Pulling back ordered rings along injective maps

@[reducible, inline]

abbrev Function.Injective.orderedSemiring {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [SMul ℕ β] [Pow β ℕ] [NatCast β] (f : β → α) (hf : Function.Injective f) [OrderedSemiring 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) (nsmul : ∀ (n : ℕ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) : OrderedSemiring β

Pullback an OrderedSemiring under an injective map.

Equations

• Function.Injective.orderedSemiring f hf zero one add mul nsmul npow natCast = OrderedSemiring.mk ⋯ ⋯ ⋯ ⋯

@[reducible, inline]

abbrev Function.Injective.orderedCommSemiring {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [SMul ℕ β] [Pow β ℕ] [NatCast β] (f : β → α) (hf : Function.Injective f) [OrderedCommSemiring 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) (nsmul : ∀ (n : ℕ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) : OrderedCommSemiring β

Pullback an OrderedCommSemiring under an injective map.

Equations

• Function.Injective.orderedCommSemiring f hf zero one add mul nsmul npow natCast = OrderedCommSemiring.mk ⋯

@[reducible, inline]

abbrev Function.Injective.orderedRing {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [SMul ℕ β] [SMul ℤ β] [Pow β ℕ] [NatCast β] [IntCast β] (f : β → α) (hf : Function.Injective f) [OrderedRing 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 : ℕ) (x : β), f (n • x) = n • f x) (zsmul : ∀ (n : ℤ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (intCast : ∀ (n : ℤ), f ↑n = ↑n) : OrderedRing β

Pullback an OrderedRing under an injective map.

Equations

• Function.Injective.orderedRing f hf zero one add mul neg sub nsmul zsmul npow natCast intCast = OrderedRing.mk ⋯ ⋯ ⋯

@[reducible, inline]

abbrev Function.Injective.orderedCommRing {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [SMul ℕ β] [SMul ℤ β] [Pow β ℕ] [NatCast β] [IntCast β] (f : β → α) (hf : Function.Injective f) [OrderedCommRing 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 : ℕ) (x : β), f (n • x) = n • f x) (zsmul : ∀ (n : ℤ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (intCast : ∀ (n : ℤ), f ↑n = ↑n) : OrderedCommRing β

Pullback an OrderedCommRing under an injective map.

Equations

• Function.Injective.orderedCommRing f hf zero one add mul neg sub nsmul zsmul npow natCast intCast = OrderedCommRing.mk ⋯

@[reducible, inline]

abbrev Function.Injective.strictOrderedSemiring {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [SMul ℕ β] [Pow β ℕ] [NatCast β] (f : β → α) (hf : Function.Injective f) [StrictOrderedSemiring 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) (nsmul : ∀ (n : ℕ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) : StrictOrderedSemiring β

Pullback a StrictOrderedSemiring under an injective map.

Equations

• Function.Injective.strictOrderedSemiring f hf zero one add mul nsmul npow natCast = StrictOrderedSemiring.mk ⋯ ⋯ ⋯ ⋯ ⋯

@[reducible, inline]

abbrev Function.Injective.strictOrderedCommSemiring {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [SMul ℕ β] [Pow β ℕ] [NatCast β] (f : β → α) (hf : Function.Injective f) [StrictOrderedCommSemiring 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) (nsmul : ∀ (n : ℕ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) : StrictOrderedCommSemiring β

Pullback a strictOrderedCommSemiring under an injective map.

Equations

• Function.Injective.strictOrderedCommSemiring f hf zero one add mul nsmul npow natCast = StrictOrderedCommSemiring.mk ⋯

@[reducible, inline]

abbrev Function.Injective.strictOrderedRing {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [SMul ℕ β] [SMul ℤ β] [Pow β ℕ] [NatCast β] [IntCast β] (f : β → α) (hf : Function.Injective f) [StrictOrderedRing 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 : ℕ) (x : β), f (n • x) = n • f x) (zsmul : ∀ (n : ℤ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (intCast : ∀ (n : ℤ), f ↑n = ↑n) : StrictOrderedRing β

Pullback a StrictOrderedRing under an injective map.

