Documentation

Mathlib.Algebra.Order.Monoid.Units

Units in ordered monoids #

Equations
instance Units.instPreorderUnits {α : Type u_1} [Monoid α] [Preorder α] :
Equations
@[simp]
theorem AddUnits.val_le_val {α : Type u_1} [AddMonoid α] [Preorder α] {a : AddUnits α} {b : AddUnits α} :
a b a b
@[simp]
theorem Units.val_le_val {α : Type u_1} [Monoid α] [Preorder α] {a : αˣ} {b : αˣ} :
a b a b
@[simp]
theorem AddUnits.val_lt_val {α : Type u_1} [AddMonoid α] [Preorder α] {a : AddUnits α} {b : AddUnits α} :
a < b a < b
@[simp]
theorem Units.val_lt_val {α : Type u_1} [Monoid α] [Preorder α] {a : αˣ} {b : αˣ} :
a < b a < b
Equations
Equations
Equations
Equations
theorem AddUnits.orderEmbeddingVal.proof_1 {α : Type u_1} [AddMonoid α] [LinearOrder α] :
∀ {a b : AddUnits α}, { toFun := AddUnits.val, inj' := (_ : Function.Injective AddUnits.val) } a { toFun := AddUnits.val, inj' := (_ : Function.Injective AddUnits.val) } b { toFun := AddUnits.val, inj' := (_ : Function.Injective AddUnits.val) } a { toFun := AddUnits.val, inj' := (_ : Function.Injective AddUnits.val) } b

val : add_units α → α as an order embedding.

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    @[simp]
    theorem AddUnits.orderEmbeddingVal_apply {α : Type u_1} [AddMonoid α] [LinearOrder α] :
    AddUnits.orderEmbeddingVal = AddUnits.val
    @[simp]
    theorem Units.orderEmbeddingVal_apply {α : Type u_1} [Monoid α] [LinearOrder α] :
    Units.orderEmbeddingVal = Units.val
    def Units.orderEmbeddingVal {α : Type u_1} [Monoid α] [LinearOrder α] :
    αˣ ↪o α

    val : αˣ → α as an order embedding.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      @[simp]
      theorem AddUnits.max_val {α : Type u_1} [AddMonoid α] [LinearOrder α] {a : AddUnits α} {b : AddUnits α} :
      (max a b) = max a b
      @[simp]
      theorem Units.max_val {α : Type u_1} [Monoid α] [LinearOrder α] {a : αˣ} {b : αˣ} :
      (max a b) = max a b
      @[simp]
      theorem AddUnits.min_val {α : Type u_1} [AddMonoid α] [LinearOrder α] {a : AddUnits α} {b : AddUnits α} :
      (min a b) = min a b
      @[simp]
      theorem Units.min_val {α : Type u_1} [Monoid α] [LinearOrder α] {a : αˣ} {b : αˣ} :
      (min a b) = min a b