# Pulling back rings along injective maps, and pushing them forward along surjective maps #

theorem Function.Injective.leftDistribClass {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Mul α] [Add α] [] (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) :

Pullback a LeftDistribClass instance along an injective function.

theorem Function.Injective.rightDistribClass {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Mul α] [Add α] (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) :

Pullback a RightDistribClass instance along an injective function.

@[reducible, inline]
abbrev Function.Injective.distrib {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [] (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) :

Pullback a Distrib instance along an injective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Injective.hasDistribNeg {α : Type u_1} {β : Type u_2} [Mul β] [Neg β] (f : βα) (hf : ) [Mul α] [] (neg : ∀ (a : β), f (-a) = -f a) (mul : ∀ (a b : β), f (a * b) = f a * f b) :

A type endowed with - and * has distributive negation, if it admits an injective map that preserves - and * to a type which has distributive negation.

Equations
Instances For
@[reducible, inline]
abbrev Function.Injective.nonUnitalNonAssocSemiring {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Zero β] [] (zero : f 0 = 0) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) :

Pullback a NonUnitalNonAssocSemiring instance along an injective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Injective.nonUnitalSemiring {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Zero β] [] (zero : f 0 = 0) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) :

Pullback a NonUnitalSemiring instance along an injective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Injective.nonAssocSemiring {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Zero β] [One β] [] [] [] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (natCast : ∀ (n : ), f n = n) :

Pullback a NonAssocSemiring instance along an injective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Injective.semiring {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Zero β] [One β] [] [Pow β ] [] [] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ), f n = n) :

Pullback a Semiring instance along an injective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Injective.nonUnitalNonAssocRing {α : Type u_1} {β : Type u_2} [Add β] [Mul β] [Zero β] [Neg β] [Sub β] [] [] (f : βα) (hf : ) (zero : f 0 = 0) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (zsmul : ∀ (n : ) (x : β), f (n x) = n f x) :

Pullback a NonUnitalNonAssocRing instance along an injective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Injective.nonUnitalRing {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Zero β] [Neg β] [Sub β] [] [] [] (zero : f 0 = 0) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (zsmul : ∀ (n : ) (x : β), f (n x) = n f x) :

Pullback a NonUnitalRing instance along an injective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Injective.nonAssocRing {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Zero β] [One β] [Neg β] [Sub β] [] [] [] [] [] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (zsmul : ∀ (n : ) (x : β), f (n x) = n f x) (natCast : ∀ (n : ), f n = n) (intCast : ∀ (n : ), f n = n) :

Pullback a NonAssocRing instance along an injective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Injective.ring {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Zero β] [One β] [Neg β] [Sub β] [] [] [Pow β ] [] [] [Ring α] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (zsmul : ∀ (n : ) (x : β), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ), f n = n) (intCast : ∀ (n : ), f n = n) :
Ring β

Pullback a Ring instance along an injective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Injective.nonUnitalNonAssocCommSemiring {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Zero β] [] (zero : f 0 = 0) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) :

Pullback a NonUnitalNonAssocCommSemiring instance along an injective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Injective.nonUnitalCommSemiring {α : Type u_1} {β : Type u_2} [Add β] [Mul β] [Zero β] [] (f : βα) (hf : ) (zero : f 0 = 0) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) :

Pullback a NonUnitalCommSemiring instance along an injective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Injective.commSemiring {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Zero β] [One β] [] [Pow β ] [] [] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ), f n = n) :

Pullback a CommSemiring instance along an injective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Injective.nonUnitalNonAssocCommRing {α : Type u_1} {β : Type u_2} [Add β] [Mul β] [Zero β] [Neg β] [Sub β] [] [] (f : βα) (hf : ) (zero : f 0 = 0) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (zsmul : ∀ (n : ) (x : β), f (n x) = n f x) :

Pullback a NonUnitalNonAssocCommRing instance along an injective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Injective.nonUnitalCommRing {α : Type u_1} {β : Type u_2} [Add β] [Mul β] [Zero β] [Neg β] [Sub β] [] [] (f : βα) (hf : ) (zero : f 0 = 0) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (zsmul : ∀ (n : ) (x : β), f (n x) = n f x) :

Pullback a NonUnitalCommRing instance along an injective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Injective.commRing {α : Type u_1} {β : Type u_2} (f : βα) (hf : ) [Add β] [Mul β] [Zero β] [One β] [Neg β] [Sub β] [] [] [Pow β ] [] [] [] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : β), f (n x) = n f x) (zsmul : ∀ (n : ) (x : β), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ), f n = n) (intCast : ∀ (n : ), f n = n) :

Pullback a CommRing instance along an injective function.

Equations
Instances For
theorem Function.Surjective.leftDistribClass {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Mul α] [Add α] [] (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) :

Pushforward a LeftDistribClass instance along a surjective function.

theorem Function.Surjective.rightDistribClass {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Mul α] [Add α] (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) :

Pushforward a RightDistribClass instance along a surjective function.

