# Units (i.e., invertible elements) of a monoid #

An element of a Monoid is a unit if it has a two-sided inverse.

## Main declarations #

• Units M: the group of units (i.e., invertible elements) of a monoid.
• IsUnit x: a predicate asserting that x is a unit (i.e., invertible element) of a monoid.

For both declarations, there is an additive counterpart: AddUnits and IsAddUnit. See also Prime, Associated, and Irreducible in Mathlib.Algebra.Associated.

## Notation #

We provide Mˣ as notation for Units M, resembling the notation $R^{\times}$ for the units of a ring, which is common in mathematics.

## TODO #

The results here should be used to golf the basic Group lemmas.

structure Units (α : Type u) [] :

Units of a Monoid, bundled version. Notation: αˣ.

An element of a Monoid is a unit if it has a two-sided inverse. This version bundles the inverse element so that it can be computed. For a predicate see IsUnit.

• val : α

The underlying value in the base Monoid.

• inv : α

The inverse value of val in the base Monoid.

• val_inv : self * self.inv = 1

inv is the right inverse of val in the base Monoid.

• inv_val : self.inv * self = 1

inv is the left inverse of val in the base Monoid.

Instances For
theorem Units.val_inv {α : Type u} [] (self : αˣ) :
self * self.inv = 1

inv is the right inverse of val in the base Monoid.

theorem Units.inv_val {α : Type u} [] (self : αˣ) :
self.inv * self = 1

inv is the left inverse of val in the base Monoid.

Units of a Monoid, bundled version. Notation: αˣ.

An element of a Monoid is a unit if it has a two-sided inverse. This version bundles the inverse element so that it can be computed. For a predicate see IsUnit.

Equations
Instances For
structure AddUnits (α : Type u) [] :

Units of an AddMonoid, bundled version.

An element of an AddMonoid is a unit if it has a two-sided additive inverse. This version bundles the inverse element so that it can be computed. For a predicate see isAddUnit.

• val : α

The underlying value in the base AddMonoid.

• neg : α

The additive inverse value of val in the base AddMonoid.

• val_neg : self + self.neg = 0

neg is the right additive inverse of val in the base AddMonoid.

• neg_val : self.neg + self = 0

neg is the left additive inverse of val in the base AddMonoid.

Instances For
theorem AddUnits.val_neg {α : Type u} [] (self : ) :
self + self.neg = 0

neg is the right additive inverse of val in the base AddMonoid.

theorem AddUnits.neg_val {α : Type u} [] (self : ) :
self.neg + self = 0

neg is the left additive inverse of val in the base AddMonoid.

theorem unique_zero {α : Type u_1} [] [Zero α] :
default = 0
theorem unique_one {α : Type u_1} [] [One α] :
default = 1

An additive unit can be interpreted as a term in the base AddMonoid.

Equations
instance Units.instCoeHead {α : Type u} [] :

A unit can be interpreted as a term in the base Monoid.

Equations
• Units.instCoeHead = { coe := Units.val }
instance AddUnits.instNeg {α : Type u} [] :

The additive inverse of an additive unit in an AddMonoid.

Equations
• AddUnits.instNeg = { neg := fun (u : ) => { val := u.neg, neg := u, val_neg := , neg_val := } }
instance Units.instInv {α : Type u} [] :

The inverse of a unit in a Monoid.

Equations
• Units.instInv = { inv := fun (u : αˣ) => { val := u.inv, inv := u, val_inv := , inv_val := } }
def AddUnits.Simps.val_neg {α : Type u} [] (u : ) :
α

See Note [custom simps projection]

Equations
• = (-u)
Instances For
def Units.Simps.val_inv {α : Type u} [] (u : αˣ) :
α

See Note [custom simps projection]

