mathlib docs: algebra.group.inj

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def function.injective.semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [semigroup M₂] (f : M₁ → M₂) :

function.injective f → (∀ (x y : M₁), f (x * y) = (f x) * f y) → semigroup M₁

A type endowed with * is a semigroup, if it admits an injective map that preserves * to a semigroup.

Equations

function.injective.semigroup f hf mul = {mul := has_mul.mul _inst_1, mul_assoc := _}

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def function.injective.add_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [add_semigroup M₂] (f : M₁ → M₂) :

function.injective f → (∀ (x y : M₁), f (x + y) = f x + f y) → add_semigroup M₁

A type endowed with + is an additive semigroup, if it admits an injective map that preserves + to an additive semigroup.

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def function.injective.add_comm_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [add_comm_semigroup M₂] (f : M₁ → M₂) :

function.injective f → (∀ (x y : M₁), f (x + y) = f x + f y) → add_comm_semigroup M₁

A type endowed with + is an additive commutative semigroup, if it admits an injective map that preserves + to an additive commutative semigroup.

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def function.injective.comm_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [comm_semigroup M₂] (f : M₁ → M₂) :

function.injective f → (∀ (x y : M₁), f (x * y) = (f x) * f y) → comm_semigroup M₁

A type endowed with * is a commutative semigroup, if it admits an injective map that preserves * to a commutative semigroup.

Equations

function.injective.comm_semigroup f hf mul = {mul := semigroup.mul (function.injective.semigroup f hf mul), mul_assoc := _, mul_comm := _}

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def function.injective.left_cancel_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [left_cancel_semigroup M₂] (f : M₁ → M₂) :

function.injective f → (∀ (x y : M₁), f (x * y) = (f x) * f y) → left_cancel_semigroup M₁

A type endowed with * is a left cancel semigroup, if it admits an injective map that preserves * to a left cancel semigroup.

Equations

function.injective.left_cancel_semigroup f hf mul = {mul := has_mul.mul _inst_1, mul_assoc := _, mul_left_cancel := _}

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def function.injective.add_left_cancel_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [add_left_cancel_semigroup M₂] (f : M₁ → M₂) :

function.injective f → (∀ (x y : M₁), f (x + y) = f x + f y) → add_left_cancel_semigroup M₁

A type endowed with + is an additive left cancel semigroup, if it admits an injective map that preserves + to an additive left cancel semigroup.

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def function.injective.right_cancel_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [right_cancel_semigroup M₂] (f : M₁ → M₂) :

function.injective f → (∀ (x y : M₁), f (x * y) = (f x) * f y) → right_cancel_semigroup M₁

A type endowed with * is a right cancel semigroup, if it admits an injective map that preserves * to a right cancel semigroup.

Equations

function.injective.right_cancel_semigroup f hf mul = {mul := has_mul.mul _inst_1, mul_assoc := _, mul_right_cancel := _}

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def function.injective.add_right_cancel_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [add_right_cancel_semigroup M₂] (f : M₁ → M₂) :

function.injective f → (∀ (x y : M₁), f (x + y) = f x + f y) → add_right_cancel_semigroup M₁

A type endowed with + is an additive right cancel semigroup, if it admits an injective map that preserves + to an additive right cancel semigroup.

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def function.injective.add_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [add_monoid M₂] (f : M₁ → M₂) :

function.injective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → add_monoid M₁

A type endowed with 0 and + is an additive monoid, if it admits an injective map that preserves 0 and + to an additive monoid.

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def function.injective.monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [monoid M₂] (f : M₁ → M₂) :

function.injective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = (f x) * f y) → monoid M₁

A type endowed with 1 and * is a monoid, if it admits an injective map that preserves 1 and * to a monoid.

Equations

function.injective.monoid f hf one mul = {mul := semigroup.mul (function.injective.semigroup f hf mul), mul_assoc := _, one := 1, one_mul := _, mul_one := _}

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def function.injective.left_cancel_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [left_cancel_monoid M₂] (f : M₁ → M₂) :

function.injective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = (f x) * f y) → left_cancel_monoid M₁

A type endowed with 1 and * is a left cancel monoid, if it admits an injective map that preserves 1 and * to a left cancel monoid.

