mathlib docs: algebra.group_action

@[nolint]

structure mul_action_hom (M : Type u_1) [monoid M] (X : Type u_2) [mul_action M X] (Y : Type u_3) [mul_action M Y] :

Type (max u_2 u_3)

to_fun : X → Y
map_smul' : ∀ (m : M) (x : X), c.to_fun (m • x) = m • c.to_fun x

Equivariant functions.

@[instance]

def mul_action_hom.has_coe_to_fun (M : Type u_1) [monoid M] (X : Type u_2) [mul_action M X] (Y : Type u_3) [mul_action M Y] :

has_coe_to_fun (X →[M] Y)

Equations

mul_action_hom.has_coe_to_fun M X Y = {F := λ (c : X →[M] Y), X → Y, coe := λ (c : X →[M] Y), c.to_fun}

source

@[simp]

theorem mul_action_hom.map_smul {M : Type u_1} [monoid M] {X : Type u_2} [mul_action M X] {Y : Type u_3} [mul_action M Y] (f : X →[M] Y) (m : M) (x : X) :

⇑f (m • x) = m • ⇑f x

source

@[ext]

theorem mul_action_hom.ext {M : Type u_1} [monoid M] {X : Type u_2} [mul_action M X] {Y : Type u_3} [mul_action M Y] {f g : X →[M] Y} :

(∀ (x : X), ⇑f x = ⇑g x) → f = g

source

theorem mul_action_hom.ext_iff {M : Type u_1} [monoid M] {X : Type u_2} [mul_action M X] {Y : Type u_3} [mul_action M Y] {f g : X →[M] Y} :

f = g ↔ ∀ (x : X), ⇑f x = ⇑g x

source

def mul_action_hom.id (M : Type u_1) [monoid M] {X : Type u_2} [mul_action M X] :

X →[M] X

The identity map as an equivariant map.

Equations

mul_action_hom.id M = {to_fun := id X, map_smul' := _}

source

@[simp]

theorem mul_action_hom.id_apply (M : Type u_1) [monoid M] {X : Type u_2} [mul_action M X] (x : X) :

⇑(mul_action_hom.id M) x = x

source

def mul_action_hom.comp {M : Type u_1} [monoid M] {X : Type u_2} [mul_action M X] {Y : Type u_3} [mul_action M Y] {Z : Type u_4} [mul_action M Z] :

(Y →[M] Z) → (X →[M] Y) → (X →[M] Z)

Composition of two equivariant maps.

Equations

g.comp f = {to_fun := ⇑g ∘ ⇑f, map_smul' := _}

source

@[simp]

theorem mul_action_hom.comp_apply {M : Type u_1} [monoid M] {X : Type u_2} [mul_action M X] {Y : Type u_3} [mul_action M Y] {Z : Type u_4} [mul_action M Z] (g : Y →[M] Z) (f : X →[M] Y) (x : X) :

⇑(g.comp f) x = ⇑g (⇑f x)

source

@[simp]

theorem mul_action_hom.id_comp {M : Type u_1} [monoid M] {X : Type u_2} [mul_action M X] {Y : Type u_3} [mul_action M Y] (f : X →[M] Y) :

(mul_action_hom.id M).comp f = f

source

@[simp]

theorem mul_action_hom.comp_id {M : Type u_1} [monoid M] {X : Type u_2} [mul_action M X] {Y : Type u_3} [mul_action M Y] (f : X →[M] Y) :

f.comp (mul_action_hom.id M) = f

source

def mul_action_hom.to_quotient {G : Type u_15} [group G] (H : subgroup G) :

G →[G] quotient_group.quotient H

The canonical map to the left cosets.

Equations

mul_action_hom.to_quotient H = {to_fun := coe coe_to_lift, map_smul' := _}

source

@[simp]

theorem mul_action_hom.to_quotient_apply {G : Type u_15} [group G] (H : subgroup G) (g : G) :

⇑(mul_action_hom.to_quotient H) g = ↑g

source

@[nolint]

structure distrib_mul_action_hom (M : Type u_1) [monoid M] (A : Type u_5) [add_monoid A] [distrib_mul_action M A] (B : Type u_7) [add_monoid B] [distrib_mul_action M B] :

Type (max u_5 u_7)

to_fun : A → B
map_smul' : ∀ (m : M) (x : A), c.to_fun (m • x) = m • c.to_fun x
map_zero' : c.to_fun 0 = 0
map_add' : ∀ (x y : A), c.to_fun (x + y) = c.to_fun x + c.to_fun y

Equivariant additive monoid homomorphisms.

source

def distrib_mul_action_hom.to_mul_action_hom {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] :

(A →+[M] B) → (A →[M] B)

Reinterpret an equivariant additive monoid homomorphism as an equivariant function.

source

def distrib_mul_action_hom.to_add_monoid_hom {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] :

(A →+[M] B) → A →+ B

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

source

@[instance]

def distrib_mul_action_hom.has_coe (M : Type u_1) [monoid M] (A : Type u_5) [add_monoid A] [distrib_mul_action M A] (B : Type u_7) [add_monoid B] [distrib_mul_action M B] :

has_coe (A →+[M] B) (A →+ B)