Equations
Instances For
theorem AddUnits.val_mk {α : Type u} [] (a : α) (b : α) (h₁ : a + b = 0) (h₂ : b + a = 0) :
{ val := a, neg := b, val_neg := h₁, neg_val := h₂ } = a
theorem Units.val_mk {α : Type u} [] (a : α) (b : α) (h₁ : a * b = 1) (h₂ : b * a = 1) :
{ val := a, inv := b, val_inv := h₁, inv_val := h₂ } = a
theorem AddUnits.ext {α : Type u} [] :
theorem Units.ext_iff {α : Type u} [] {a₁ : αˣ} {a₂ : αˣ} :
a₁ = a₂ a₁ = a₂
theorem AddUnits.ext_iff {α : Type u} [] {a₁ : } {a₂ : } :
a₁ = a₂ a₁ = a₂
theorem Units.ext {α : Type u} [] :
theorem AddUnits.eq_iff {α : Type u} [] {a : } {b : } :
a = b a = b
theorem Units.eq_iff {α : Type u} [] {a : αˣ} {b : αˣ} :
a = b a = b
instance AddUnits.instDecidableEq {α : Type u} [] [] :

Additive units have decidable equality if the base AddMonoid has deciable equality.

Equations
instance Units.instDecidableEq {α : Type u} [] [] :

Units have decidable equality if the base Monoid has decidable equality.

Equations
@[simp]
theorem AddUnits.mk_val {α : Type u} [] (u : ) (y : α) (h₁ : u + y = 0) (h₂ : y + u = 0) :
{ val := u, neg := y, val_neg := h₁, neg_val := h₂ } = u
@[simp]
theorem Units.mk_val {α : Type u} [] (u : αˣ) (y : α) (h₁ : u * y = 1) (h₂ : y * u = 1) :
{ val := u, inv := y, val_inv := h₁, inv_val := h₂ } = u
theorem AddUnits.copy.proof_2 {α : Type u_1} [] (u : ) (val : α) (hv : val = u) (inv : α) (hi : inv = (-u)) :
inv + val = 0
def AddUnits.copy {α : Type u} [] (u : ) (val : α) (hv : val = u) (inv : α) (hi : inv = (-u)) :

Copy an AddUnit, adjusting definitional equalities.

Equations
• u.copy val hv inv hi = { val := val, neg := inv, val_neg := , neg_val := }
Instances For
theorem AddUnits.copy.proof_1 {α : Type u_1} [] (u : ) (val : α) (hv : val = u) (inv : α) (hi : inv = (-u)) :
val + inv = 0
@[simp]
theorem Units.val_inv_copy {α : Type u} [] (u : αˣ) (val : α) (hv : val = u) (inv : α) (hi : inv = u⁻¹) :
(u.copy val hv inv hi)⁻¹ = inv
@[simp]
theorem Units.val_copy {α : Type u} [] (u : αˣ) (val : α) (hv : val = u) (inv : α) (hi : inv = u⁻¹) :
(u.copy val hv inv hi) = val
@[simp]
theorem AddUnits.val_copy {α : Type u} [] (u : ) (val : α) (hv : val = u) (inv : α) (hi : inv = (-u)) :
(u.copy val hv inv hi) = val
@[simp]
theorem AddUnits.val_neg_copy {α : Type u} [] (u : ) (val : α) (hv : val = u) (inv : α) (hi : inv = (-u)) :
(-u.copy val hv inv hi) = inv
def Units.copy {α : Type u} [] (u : αˣ) (val : α) (hv : val = u) (inv : α) (hi : inv = u⁻¹) :
αˣ

Copy a unit, adjusting definition equalities.

Equations
• u.copy val hv inv hi = { val := val, inv := inv, val_inv := , inv_val := }
Instances For
theorem AddUnits.copy_eq {α : Type u} [] (u : ) (val : α) (hv : val = u) (inv : α) (hi : inv = (-u)) :
u.copy val hv inv hi = u
theorem Units.copy_eq {α : Type u} [] (u : αˣ) (val : α) (hv : val = u) (inv : α) (hi : inv = u⁻¹) :
u.copy val hv inv hi = u
theorem AddUnits.instAdd.proof_2 {α : Type u_1} [] (u₁ : ) (u₂ : ) :
u₂.neg + u₁.neg + (u₁ + u₂) = 0
theorem AddUnits.instAdd.proof_1 {α : Type u_1} [] (u₁ : ) (u₂ : ) :
u₁ + u₂ + (u₂.neg + u₁.neg) = 0

Equations
• AddUnits.instAdd = { add := fun (u₁ u₂ : ) => { val := u₁ + u₂, neg := u₂.neg + u₁.neg, val_neg := , neg_val := } }
instance Units.instMul {α : Type u} [] :

Units of a monoid have an induced multiplication.