Equations

function.injective.left_cancel_monoid f hf one mul = {mul := left_cancel_semigroup.mul (function.injective.left_cancel_semigroup f hf mul), mul_assoc := _, mul_left_cancel := _, one := monoid.one (function.injective.monoid f hf one mul), one_mul := _, mul_one := _}

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def function.injective.add_left_cancel_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [add_left_cancel_monoid M₂] (f : M₁ → M₂) :

function.injective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → add_left_cancel_monoid M₁

A type endowed with 0 and + is an additive left cancel monoid, if it admits an injective map that preserves 0 and + to an additive left cancel monoid.

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def function.injective.add_comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [add_comm_monoid M₂] (f : M₁ → M₂) :

function.injective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → add_comm_monoid M₁

A type endowed with 0 and + is an additive commutative monoid, if it admits an injective map that preserves 0 and + to an additive commutative monoid.

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def function.injective.comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [comm_monoid M₂] (f : M₁ → M₂) :

function.injective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = (f x) * f y) → comm_monoid M₁

A type endowed with 1 and * is a commutative monoid, if it admits an injective map that preserves 1 and * to a commutative monoid.

Equations

function.injective.comm_monoid f hf one mul = {mul := comm_semigroup.mul (function.injective.comm_semigroup f hf mul), mul_assoc := _, one := monoid.one (function.injective.monoid f hf one mul), one_mul := _, mul_one := _, mul_comm := _}

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def function.injective.group {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_inv M₁] [group M₂] (f : M₁ → M₂) :

function.injective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = (f x) * f y) → (∀ (x : M₁), f x⁻¹ = (f x)⁻¹) → group M₁

A type endowed with 1, * and ⁻¹ is a group, if it admits an injective map that preserves 1, * and ⁻¹ to a group.

Equations

function.injective.group f hf one mul inv = {mul := monoid.mul (function.injective.monoid f hf one mul), mul_assoc := _, one := monoid.one (function.injective.monoid f hf one mul), one_mul := _, mul_one := _, inv := has_inv.inv _inst_3, mul_left_inv := _}

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def function.injective.add_group {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_neg M₁] [add_group M₂] (f : M₁ → M₂) :

function.injective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁), f (-x) = -f x) → add_group M₁

A type endowed with 0 and + is an additive group, if it admits an injective map that preserves 0 and + to an additive group.

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def function.injective.comm_group {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_inv M₁] [comm_group M₂] (f : M₁ → M₂) :

function.injective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = (f x) * f y) → (∀ (x : M₁), f x⁻¹ = (f x)⁻¹) → comm_group M₁

A type endowed with 1, * and ⁻¹ is a commutative group, if it admits an injective map that preserves 1, * and ⁻¹ to a commutative group.

Equations

function.injective.comm_group f hf one mul inv = {mul := comm_monoid.mul (function.injective.comm_monoid f hf one mul), mul_assoc := _, one := comm_monoid.one (function.injective.comm_monoid f hf one mul), one_mul := _, mul_one := _, inv := group.inv (function.injective.group f hf one mul inv), mul_left_inv := _, mul_comm := _}

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def function.injective.add_comm_group {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_neg M₁] [add_comm_group M₂] (f : M₁ → M₂) :

function.injective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁), f (-x) = -f x) → add_comm_group M₁

A type endowed with 0 and + is an additive commutative group, if it admits an injective map that preserves 0 and + to an additive commutative group.

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def function.surjective.add_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [add_semigroup M₁] (f : M₁ → M₂) :

function.surjective f → (∀ (x y : M₁), f (x + y) = f x + f y) → add_semigroup M₂

A type endowed with + is an additive semigroup, if it admits a surjective map that preserves + from an additive semigroup.

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def function.surjective.semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [semigroup M₁] (f : M₁ → M₂) :

function.surjective f → (∀ (x y : M₁), f (x * y) = (f x) * f y) → semigroup M₂

A type endowed with * is a semigroup, if it admits a surjective map that preserves * from a semigroup.

Equations

function.surjective.semigroup f hf mul = {mul := has_mul.mul _inst_1, mul_assoc := _}

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def function.surjective.comm_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [comm_semigroup M₁] (f : M₁ → M₂) :

function.surjective f → (∀ (x y : M₁), f (x * y) = (f x) * f y) → comm_semigroup M₂

A type endowed with * is a commutative semigroup, if it admits a surjective map that preserves * from a commutative semigroup.