Equations

distrib_mul_action_hom.has_coe M A B = {coe := distrib_mul_action_hom.to_add_monoid_hom _inst_10}

source

@[instance]

def distrib_mul_action_hom.has_coe' (M : Type u_1) [monoid M] (A : Type u_5) [add_monoid A] [distrib_mul_action M A] (B : Type u_7) [add_monoid B] [distrib_mul_action M B] :

has_coe (A →+[M] B) (A →[M] B)

Equations

distrib_mul_action_hom.has_coe' M A B = {coe := distrib_mul_action_hom.to_mul_action_hom _inst_10}

source

@[instance]

def distrib_mul_action_hom.has_coe_to_fun (M : Type u_1) [monoid M] (A : Type u_5) [add_monoid A] [distrib_mul_action M A] (B : Type u_7) [add_monoid B] [distrib_mul_action M B] :

has_coe_to_fun (A →+[M] B)

Equations

distrib_mul_action_hom.has_coe_to_fun M A B = {F := λ (c : A →+[M] B), A → B, coe := λ (c : A →+[M] B), c.to_fun}

source

theorem distrib_mul_action_hom.coe_fn_coe {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) :

⇑↑f = ⇑f

source

theorem distrib_mul_action_hom.coe_fn_coe' {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) :

⇑↑f = ⇑f

source

@[ext]

theorem distrib_mul_action_hom.ext {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] {f g : A →+[M] B} :

(∀ (x : A), ⇑f x = ⇑g x) → f = g

source

theorem distrib_mul_action_hom.ext_iff {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] {f g : A →+[M] B} :

f = g ↔ ∀ (x : A), ⇑f x = ⇑g x

source

@[simp]

theorem distrib_mul_action_hom.map_zero {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) :

⇑f 0 = 0

source

@[simp]

theorem distrib_mul_action_hom.map_add {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) (x y : A) :

⇑f (x + y) = ⇑f x + ⇑f y

source

@[simp]

theorem distrib_mul_action_hom.map_neg {M : Type u_1} [monoid M] (A' : Type u_6) [add_group A'] [distrib_mul_action M A'] (B' : Type u_8) [add_group B'] [distrib_mul_action M B'] (f : A' →+[M] B') (x : A') :

⇑f (-x) = -⇑f x

source

@[simp]

theorem distrib_mul_action_hom.map_sub {M : Type u_1} [monoid M] (A' : Type u_6) [add_group A'] [distrib_mul_action M A'] (B' : Type u_8) [add_group B'] [distrib_mul_action M B'] (f : A' →+[M] B') (x y : A') :

⇑f (x - y) = ⇑f x - ⇑f y

source

@[simp]

theorem distrib_mul_action_hom.map_smul {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) (m : M) (x : A) :

⇑f (m • x) = m • ⇑f x

source

def distrib_mul_action_hom.id (M : Type u_1) [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] :

A →+[M] A

The identity map as an equivariant additive monoid homomorphism.

Equations

distrib_mul_action_hom.id M = {to_fun := id A, map_smul' := _, map_zero' := _, map_add' := _}

source

@[simp]

theorem distrib_mul_action_hom.id_apply (M : Type u_1) [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] (x : A) :

⇑(distrib_mul_action_hom.id M) x = x

source

def distrib_mul_action_hom.comp {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] {C : Type u_9} [add_monoid C] [distrib_mul_action M C] :

(B →+[M] C) → (A →+[M] B) → (A →+[M] C)

Composition of two equivariant additive monoid homomorphisms.

Equations

g.comp f = {to_fun := (↑g.comp ↑f).to_fun, map_smul' := _, map_zero' := _, map_add' := _}

source

@[simp]

theorem distrib_mul_action_hom.comp_apply {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] {C : Type u_9} [add_monoid C] [distrib_mul_action M C] (g : B →+[M] C) (f : A →+[M] B) (x : A) :

⇑(g.comp f) x = ⇑g (⇑f x)

source

@[simp]

theorem distrib_mul_action_hom.id_comp {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) :

(distrib_mul_action_hom.id M).comp f = f

source

@[simp]

theorem distrib_mul_action_hom.comp_id {M : Type u_1} [monoid M] {A : Type u_5} [add_monoid A] [distrib_mul_action M A] {B : Type u_7} [add_monoid B] [distrib_mul_action M B] (f : A →+[M] B) :

f.comp (distrib_mul_action_hom.id M) = f

source

def mul_semiring_action_hom.to_distrib_mul_action_hom {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] :

(R →+*[M] S) → (R →+[M] S)

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

source

@[nolint]

structure mul_semiring_action_hom (M : Type u_1) [monoid M] (R : Type u_10) [semiring R] [mul_semiring_action M R] (S : Type u_12) [semiring S] [mul_semiring_action M S] :

Type (max u_10 u_12)

to_fun : R → S
map_smul' : ∀ (m : M) (x : R), c.to_fun (m • x) = m • c.to_fun x
map_zero' : c.to_fun 0 = 0
map_add' : ∀ (x y : R), c.to_fun (x + y) = c.to_fun x + c.to_fun y
map_one' : c.to_fun 1 = 1
map_mul' : ∀ (x y : R), c.to_fun (x * y) = c.to_fun x * c.to_fun y

