Lifting groups with zero along injective/surjective maps

@[reducible, inline]

abbrev Function.Injective.mulZeroClass {M₀ : Type u_1} {M₀' : Type u_3} [MulZeroClass M₀] [Mul M₀'] [Zero M₀'] (f : M₀' → M₀) (hf : Injective f) (zero : f 0 = 0) (mul : ∀ (a b : M₀'), f (a * b) = f a * f b) : MulZeroClass M₀'

Pull back a MulZeroClass instance along an injective function. See note [reducible non-instances].

Equations

• Function.Injective.mulZeroClass f hf zero mul = { mul := fun (x1 x2 : M₀') => x1 * x2, zero := 0, zero_mul := ⋯, mul_zero := ⋯ }

@[reducible, inline]

abbrev Function.Surjective.mulZeroClass {M₀ : Type u_1} {M₀' : Type u_3} [MulZeroClass M₀] [Mul M₀'] [Zero M₀'] (f : M₀ → M₀') (hf : Surjective f) (zero : f 0 = 0) (mul : ∀ (a b : M₀), f (a * b) = f a * f b) : MulZeroClass M₀'

Push forward a MulZeroClass instance along a surjective function. See note [reducible non-instances].

Equations

• Function.Surjective.mulZeroClass f hf zero mul = { mul := fun (x1 x2 : M₀') => x1 * x2, zero := 0, zero_mul := ⋯, mul_zero := ⋯ }

theorem Function.Injective.noZeroDivisors {M₀ : Type u_1} {M₀' : Type u_3} [Mul M₀] [Zero M₀] [Mul M₀'] [Zero M₀'] (f : M₀ → M₀') (hf : Injective f) (zero : f 0 = 0) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) [NoZeroDivisors M₀'] : NoZeroDivisors M₀

Pull back a NoZeroDivisors instance along an injective function.

theorem Function.Injective.isLeftCancelMulZero {M₀ : Type u_1} {M₀' : Type u_3} [Mul M₀] [Zero M₀] [Mul M₀'] [Zero M₀'] (f : M₀ → M₀') (hf : Injective f) (zero : f 0 = 0) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) [IsLeftCancelMulZero M₀'] : IsLeftCancelMulZero M₀

theorem Function.Injective.isRightCancelMulZero {M₀ : Type u_1} {M₀' : Type u_3} [Mul M₀] [Zero M₀] [Mul M₀'] [Zero M₀'] (f : M₀ → M₀') (hf : Injective f) (zero : f 0 = 0) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) [IsRightCancelMulZero M₀'] : IsRightCancelMulZero M₀

theorem Function.Injective.isCancelMulZero {M₀ : Type u_1} {M₀' : Type u_3} [Mul M₀] [Zero M₀] [Mul M₀'] [Zero M₀'] (f : M₀ → M₀') (hf : Injective f) (zero : f 0 = 0) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) [IsCancelMulZero M₀'] : IsCancelMulZero M₀

@[reducible, inline]

abbrev Function.Injective.mulZeroOneClass {M₀ : Type u_1} {M₀' : Type u_3} [MulZeroOneClass M₀] [Mul M₀'] [Zero M₀'] [One M₀'] (f : M₀' → M₀) (hf : Injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (a b : M₀'), f (a * b) = f a * f b) : MulZeroOneClass M₀'

Pull back a MulZeroOneClass instance along an injective function. See note [reducible non-instances].

Equations

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

@[reducible, inline]

abbrev Function.Surjective.mulZeroOneClass {M₀ : Type u_1} {M₀' : Type u_3} [MulZeroOneClass M₀] [Mul M₀'] [Zero M₀'] [One M₀'] (f : M₀ → M₀') (hf : Surjective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (a b : M₀), f (a * b) = f a * f b) : MulZeroOneClass M₀'

Push forward a MulZeroOneClass instance along a surjective function. See note [reducible non-instances].