Equations

function.surjective.comm_semigroup f hf mul = {mul := semigroup.mul (function.surjective.semigroup f hf mul), mul_assoc := _, mul_comm := _}

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def function.surjective.add_comm_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [add_comm_semigroup M₁] (f : M₁ → M₂) :

function.surjective f → (∀ (x y : M₁), f (x + y) = f x + f y) → add_comm_semigroup M₂

A type endowed with + is an additive commutative semigroup, if it admits a surjective map that preserves + from an additive commutative semigroup.

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def function.surjective.monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [has_one M₂] [monoid M₁] (f : M₁ → M₂) :

function.surjective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = (f x) * f y) → monoid M₂

A type endowed with 1 and * is a monoid, if it admits a surjective map that preserves 1 and * from a monoid.

Equations

function.surjective.monoid f hf one mul = {mul := semigroup.mul (function.surjective.semigroup f hf mul), mul_assoc := _, one := 1, one_mul := _, mul_one := _}

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def function.surjective.add_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [has_zero M₂] [add_monoid M₁] (f : M₁ → M₂) :

function.surjective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → add_monoid M₂

A type endowed with 0 and + is an additive monoid, if it admits a surjective map that preserves 0 and + to an additive monoid.

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def function.surjective.add_comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [has_zero M₂] [add_comm_monoid M₁] (f : M₁ → M₂) :

function.surjective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → add_comm_monoid M₂

A type endowed with 0 and + is an additive commutative monoid, if it admits a surjective map that preserves 0 and + to an additive commutative monoid.

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def function.surjective.comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [has_one M₂] [comm_monoid M₁] (f : M₁ → M₂) :

function.surjective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = (f x) * f y) → comm_monoid M₂

A type endowed with 1 and * is a commutative monoid, if it admits a surjective map that preserves 1 and * from a commutative monoid.

Equations

function.surjective.comm_monoid f hf one mul = {mul := comm_semigroup.mul (function.surjective.comm_semigroup f hf mul), mul_assoc := _, one := monoid.one (function.surjective.monoid f hf one mul), one_mul := _, mul_one := _, mul_comm := _}

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def function.surjective.group {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [has_one M₂] [has_inv M₂] [group M₁] (f : M₁ → M₂) :

function.surjective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = (f x) * f y) → (∀ (x : M₁), f x⁻¹ = (f x)⁻¹) → group M₂

A type endowed with 1, * and ⁻¹ is a group, if it admits a surjective map that preserves 1, * and ⁻¹ from a group.

Equations

function.surjective.group f hf one mul inv = {mul := monoid.mul (function.surjective.monoid f hf one mul), mul_assoc := _, one := monoid.one (function.surjective.monoid f hf one mul), one_mul := _, mul_one := _, inv := has_inv.inv _inst_3, mul_left_inv := _}

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def function.surjective.add_group {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [has_zero M₂] [has_neg M₂] [add_group M₁] (f : M₁ → M₂) :

function.surjective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁), f (-x) = -f x) → add_group M₂

A type endowed with 0 and + is an additive group, if it admits a surjective map that preserves 0 and + to an additive group.

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def function.surjective.comm_group {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [has_one M₂] [has_inv M₂] [comm_group M₁] (f : M₁ → M₂) :

function.surjective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = (f x) * f y) → (∀ (x : M₁), f x⁻¹ = (f x)⁻¹) → comm_group M₂

A type endowed with 1, * and ⁻¹ is a commutative group, if it admits a surjective map that preserves 1, * and ⁻¹ from a commutative group.

Equations

function.surjective.comm_group f hf one mul inv = {mul := comm_monoid.mul (function.surjective.comm_monoid f hf one mul), mul_assoc := _, one := comm_monoid.one (function.surjective.comm_monoid f hf one mul), one_mul := _, mul_one := _, inv := group.inv (function.surjective.group f hf one mul inv), mul_left_inv := _, mul_comm := _}

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def function.surjective.add_comm_group {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [has_zero M₂] [has_neg M₂] [add_comm_group M₁] (f : M₁ → M₂) :

function.surjective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁), f (-x) = -f x) → add_comm_group M₂

A type endowed with 0 and + is an additive commutative group, if it admits a surjective map that preserves 0 and + to an additive commutative group.

mathlib documentation

Lifting algebraic data classes along injective/surjective maps

Injective

Surjective