoid C] [DistribMulAction P C] (g : B →ₑ+[ψ] C) (f : A →ₑ+[φ] B) [κ : φ.CompTriple ψ χ] :
        A →ₑ+[χ] C

        Composition of two equivariant additive monoid homomorphisms.

        Equations
        • g.comp f = { toMulActionHom := (↑g).comp f, map_zero' := , map_add' := }
        @[simp]
        theorem DistribMulActionHom.comp_apply {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {P : Type u_3} [Monoid P] {φ : M →* N} {ψ : N →* P} {χ : M →* P} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {C : Type u_7} [AddMonoid C] [DistribMulAction P C] (g : B →ₑ+[ψ] C) (f : A →ₑ+[φ] B) [φ.CompTriple ψ χ] (x : A) :
        (g.comp f) x = g (f x)
        @[simp]
        theorem DistribMulActionHom.id_comp {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B) :
        @[simp]
        theorem DistribMulActionHom.comp_id {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B) :
        @[simp]
        theorem DistribMulActionHom.comp_assoc {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {P : Type u_3} [Monoid P] {φ : M →* N} {ψ : N →* P} {χ : M →* P} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {C : Type u_7} [AddMonoid C] [DistribMulAction P C] {Q : Type u_11} {D : Type u_12} [Monoid Q] [AddMonoid D] [DistribMulAction Q D] {η : P →* Q} {θ : M →* Q} {ζ : N →* Q} (h : C →ₑ+[η] D) (g : B →ₑ+[ψ] C) (f : A →ₑ+[φ] B) [φ.CompTriple ψ χ] [χ.CompTriple η θ] [ψ.CompTriple η ζ] [φ.CompTriple ζ θ] :
        h.comp (g.comp f) = (h.comp g).comp f
        @[simp]
        theorem DistribMulActionHom.inverse_toFun {M : Type u_1} [Monoid M] {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B₁ : Type u_6} [AddMonoid B₁] [DistribMulAction M B₁] (f : A →+[M] B₁) (g : B₁A) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
        ∀ (a : B₁), (f.inverse g h₁ h₂) a = g a
        def DistribMulActionHom.inverse {M : Type u_1} [Monoid M] {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B₁ : Type u_6} [AddMonoid B₁] [DistribMulAction M B₁] (f : A →+[M] B₁) (g : B₁A) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
        B₁ →+[M] A

        The inverse of a bijective DistribMulActionHom is a DistribMulActionHom.

        Equations
        • f.inverse g h₁ h₂ = { toFun := g, map_smul' := , map_zero' := , map_add' := }
        theorem DistribMulActionHom.ext_ring_iff {R : Type u_11} [Semiring R] {S : Type u_12} [Semiring S] {N' : Type u_14} [AddMonoid N'] [DistribMulAction S N'] {σ : R →* S} {f : R →ₑ+[σ] N'} {g : R →ₑ+[σ] N'} :
        f = g f 1 = g 1
        theorem DistribMulActionHom.ext_ring {R : Type u_11} [Semiring R] {S : Type u_12} [Semiring S] {N' : Type u_14} [AddMonoid N'] [DistribMulAction S N'] {σ : R →* S} {f : R →ₑ+[σ] N'} {g : R →ₑ+[σ] N'} (h : f 1 = g 1) :
        f = g
        structure MulSemiringActionHom {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] (φ : M →* N) (R : Type u_10) [Semiring R] [MulSemiringAction M R] (S : Type u_12) [Semiring S] [MulSemiringAction N S] extends DistribMulActionHom :
        Type (max u_10 u_12)

        Equivariant ring homomorphisms.

        • toFun : RS
        • map_smul' : ∀ (m : M) (x : R), self.toFun (m x) = φ m self.toFun x
        • map_zero' : self.toFun 0 = 0
        • map_add' : ∀ (x y : R), self.toFun (x + y) = self.toFun x + self.toFun y
        • map_one' : self.toFun 1 = 1

          The proposition that the function preserves 1

        • map_mul' : ∀ (x y : R), self.toFun (x * y) = self.toFun x * self.toFun y

          The proposition that the function preserves multiplication

        @[reducible]
        abbrev MulSemiringActionHom.toRingHom {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (self : R →ₑ+*[φ] S) :
        R →+* S

        Reinterpret an equivariant ring homomorphism as a ring homomorphism.