Equations
• Units.instMul = { mul := fun (u₁ u₂ : αˣ) => { val := u₁ * u₂, inv := u₂.inv * u₁.inv, val_inv := , inv_val := } }
theorem AddUnits.instZero.proof_1 {α : Type u_1} [] :
0 + 0 = 0
instance AddUnits.instZero {α : Type u} [] :

Equations
• AddUnits.instZero = { zero := { val := 0, neg := 0, val_neg := , neg_val := } }
instance Units.instOne {α : Type u} [] :

Units of a monoid have a unit

Equations
• Units.instOne = { one := { val := 1, inv := 1, val_inv := , inv_val := } }

Equations
u + 0 = u
0 + u = u
instance Units.instMulOneClass {α : Type u} [] :

Units of a monoid have a multiplication and multiplicative identity.

Equations
• Units.instMulOneClass =
instance AddUnits.instInhabited {α : Type u} [] :

Additive units of an additive monoid are inhabited because 0 is an additive unit.

Equations
• AddUnits.instInhabited = { default := 0 }
instance Units.instInhabited {α : Type u} [] :

Units of a monoid are inhabited because 1 is a unit.

Equations
• Units.instInhabited = { default := 1 }
instance AddUnits.instRepr {α : Type u} [] [Repr α] :

Additive units of an additive monoid have a representation of the base value in the AddMonoid.

Equations
instance Units.instRepr {α : Type u} [] [Repr α] :

Units of a monoid have a representation of the base value in the Monoid.