Equations

• Function.Injective.strictOrderedRing f hf zero one add mul neg sub nsmul zsmul npow natCast intCast = StrictOrderedRing.mk ⋯ ⋯ ⋯

@[reducible, inline]

abbrev Function.Injective.strictOrderedCommRing {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [SMul ℕ β] [SMul ℤ β] [Pow β ℕ] [NatCast β] [IntCast β] (f : β → α) (hf : Function.Injective f) [StrictOrderedCommRing 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 : ℕ) (x : β), f (n • x) = n • f x) (zsmul : ∀ (n : ℤ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (intCast : ∀ (n : ℤ), f ↑n = ↑n) : StrictOrderedCommRing β

Pullback a StrictOrderedCommRing under an injective map.

Equations

• Function.Injective.strictOrderedCommRing f hf zero one add mul neg sub nsmul zsmul npow natCast intCast = StrictOrderedCommRing.mk ⋯

@[reducible, inline]

abbrev Function.Injective.linearOrderedSemiring {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [SMul ℕ β] [Pow β ℕ] [NatCast β] [Max β] [Min β] (f : β → α) (hf : Function.Injective f) [LinearOrderedSemiring 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) (nsmul : ∀ (n : ℕ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (sup : ∀ (x y : β), f (x ⊔ y) = f x ⊔ f y) (inf : ∀ (x y : β), f (x ⊓ y) = f x ⊓ f y) : LinearOrderedSemiring β

Pullback a LinearOrderedSemiring under an injective map.

Equations

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

@[reducible, inline]

abbrev Function.Injective.linearOrderedCommSemiring {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [SMul ℕ β] [Pow β ℕ] [NatCast β] [Max β] [Min β] (f : β → α) (hf : Function.Injective f) [LinearOrderedCommSemiring 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) (nsmul : ∀ (n : ℕ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (hsup : ∀ (x y : β), f (x ⊔ y) = f x ⊔ f y) (hinf : ∀ (x y : β), f (x ⊓ y) = f x ⊓ f y) : LinearOrderedCommSemiring β

Pullback a LinearOrderedSemiring under an injective map.

Equations

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

@[reducible, inline]

abbrev Function.Injective.linearOrderedRing {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [SMul ℕ β] [SMul ℤ β] [Pow β ℕ] [NatCast β] [IntCast β] [Max β] [Min β] (f : β → α) (hf : Function.Injective f) [LinearOrderedRing 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 : ℕ) (x : β), f (n • x) = n • f x) (zsmul : ∀ (n : ℤ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (intCast : ∀ (n : ℤ), f ↑n = ↑n) (hsup : ∀ (x y : β), f (x ⊔ y) = f x ⊔ f y) (hinf : ∀ (x y : β), f (x ⊓ y) = f x ⊓ f y) : LinearOrderedRing β

Pullback a LinearOrderedRing under an injective map.

Equations

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

@[reducible, inline]

abbrev Function.Injective.linearOrderedCommRing {α : Type u_1} {β : Type u_2} [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [SMul ℕ β] [SMul ℤ β] [Pow β ℕ] [NatCast β] [IntCast β] [Max β] [Min β] (f : β → α) (hf : Function.Injective f) [LinearOrderedCommRing 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 : ℕ) (x : β), f (n • x) = n • f x) (zsmul : ∀ (n : ℤ) (x : β), f (n • x) = n • f x) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (intCast : ∀ (n : ℤ), f ↑n = ↑n) (sup : ∀ (x y : β), f (x ⊔ y) = f x ⊔ f y) (inf : ∀ (x y : β), f (x ⊓ y) = f x ⊓ f y) : LinearOrderedCommRing β

Pullback a LinearOrderedCommRing under an injective map.

Equations

• Function.Injective.linearOrderedCommRing f hf zero one add mul neg sub nsmul zsmul npow natCast intCast sup inf = LinearOrderedCommRing.mk ⋯