        Equations
        • self.toRingHom = { toFun := self.toFun, map_one' := , map_mul' := , map_zero' := , map_add' := }

        Equivariant ring homomorphisms.

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

        Equivariant ring homomorphisms.

        Equations
        • One or more equations did not get rendered due to their size.
        class MulSemiringActionSemiHomClass (F : Type u_15) {M : outParam (Type u_16)} {N : outParam (Type u_17)} [Monoid M] [Monoid N] (φ : outParam (MN)) (R : outParam (Type u_18)) (S : outParam (Type u_19)) [Semiring R] [Semiring S] [DistribMulAction M R] [DistribMulAction N S] [FunLike F R S] extends DistribMulActionSemiHomClass , MonoidHomClass :

        MulSemiringActionHomClass F φ R S states that F is a type of morphisms preserving the ring structure and equivariant with respect to φ.

        You should extend this class when you extend MulSemiringActionHom.

          Instances
            @[reducible, inline]
            abbrev MulSemiringActionHomClass (F : Type u_15) {M : outParam (Type u_16)} [Monoid M] (R : outParam (Type u_17)) (S : outParam (Type u_18)) [Semiring R] [Semiring S] [DistribMulAction M R] [DistribMulAction M S] [FunLike F R S] :

            MulSemiringActionHomClass F M R S states that F is a type of morphisms preserving the ring structure and equivariant with respect to a DistribMulActionof M on R and S .

            Equations
            instance MulSemiringActionHom.instFunLike {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] (φ : M →* N) (R : Type u_10) [Semiring R] [MulSemiringAction M R] (S : Type u_12) [Semiring S] [MulSemiringAction N S] :
            FunLike (R →ₑ+*[φ] S) R S
            Equations
            Equations
            • =
            def MulSemiringActionHomClass.toMulSemiringActionHom {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] {F : Type u_15} [FunLike F R S] [MulSemiringActionSemiHomClass F (⇑φ) R S] (f : F) :

            Turn an element of a type F satisfying MulSemiringActionHomClass F M R S into an actual MulSemiringActionHom. This is declared as the default coercion from F to MulSemiringActionHom M X Y.

            Equations
            • f = { toFun := (↑f).toFun, map_smul' := , map_zero' := , map_add' := , map_one' := , map_mul' := }
            instance MulSemiringActionHom.instCoeTCOfMulSemiringActionSemiHomClassCoeMonoidHom {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] {F : Type u_15} [FunLike F R S] [MulSemiringActionSemiHomClass F (⇑φ) R S] :
            CoeTC F (R →ₑ+*[φ] S)

            Any type satisfying MulSemiringActionHomClass can be cast into MulSemiringActionHom via MulSemiringActionHomClass.toMulSemiringActionHom.