Equations
• Units.instRepr = { reprPrec := reprPrec Units.val }
@[simp]
theorem AddUnits.val_add {α : Type u} [] (a : ) (b : ) :
(a + b) = a + b
@[simp]
theorem Units.val_mul {α : Type u} [] (a : αˣ) (b : αˣ) :
(a * b) = a * b
@[simp]
theorem AddUnits.val_zero {α : Type u} [] :
0 = 0
@[simp]
theorem Units.val_one {α : Type u} [] :
1 = 1
@[simp]
theorem AddUnits.val_eq_zero {α : Type u} [] {a : } :
a = 0 a = 0
@[simp]
theorem Units.val_eq_one {α : Type u} [] {a : αˣ} :
a = 1 a = 1
@[simp]
theorem AddUnits.neg_mk {α : Type u} [] (x : α) (y : α) (h₁ : x + y = 0) (h₂ : y + x = 0) :
-{ val := x, neg := y, val_neg := h₁, neg_val := h₂ } = { val := y, neg := x, val_neg := h₂, neg_val := h₁ }
@[simp]
theorem Units.inv_mk {α : Type u} [] (x : α) (y : α) (h₁ : x * y = 1) (h₂ : y * x = 1) :
{ val := x, inv := y, val_inv := h₁, inv_val := h₂ }⁻¹ = { val := y, inv := x, val_inv := h₂, inv_val := h₁ }
@[simp]
theorem AddUnits.neg_eq_val_neg {α : Type u} [] (a : ) :
a.neg = (-a)
@[simp]
theorem Units.inv_eq_val_inv {α : Type u} [] (a : αˣ) :
a.inv = a⁻¹
@[simp]
(-a) + a = 0
@[simp]
theorem Units.inv_mul {α : Type u} [] (a : αˣ) :
a⁻¹ * a = 1
@[simp]
a + (-a) = 0
@[simp]
theorem Units.mul_inv {α : Type u} [] (a : αˣ) :
a * a⁻¹ = 1
theorem Units.commute_coe_inv {α : Type u} [] (a : αˣ) :
Commute a a⁻¹
theorem Units.commute_inv_coe {α : Type u} [] (a : αˣ) :
Commute a⁻¹ a
theorem AddUnits.neg_add_of_eq {α : Type u} [] {u : } {a : α} (h : u = a) :
(-u) + a = 0
theorem Units.inv_mul_of_eq {α : Type u} [] {u : αˣ} {a : α} (h : u = a) :
u⁻¹ * a = 1
theorem AddUnits.add_neg_of_eq {α : Type u} [] {u : } {a : α} (h : u = a) :
a + (-u) = 0
theorem Units.mul_inv_of_eq {α : Type u} [] {u : αˣ} {a : α} (h : u = a) :
a * u⁻¹ = 1
@[simp]
theorem AddUnits.add_neg_cancel_left {α : Type u} [] (a : ) (b : α) :
a + ((-a) + b) = b
@[simp]
theorem Units.mul_inv_cancel_left {α : Type u} [] (a : αˣ) (b : α) :
a * (a⁻¹ * b) = b
@[simp]
theorem AddUnits.neg_add_cancel_left {α : Type u} [] (a : ) (b : α) :
(-a) + (a + b) = b
@[simp]
theorem Units.inv_mul_cancel_left {α : Type u} [] (a : αˣ) (b : α) :
a⁻¹ * (a * b) = b
@[simp]
theorem AddUnits.add_neg_cancel_right {α : Type u} [] (a : α) (b : ) :
a + b + (-b) = a
@[simp]
theorem Units.mul_inv_cancel_right {α : Type u} [] (a : α) (b : αˣ) :
a * b * b⁻¹ = a
@[simp]
theorem AddUnits.neg_add_cancel_right {α : Type u} [] (a : α) (b : ) :
a + (-b) + b = a
@[simp]
theorem Units.inv_mul_cancel_right {α : Type u} [] (a : α) (b : αˣ) :
a * b⁻¹ * b = a
@[simp]
theorem AddUnits.add_right_inj {α : Type u} [] (a : ) {b : α} {c : α} :
a + b = a + c b = c
@[simp]
theorem Units.mul_right_inj {α : Type u} [] (a : αˣ) {b : α} {c : α} :
a * b = a * c b = c
@[simp]
theorem AddUnits.add_left_inj {α : Type u} [] (a : ) {b : α} {c : α} :
b + a = c + a b = c
@[simp]
theorem Units.mul_left_inj {α : Type u} [] (a : αˣ) {b : α} {c : α} :
b * a = c * a b = c
theorem AddUnits.eq_add_neg_iff_add_eq {α : Type u} [] (c : ) {a : α} {b : α} :
a = b + (-c) a + c = b
theorem Units.eq_mul_inv_iff_mul_eq {α : Type u} [] (c : αˣ) {a : α} {b : α} :
a = b * c⁻¹ a * c = b
theorem AddUnits.eq_neg_add_iff_add_eq {α : Type u} [] (b : ) {a : α} {c : α} :
a = (-b) + c b + a = c
theorem Units.eq_inv_mul_iff_mul_eq {α : Type u} [] (b : αˣ) {a : α} {c : α} :
a = b⁻¹ * c b * a = c
theorem AddUnits.neg_add_eq_iff_eq_add {α : Type u} [] (a : ) {b : α} {c : α} :
(-a) + b = c b = a + c
theorem Units.inv_mul_eq_iff_eq_mul {α : Type u} [] (a : αˣ) {b : α} {c : α} :
a⁻¹ * b = c b = a * c
theorem AddUnits.add_neg_eq_iff_eq_add {α : Type u} [] (b : ) {a : α} {c : α} :
a + (-b) = c a = c + b
theorem Units.mul_inv_eq_iff_eq_mul {α : Type u} [] (b : αˣ) {a : α} {c : α} :
a * b⁻¹ = c a = c * b
theorem AddUnits.neg_eq_of_add_eq_zero_left {α : Type u} [] {u : } {a : α} (h : a + u = 0) :
(-u) = a
theorem Units.inv_eq_of_mul_eq_one_left {α : Type u} [] {u : αˣ} {a : α} (h : a * u = 1) :
u⁻¹ = a
theorem AddUnits.neg_eq_of_add_eq_zero_right {α : Type u} [] {u : } {a : α} (h : u + a = 0) :
(-u) = a
theorem Units.inv_eq_of_mul_eq_one_right {α : Type u} [] {u : αˣ} {a : α} (h : u * a = 1) :
u⁻¹ = a
theorem AddUnits.eq_neg_of_add_eq_zero_left {α : Type u} [] {u : } {a : α} (h : u + a = 0) :
a = (-u)
theorem Units.eq_inv_of_mul_eq_one_left {α : Type u} [] {u : αˣ} {a : α} (h : u * a = 1) :
a = u⁻¹
theorem AddUnits.eq_neg_of_add_eq_zero_right {α : Type u} [] {u : } {a : α} (h : a + u = 0) :
a = (-u)
theorem Units.eq_inv_of_mul_eq_one_right {α : Type u} [] {u : αˣ} {a : α} (h : a * u = 1) :
a = u⁻¹
theorem AddUnits.instAddMonoid.proof_7 {α : Type u_1} [] (n : ) (a : ) :
(fun (n : ) (a : ) => { val := n a, neg := n (-a), val_neg := , neg_val := }) (n + 1) a = (fun (n : ) (a : ) => { val := n a, neg := n (-a), val_neg := , neg_val := }) n a + a
∀ (x x_1 x_2 : ), x + x_1 + x_2 = x + (x_1 + x_2)
Equations
• AddUnits.instAddMonoid = AddMonoid.mk (fun (n : ) (a : ) => { val := n a, neg := n (-a), val_neg := , neg_val := })
a + 0 = a
0 + a = a
theorem AddUnits.instAddMonoid.proof_4 {α : Type u_1} [] (n : ) (a : ) :
n a + n (-a) = 0
theorem AddUnits.instAddMonoid.proof_5 {α : Type u_1} [] (n : ) (a : ) :
n (-a) + n a = 0
(fun (n : ) (a : ) => { val := n a, neg := n (-a), val_neg := , neg_val := }) 0 a = 0
instance Units.instMonoid {α : Type u} [] :
Equations
• Units.instMonoid = Monoid.mk (fun (n : ) (a : αˣ) => { val := a ^ n, inv := a⁻¹ ^ n, val_inv := , inv_val := })
theorem AddUnits.instSub.proof_1 {α : Type u_1} [] (a : ) (b : ) :
a + (-b) + (b + (-a)) = 0
instance AddUnits.instSub {α : Type u} [] :