Equivariant ring homomorphisms.

source

def mul_semiring_action_hom.to_ring_hom {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] :

(R →+*[M] S) → R →+* S

Reinterpret an equivariant ring homomorphism as a ring homomorphism.

source

@[instance]

def mul_semiring_action_hom.has_coe (M : Type u_1) [monoid M] (R : Type u_10) [semiring R] [mul_semiring_action M R] (S : Type u_12) [semiring S] [mul_semiring_action M S] :

has_coe (R →+*[M] S) (R →+* S)

Equations

mul_semiring_action_hom.has_coe M R S = {coe := mul_semiring_action_hom.to_ring_hom _inst_20}

source

@[instance]

def mul_semiring_action_hom.has_coe' (M : Type u_1) [monoid M] (R : Type u_10) [semiring R] [mul_semiring_action M R] (S : Type u_12) [semiring S] [mul_semiring_action M S] :

has_coe (R →+*[M] S) (R →+[M] S)

Equations

mul_semiring_action_hom.has_coe' M R S = {coe := mul_semiring_action_hom.to_distrib_mul_action_hom _inst_20}

source

@[instance]

def mul_semiring_action_hom.has_coe_to_fun (M : Type u_1) [monoid M] (R : Type u_10) [semiring R] [mul_semiring_action M R] (S : Type u_12) [semiring S] [mul_semiring_action M S] :

has_coe_to_fun (R →+*[M] S)

Equations

mul_semiring_action_hom.has_coe_to_fun M R S = {F := λ (c : R →+*[M] S), R → S, coe := λ (c : R →+*[M] S), c.to_fun}

source

theorem mul_semiring_action_hom.coe_fn_coe {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) :

⇑↑f = ⇑f

source

theorem mul_semiring_action_hom.coe_fn_coe' {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) :

⇑↑f = ⇑f

source

@[ext]

theorem mul_semiring_action_hom.ext {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] {f g : R →+*[M] S} :

(∀ (x : R), ⇑f x = ⇑g x) → f = g

source

theorem mul_semiring_action_hom.ext_iff {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] {f g : R →+*[M] S} :

f = g ↔ ∀ (x : R), ⇑f x = ⇑g x

source

@[simp]

theorem mul_semiring_action_hom.map_zero {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) :

⇑f 0 = 0

source

@[simp]

theorem mul_semiring_action_hom.map_add {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) (x y : R) :

⇑f (x + y) = ⇑f x + ⇑f y

source

@[simp]

theorem mul_semiring_action_hom.map_neg {M : Type u_1} [monoid M] (R' : Type u_11) [ring R'] [mul_semiring_action M R'] (S' : Type u_13) [ring S'] [mul_semiring_action M S'] (f : R' →+*[M] S') (x : R') :

⇑f (-x) = -⇑f x

source

@[simp]

theorem mul_semiring_action_hom.map_sub {M : Type u_1} [monoid M] (R' : Type u_11) [ring R'] [mul_semiring_action M R'] (S' : Type u_13) [ring S'] [mul_semiring_action M S'] (f : R' →+*[M] S') (x y : R') :

⇑f (x - y) = ⇑f x - ⇑f y

source

@[simp]

theorem mul_semiring_action_hom.map_one {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) :

⇑f 1 = 1

source

@[simp]

theorem mul_semiring_action_hom.map_mul {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) (x y : R) :

⇑f (x * y) = ⇑f x * ⇑f y

source

@[simp]

theorem mul_semiring_action_hom.map_smul {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] (f : R →+*[M] S) (m : M) (x : R) :

⇑f (m • x) = m • ⇑f x

source

def mul_semiring_action_hom.id (M : Type u_1) [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] :

R →+*[M] R

The identity map as an equivariant ring homomorphism.

Equations

mul_semiring_action_hom.id M = {to_fun := id R, map_smul' := _, map_zero' := _, map_add' := _, map_one' := _, map_mul' := _}

source

@[simp]

theorem mul_semiring_action_hom.id_apply (M : Type u_1) [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] (x : R) :

⇑(mul_semiring_action_hom.id M) x = x

source

def mul_semiring_action_hom.comp {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] {T : Type u_14} [semiring T] [mul_semiring_action M T] :

(S →+*[M] T) → (R →+*[M] S) → (R →+*[M] T)

Composition of two equivariant additive monoid homomorphisms.

Equations

g.comp f = {to_fun := (↑g.comp ↑f).to_fun, map_smul' := _, map_zero' := _, map_add' := _, map_one' := _, map_mul' := _}

source

@[simp]

theorem mul_semiring_action_hom.comp_apply {M : Type u_1} [monoid M] {R : Type u_10} [semiring R] [mul_semiring_action M R] {S : Type u_12} [semiring S] [mul_semiring_action M S] {T : Type u_14} [semiring T] [mul_semiring_action M T] (g : S →+*[M] T) (f : R →+*[M] S) (x : R) :

⇑(g.comp f) x = ⇑g (⇑f x)

source

@[simp]