            Equations
            • MulSemiringActionHom.instCoeTCOfMulSemiringActionSemiHomClassCoeMonoidHom = { coe := MulSemiringActionHomClass.toMulSemiringActionHom }
            theorem MulSemiringActionHom.coe_fn_coe {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) :
            f = f
            theorem MulSemiringActionHom.coe_fn_coe' {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) :
            f = f
            theorem MulSemiringActionHom.ext_iff {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] {f : R →ₑ+*[φ] S} {g : R →ₑ+*[φ] S} :
            f = g ∀ (x : R), f x = g x
            theorem MulSemiringActionHom.ext {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] {f : R →ₑ+*[φ] S} {g : R →ₑ+*[φ] S} :
            (∀ (x : R), f x = g x)f = g
            theorem MulSemiringActionHom.map_zero {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) :
            f 0 = 0
            theorem MulSemiringActionHom.map_add {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) (x : R) (y : R) :
            f (x + y) = f x + f y
            theorem MulSemiringActionHom.map_neg {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} (R' : Type u_11) [Ring R'] [MulSemiringAction M R'] (S' : Type u_13) [Ring S'] [MulSemiringAction N S'] (f : R' →ₑ+*[φ] S') (x : R') :
            f (-x) = -f x
            theorem MulSemiringActionHom.map_sub {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} (R' : Type u_11) [Ring R'] [MulSemiringAction M R'] (S' : Type u_13) [Ring S'] [MulSemiringAction N S'] (f : R' →ₑ+*[φ] S') (x : R') (y : R') :
            f (x - y) = f x - f y
            theorem MulSemiringActionHom.map_one {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) :
            f 1 = 1
            theorem MulSemiringActionHom.map_mul {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) (x : R) (y : R) :
            f (x * y) = f x * f y
            theorem MulSemiringActionHom.map_smulₛₗ {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) (m : M) (x : R) :
            f (m x) = φ m f x
            theorem MulSemiringActionHom.map_smul {M : Type u_1} [Monoid M] {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction M S] (f : R →+*[M] S) (m : M) (x : R) :
            f (m x) = m f x
            def MulSemiringActionHom.id (M : Type u_1) [Monoid M] {R : Type u_10} [Semiring R] [MulSemiringAction M R] :
            R →+*[M] R

            The identity map as an equivariant ring homomorphism.

            Equations
            @[simp]
            theorem MulSemiringActionHom.id_apply (M : Type u_1) [Monoid M] {R : Type u_10} [Semiring R] [MulSemiringAction M R] (x : R) :
            def MulSemiringActionHom.comp {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {P : Type u_3} [Monoid P] {φ : M →* N} {ψ : N →* P} {χ : M →* P} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] {T : Type u_14} [Semiring T] [MulSemiringAction P T] (g : S →ₑ+*[ψ] T) (f : R →ₑ+*[φ] S) [κ : φ.CompTriple ψ χ] :

            Composition of two equivariant additive ring homomorphisms.

            Equations
            • g.comp f = { toDistribMulActionHom := (↑g).comp f, map_one' := , map_mul' := }
            @[simp]
            theorem MulSemiringActionHom.comp_apply {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {P : Type u_3} [Monoid P] {φ : M →* N} {ψ : N →* P} {χ : M →* P} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] {T : Type u_14} [Semiring T] [MulSemiringAction P T] (g : S →ₑ+*[ψ] T) (f : R →ₑ+*[φ] S) [φ.CompTriple ψ χ] (x : R) :
            (g.comp f) x = g (f x)
            @[simp]
            theorem MulSemiringActionHom.id_comp {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) :
            @[simp]
            theorem MulSemiringActionHom.comp_id {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) :
            @[simp]
            theorem MulSemiringActionHom.inverse'_toFun {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {φ' : N →* M} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) (g : SR) (k : Function.RightInverse φ' φ) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
            ∀ (a : S), (f.inverse' g k h₁ h₂) a = g a
            def MulSemiringActionHom.inverse' {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {φ' : N →* M} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) (g : SR) (k : Function.RightInverse φ' φ) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
            S →ₑ+*[φ'] R

            The inverse of a bijective MulSemiringActionHom is a MulSemiringActionHom.

            Equations
            • f.inverse' g k h₁ h₂ = { toFun := g, map_smul' := , map_zero' := , map_add' := , map_one' := , map_mul' := }
            @[simp]
            theorem MulSemiringActionHom.inverse_toFun {M : Type u_1} [Monoid M] {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S₁ : Type u_15} [Semiring S₁] [MulSemiringAction M S₁] (f : R →+*[M] S₁) (g : S₁R) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
            ∀ (a : S₁), (f.inverse g h₁ h₂) a = g a
            def MulSemiringActionHom.inverse {M : Type u_1} [Monoid M] {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S₁ : Type u_15} [Semiring S₁] [MulSemiringAction M S₁] (f : R →+*[M] S₁) (g : S₁R) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
            S₁ →+*[M] R

            The inverse of a bijective MulSemiringActionHom is a MulSemiringActionHom.

            Equations
            • f.inverse g h₁ h₂ = { toFun := g, map_smul' := , map_zero' := , map_add' := , map_one' := , map_mul' := }