Equations
• AddUnits.instSub = { sub := fun (a b : ) => { val := a + (-b), neg := b + (-a), val_neg := , neg_val := } }
theorem AddUnits.instSub.proof_2 {α : Type u_1} [] (a : ) (b : ) :
b + (-a) + (a + (-b)) = 0
instance Units.instDiv {α : Type u} [] :

Units of a monoid have division

Equations
• Units.instDiv = { div := fun (a b : αˣ) => { val := a * b⁻¹, inv := b * a⁻¹, val_inv := , inv_val := } }
theorem AddUnits.instSubNegAddMonoid.proof_4 {α : Type u_1} [] (n : ) (a : ) :
-((n + 1) a) = -((n + 1) a)
theorem AddUnits.instSubNegAddMonoid.proof_3 {α : Type u_1} [] (n : ) (a : ) :
(n + 1) a = n a + a

Additive units of an additive monoid form a SubNegMonoid.

Equations
• AddUnits.instSubNegAddMonoid = SubNegMonoid.mk (fun (n : ) (a : ) => match n, a with | , a => n a | , a => -(n.succ a))
∀ (a b : ), a - b = a - b
0 a = 0
instance Units.instDivInvMonoid {α : Type u} [] :

Units of a monoid form a DivInvMonoid.

Equations
• Units.instDivInvMonoid = DivInvMonoid.mk (fun (n : ) (a : αˣ) => match n, a with | , a => a ^ n | , a => (a ^ n.succ)⁻¹)

Equations
-u + u = 0
instance Units.instGroup {α : Type u} [] :

Units of a monoid form a group.

Equations
• Units.instGroup =
∀ (x x_1 : ), x + x_1 = x_1 + x

Equations
instance Units.instCommGroupUnits {α : Type u_1} [] :

Units of a commutative monoid form a commutative group.

