# Documentation

Mathlib.Algebra.Order.Monoid.Units

# Units in ordered monoids #

instance AddUnits.instPreorderAddUnits {α : Type u_1} [inst : ] [inst : ] :
Equations
instance Units.instPreorderUnits {α : Type u_1} [inst : ] [inst : ] :
Equations
@[simp]
theorem AddUnits.val_le_val {α : Type u_1} [inst : ] [inst : ] {a : } {b : } :
a b a b
@[simp]
theorem Units.val_le_val {α : Type u_1} [inst : ] [inst : ] {a : αˣ} {b : αˣ} :
a b a b
@[simp]
theorem AddUnits.val_lt_val {α : Type u_1} [inst : ] [inst : ] {a : } {b : } :
a < b a < b
@[simp]
theorem Units.val_lt_val {α : Type u_1} [inst : ] [inst : ] {a : αˣ} {b : αˣ} :
a < b a < b
instance AddUnits.instPartialOrderAddUnits {α : Type u_1} [inst : ] [inst : ] :
Equations
instance Units.instPartialOrderUnits {α : Type u_1} [inst : ] [inst : ] :
Equations
instance AddUnits.instLinearOrderAddUnits {α : Type u_1} [inst : ] [inst : ] :
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instance Units.instLinearOrderUnits {α : Type u_1} [inst : ] [inst : ] :
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def AddUnits.orderEmbeddingVal.proof_1 {α : Type u_1} [inst : ] [inst : ] :
∀ {a b : }, { toFun := AddUnits.val, inj' := (_ : Function.Injective fun u => u) } a { toFun := AddUnits.val, inj' := (_ : Function.Injective fun u => u) } b { toFun := AddUnits.val, inj' := (_ : Function.Injective fun u => u) } a { toFun := AddUnits.val, inj' := (_ : Function.Injective fun u => u) } b
Equations
• One or more equations did not get rendered due to their size.
def AddUnits.orderEmbeddingVal {α : Type u_1} [inst : ] [inst : ] :
↪o α

val : add_units α → α as an order embedding.

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

val : αˣ → α as an order embedding.

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