Equations

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

@[reducible, inline]

abbrev Function.Injective.semigroupWithZero {M₀ : Type u_1} {M₀' : Type u_3} [Zero M₀'] [Mul M₀'] [SemigroupWithZero M₀] (f : M₀' → M₀) (hf : Injective f) (zero : f 0 = 0) (mul : ∀ (x y : M₀'), f (x * y) = f x * f y) : SemigroupWithZero M₀'

Pull back a SemigroupWithZero along an injective function. See note [reducible non-instances].

Equations

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

@[reducible, inline]

abbrev Function.Surjective.semigroupWithZero {M₀ : Type u_1} {M₀' : Type u_3} [SemigroupWithZero M₀] [Zero M₀'] [Mul M₀'] (f : M₀ → M₀') (hf : Surjective f) (zero : f 0 = 0) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) : SemigroupWithZero M₀'

Push forward a SemigroupWithZero along a surjective function. See note [reducible non-instances].

Equations

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

@[reducible, inline]

abbrev Function.Injective.monoidWithZero {M₀ : Type u_1} {M₀' : Type u_3} [Zero M₀'] [Mul M₀'] [One M₀'] [Pow M₀' ℕ] [MonoidWithZero M₀] (f : M₀' → M₀) (hf : Injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀'), f (x * y) = f x * f y) (npow : ∀ (x : M₀') (n : ℕ), f (x ^ n) = f x ^ n) : MonoidWithZero M₀'

Pull back a MonoidWithZero along an injective function. See note [reducible non-instances].

Equations

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

@[reducible, inline]

abbrev Function.Surjective.monoidWithZero {M₀ : Type u_1} {M₀' : Type u_3} [Zero M₀'] [Mul M₀'] [One M₀'] [Pow M₀' ℕ] [MonoidWithZero M₀] (f : M₀ → M₀') (hf : Surjective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) (npow : ∀ (x : M₀) (n : ℕ), f (x ^ n) = f x ^ n) : MonoidWithZero M₀'

Push forward a MonoidWithZero along a surjective function. See note [reducible non-instances].

Equations

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

@[reducible, inline]

abbrev Function.Injective.commMonoidWithZero {M₀ : Type u_1} {M₀' : Type u_3} [Zero M₀'] [Mul M₀'] [One M₀'] [Pow M₀' ℕ] [CommMonoidWithZero M₀] (f : M₀' → M₀) (hf : Injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀'), f (x * y) = f x * f y) (npow : ∀ (x : M₀') (n : ℕ), f (x ^ n) = f x ^ n) : CommMonoidWithZero M₀'

Pull back a CommMonoidWithZero along an injective function. See note [reducible non-instances].

Equations

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

@[reducible, inline]

abbrev Function.Surjective.commMonoidWithZero {M₀ : Type u_1} {M₀' : Type u_3} [Zero M₀'] [Mul M₀'] [One M₀'] [Pow M₀' ℕ] [CommMonoidWithZero M₀] (f : M₀ → M₀') (hf : Surjective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) (npow : ∀ (x : M₀) (n : ℕ), f (x ^ n) = f x ^ n) : CommMonoidWithZero M₀'

Push forward a CommMonoidWithZero along a surjective function. See note [reducible non-instances].

Equations

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

@[reducible, inline]

abbrev Function.Injective.cancelMonoidWithZero {M₀ : Type u_1} {M₀' : Type u_3} [CancelMonoidWithZero M₀] [Zero M₀'] [Mul M₀'] [One M₀'] [Pow M₀' ℕ] (f : M₀' → M₀) (hf : Injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀'), f (x * y) = f x * f y) (npow : ∀ (x : M₀') (n : ℕ), f (x ^ n) = f x ^ n) : CancelMonoidWithZero M₀'

Pull back a CancelMonoidWithZero along an injective function. See note [reducible non-instances].

