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Mathlib.Algebra.Order.Field.InjSurj

Pulling back linearly ordered fields along injective maps. #

@[reducible]
def Function.Injective.linearOrderedSemifield {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [Zero β] [One β] [Add β] [Mul β] [Pow β ] [SMul β] [NatCast β] [Inv β] [Div β] [Pow β ] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (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) (inv : ∀ (x : β), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : β), f (x / y) = f x / f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) (hsup : ∀ (x y : β), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

Pullback a LinearOrderedSemifield under an injective map.

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    @[reducible]
    def Function.Injective.linearOrderedField {α : Type u_2} {β : Type u_3} [LinearOrderedField α] [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [Pow β ] [SMul β] [SMul β] [SMul β] [NatCast β] [IntCast β] [RatCast β] [Inv β] [Div β] [Pow β ] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (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) (inv : ∀ (x : β), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : β), f (x / y) = f x / f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (zsmul : ∀ (x : β) (n : ), f (n x) = n f x) (qsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) (int_cast : ∀ (n : ), f n = n) (rat_cast : ∀ (n : ), f n = n) (hsup : ∀ (x y : β), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

    Pullback a LinearOrderedField under an injective map.

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