Documentation

Mathlib.Algebra.Order.Monoid.Units

Units in ordered monoids #

source
instance AddUnits.instPreorderAddUnits {α : Type u_1} [AddMonoid α] [Preorder α] :
Preorder (AddUnits α)
source
instance Units.instPreorderUnits {α : Type u_1} [Monoid α] [Preorder α] :
Preorder αˣ
source
@[simp]
theorem AddUnits.val_le_val {α : Type u_1} [AddMonoid α] [Preorder α] {a : AddUnits α} {b : AddUnits α} :
↑a ≤ ↑b ↔ a ≤ b
source
@[simp]
theorem Units.val_le_val {α : Type u_1} [Monoid α] [Preorder α] {a : αˣ} {b : αˣ} :
↑a ≤ ↑b ↔ a ≤ b
source
@[simp]
theorem AddUnits.val_lt_val {α : Type u_1} [AddMonoid α] [Preorder α] {a : AddUnits α} {b : AddUnits α} :
↑a < ↑b ↔ a < b
source
@[simp]
theorem Units.val_lt_val {α : Type u_1} [Monoid α] [Preorder α] {a : αˣ} {b : αˣ} :
↑a < ↑b ↔ a < b
source
instance AddUnits.instPartialOrderAddUnits {α : Type u_1} [AddMonoid α] [PartialOrder α] :
PartialOrder (AddUnits α)
source
instance Units.instPartialOrderUnits {α : Type u_1} [Monoid α] [PartialOrder α] :
PartialOrder αˣ
source
instance AddUnits.instLinearOrderAddUnits {α : Type u_1} [AddMonoid α] [LinearOrder α] :
LinearOrder (AddUnits α)
source
instance Units.instLinearOrderUnits {α : Type u_1} [Monoid α] [LinearOrder α] :
LinearOrder αˣ
source
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
source
def AddUnits.orderEmbeddingVal {α : Type u_1} [AddMonoid α] [LinearOrder α] :
AddUnits α ↪o α

val : add_units α → α as an order embedding.

Instances For
    source
    @[simp]
    theorem AddUnits.orderEmbeddingVal_apply {α : Type u_1} [AddMonoid α] [LinearOrder α] :
    ↑AddUnits.orderEmbeddingVal = AddUnits.val
    source
    @[simp]
    theorem Units.orderEmbeddingVal_apply {α : Type u_1} [Monoid α] [LinearOrder α] :
    ↑Units.orderEmbeddingVal = Units.val
    source
    def Units.orderEmbeddingVal {α : Type u_1} [Monoid α] [LinearOrder α] :
    αˣ ↪o α

    val : αˣ → α as an order embedding.

    Instances For
      source
      @[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
      source
      @[simp]
      theorem Units.max_val {α : Type u_1} [Monoid α] [LinearOrder α] {a : αˣ} {b : αˣ} :
      ↑(max a b) = max ↑a ↑b
      source
      @[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
      source
      @[simp]
      theorem Units.min_val {α : Type u_1} [Monoid α] [LinearOrder α] {a : αˣ} {b : αˣ} :
      ↑(min a b) = min ↑a ↑b