@[reducible, inline]
abbrev Function.Surjective.distrib {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [] (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) :

Pushforward a Distrib instance along a surjective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Surjective.hasDistribNeg {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Mul β] [Neg β] [Mul α] [] (neg : ∀ (a : α), f (-a) = -f a) (mul : ∀ (a b : α), f (a * b) = f a * f b) :

A type endowed with - and * has distributive negation, if it admits a surjective map that preserves - and * from a type which has distributive negation.

Equations
Instances For
@[reducible, inline]
abbrev Function.Surjective.nonUnitalNonAssocSemiring {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [] (zero : f 0 = 0) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) :

Pushforward a NonUnitalNonAssocSemiring instance along a surjective function. See note [reducible non-instances].

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Surjective.nonUnitalSemiring {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [] (zero : f 0 = 0) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) :

Pushforward a NonUnitalSemiring instance along a surjective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Surjective.nonAssocSemiring {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [One β] [] [] [] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) (natCast : ∀ (n : ), f n = n) :

Pushforward a NonAssocSemiring instance along a surjective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Surjective.semiring {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [One β] [] [Pow β ] [] [] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) (npow : ∀ (x : α) (n : ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ), f n = n) :

Pushforward a Semiring instance along a surjective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Surjective.nonUnitalNonAssocRing {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [Neg β] [Sub β] [] [] (zero : f 0 = 0) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (neg : ∀ (x : α), f (-x) = -f x) (sub : ∀ (x y : α), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) (zsmul : ∀ (n : ) (x : α), f (n x) = n f x) :

Pushforward a NonUnitalNonAssocRing instance along a surjective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Surjective.nonUnitalRing {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [Neg β] [Sub β] [] [] [] (zero : f 0 = 0) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (neg : ∀ (x : α), f (-x) = -f x) (sub : ∀ (x y : α), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) (zsmul : ∀ (n : ) (x : α), f (n x) = n f x) :

Pushforward a NonUnitalRing instance along a surjective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Surjective.nonAssocRing {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [One β] [Neg β] [Sub β] [] [] [] [] [] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (neg : ∀ (x : α), f (-x) = -f x) (sub : ∀ (x y : α), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) (zsmul : ∀ (n : ) (x : α), f (n x) = n f x) (natCast : ∀ (n : ), f n = n) (intCast : ∀ (n : ), f n = n) :

Pushforward a NonAssocRing instance along a surjective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Surjective.ring {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [One β] [Neg β] [Sub β] [] [] [Pow β ] [] [] [Ring α] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (neg : ∀ (x : α), f (-x) = -f x) (sub : ∀ (x y : α), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) (zsmul : ∀ (n : ) (x : α), f (n x) = n f x) (npow : ∀ (x : α) (n : ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ), f n = n) (intCast : ∀ (n : ), f n = n) :
Ring β

Pushforward a Ring instance along a surjective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Surjective.nonUnitalNonAssocCommSemiring {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [] (zero : f 0 = 0) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) :

Pushforward a NonUnitalNonAssocCommSemiring instance along a surjective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Surjective.nonUnitalCommSemiring {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [] (zero : f 0 = 0) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) :

Pushforward a NonUnitalCommSemiring instance along a surjective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Surjective.commSemiring {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [One β] [] [Pow β ] [] [] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) (npow : ∀ (x : α) (n : ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ), f n = n) :

Pushforward a CommSemiring instance along a surjective function.

Equations
Instances For
@[reducible, inline]
abbrev Function.Surjective.nonUnitalNonAssocCommRing {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [Neg β] [Sub β] [] [] (zero : f 0 = 0) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (neg : ∀ (x : α), f (-x) = -f x) (sub : ∀ (x y : α), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) (zsmul : ∀ (n : ) (x : α), f (n x) = n f x) :

Pushforward a NonUnitalNonAssocCommRing instance along a surjective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Surjective.nonUnitalCommRing {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [Neg β] [Sub β] [] [] (zero : f 0 = 0) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (neg : ∀ (x : α), f (-x) = -f x) (sub : ∀ (x y : α), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) (zsmul : ∀ (n : ) (x : α), f (n x) = n f x) :

Pushforward a NonUnitalCommRing instance along a surjective function.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev Function.Surjective.commRing {α : Type u_1} {β : Type u_2} (f : αβ) (hf : ) [Add β] [Mul β] [Zero β] [One β] [Neg β] [Sub β] [] [] [Pow β ] [] [] [] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (neg : ∀ (x : α), f (-x) = -f x) (sub : ∀ (x y : α), f (x - y) = f x - f y) (nsmul : ∀ (n : ) (x : α), f (n x) = n f x) (zsmul : ∀ (n : ) (x : α), f (n x) = n f x) (npow : ∀ (x : α) (n : ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ), f n = n) (intCast : ∀ (n : ), f n = n) :

Pushforward a CommRing instance along a surjective function.

Equations
Instances For
instance AddOpposite.instHasDistribNeg {α : Type u_1} [Mul α] [] :
Equations