Equations

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

@[reducible, inline]

abbrev Function.Injective.cancelCommMonoidWithZero {M₀ : Type u_1} {M₀' : Type u_3} [CancelCommMonoidWithZero M₀] [Zero M₀'] [Mul M₀'] [One M₀'] [Pow M₀' ℕ] (f : M₀' → M₀) (hf : Injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀'), f (x * y) = f x * f y) (npow : ∀ (x : M₀') (n : ℕ), f (x ^ n) = f x ^ n) : CancelCommMonoidWithZero M₀'

Pull back a CancelCommMonoidWithZero along an injective function. See note [reducible non-instances].

Equations

• Function.Injective.cancelCommMonoidWithZero f hf zero one mul npow = { toCommMonoidWithZero := Function.Injective.commMonoidWithZero f hf zero one mul npow, toIsLeftCancelMulZero := ⋯ }

@[reducible, inline]

abbrev Function.Injective.groupWithZero {G₀ : Type u_2} {G₀' : Type u_4} [GroupWithZero G₀] [Zero G₀'] [Mul G₀'] [One G₀'] [Inv G₀'] [Div G₀'] [Pow G₀' ℕ] [Pow G₀' ℤ] (f : G₀' → G₀) (hf : Injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀'), f (x * y) = f x * f y) (inv : ∀ (x : G₀'), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : G₀'), f (x / y) = f x / f y) (npow : ∀ (x : G₀') (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : G₀') (n : ℤ), f (x ^ n) = f x ^ n) : GroupWithZero G₀'

Pull back a GroupWithZero along an injective function. See note [reducible non-instances].

Equations

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

@[reducible, inline]

abbrev Function.Surjective.groupWithZero {G₀ : Type u_2} {G₀' : Type u_4} [GroupWithZero G₀] [Zero G₀'] [Mul G₀'] [One G₀'] [Inv G₀'] [Div G₀'] [Pow G₀' ℕ] [Pow G₀' ℤ] (h01 : 0 ≠ 1) (f : G₀ → G₀') (hf : Surjective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀), f (x * y) = f x * f y) (inv : ∀ (x : G₀), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : G₀), f (x / y) = f x / f y) (npow : ∀ (x : G₀) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : G₀) (n : ℤ), f (x ^ n) = f x ^ n) : GroupWithZero G₀'

Push forward a GroupWithZero along a surjective function. See note [reducible non-instances].

Equations

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

@[reducible, inline]

abbrev Function.Injective.commGroupWithZero {G₀ : Type u_2} {G₀' : Type u_4} [CommGroupWithZero G₀] [Zero G₀'] [Mul G₀'] [One G₀'] [Inv G₀'] [Div G₀'] [Pow G₀' ℕ] [Pow G₀' ℤ] (f : G₀' → G₀) (hf : Injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀'), f (x * y) = f x * f y) (inv : ∀ (x : G₀'), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : G₀'), f (x / y) = f x / f y) (npow : ∀ (x : G₀') (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : G₀') (n : ℤ), f (x ^ n) = f x ^ n) : CommGroupWithZero G₀'

Pull back a CommGroupWithZero along an injective function. See note [reducible non-instances].

Equations

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

def Function.Surjective.commGroupWithZero {G₀ : Type u_2} {G₀' : Type u_4} [CommGroupWithZero G₀] [Zero G₀'] [Mul G₀'] [One G₀'] [Inv G₀'] [Div G₀'] [Pow G₀' ℕ] [Pow G₀' ℤ] (h01 : 0 ≠ 1) (f : G₀ → G₀') (hf : Surjective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀), f (x * y) = f x * f y) (inv : ∀ (x : G₀), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : G₀), f (x / y) = f x / f y) (npow : ∀ (x : G₀) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : G₀) (n : ℤ), f (x ^ n) = f x ^ n) : CommGroupWithZero G₀'

Push forward a CommGroupWithZero along a surjective function. See note [reducible non-instances].

Equations

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