Equations
• Units.instCommGroupUnits =
@[simp]
theorem AddUnits.val_nsmul_eq_nsmul_val {α : Type u} [] (a : ) (n : ) :
(n a) = n a
@[simp]
theorem Units.val_pow_eq_pow_val {α : Type u} [] (a : αˣ) (n : ) :
(a ^ n) = a ^ n
@[simp]
theorem AddUnits.add_neg_eq_zero {α : Type u} [] {u : } {a : α} :
a + (-u) = 0 a = u
@[simp]
theorem Units.mul_inv_eq_one {α : Type u} [] {u : αˣ} {a : α} :
a * u⁻¹ = 1 a = u
@[simp]
theorem AddUnits.neg_add_eq_zero {α : Type u} [] {u : } {a : α} :
(-u) + a = 0 u = a
@[simp]
theorem Units.inv_mul_eq_one {α : Type u} [] {u : αˣ} {a : α} :
u⁻¹ * a = 1 u = a
theorem AddUnits.add_eq_zero_iff_eq_neg {α : Type u} [] {u : } {a : α} :
a + u = 0 a = (-u)
theorem Units.mul_eq_one_iff_eq_inv {α : Type u} [] {u : αˣ} {a : α} :
a * u = 1 a = u⁻¹
theorem AddUnits.add_eq_zero_iff_neg_eq {α : Type u} [] {u : } {a : α} :
u + a = 0 (-u) = a
theorem Units.mul_eq_one_iff_inv_eq {α : Type u} [] {u : αˣ} {a : α} :
u * a = 1 u⁻¹ = a
theorem AddUnits.neg_unique {α : Type u} [] {u₁ : } {u₂ : } (h : u₁ = u₂) :
(-u₁) = (-u₂)
theorem Units.inv_unique {α : Type u} [] {u₁ : αˣ} {u₂ : αˣ} (h : u₁ = u₂) :
u₁⁻¹ = u₂⁻¹
@[simp]
theorem AddUnits.val_neg_eq_neg_val {α : Type u} (u : ) :
(-u) = -u
@[simp]
theorem Units.val_inv_eq_inv_val {α : Type u} [] (u : αˣ) :
u⁻¹ = (↑u)⁻¹
@[simp]
theorem AddUnits.val_sub_eq_sub_val {α : Type u} (u₁ : ) (u₂ : ) :
(u₁ - u₂) = u₁ - u₂
@[simp]
theorem Units.val_div_eq_div_val {α : Type u} [] (u₁ : αˣ) (u₂ : αˣ) :
(u₁ / u₂) = u₁ / u₂
theorem AddUnits.mkOfAddEqZero.proof_1 {α : Type u_1} [] (a : α) (b : α) (hab : a + b = 0) :
b + a = 0
def AddUnits.mkOfAddEqZero {α : Type u} [] (a : α) (b : α) (hab : a + b = 0) :

For a, b in an AddCommMonoid such that a + b = 0, makes an addUnit out of a.

Equations
• = { val := a, neg := b, val_neg := hab, neg_val := }
Instances For
def Units.mkOfMulEqOne {α : Type u} [] (a : α) (b : α) (hab : a * b = 1) :
αˣ

For a, b in a CommMonoid such that a * b = 1, makes a unit out of a.

Equations
Instances For
@[simp]
theorem AddUnits.val_mkOfAddEqZero {α : Type u} [] {a : α} {b : α} (h : a + b = 0) :
= a
@[simp]
theorem Units.val_mkOfMulEqOne {α : Type u} [] {a : α} {b : α} (h : a * b = 1) :
(Units.mkOfMulEqOne a b h) = a
def divp {α : Type u} [] (a : α) (u : αˣ) :
α

Partial division. It is defined when the second argument is invertible, and unlike the division operator in DivisionRing it is not totalized at zero.

Equations
Instances For

Partial division. It is defined when the second argument is invertible, and unlike the division operator in DivisionRing it is not totalized at zero.

Equations
Instances For
@[simp]
theorem divp_self {α : Type u} [] (u : αˣ) :
u /ₚ u = 1
@[simp]
theorem divp_one {α : Type u} [] (a : α) :
a /ₚ 1 = a
theorem divp_assoc {α : Type u} [] (a : α) (b : α) (u : αˣ) :
a * b /ₚ u = a * (b /ₚ u)
theorem divp_assoc' {α : Type u} [] (x : α) (y : α) (u : αˣ) :
x * (y /ₚ u) = x * y /ₚ u

field_simp needs the reverse direction of divp_assoc to move all /ₚ to the right.

@[simp]
theorem divp_inv {α : Type u} [] {a : α} (u : αˣ) :
a /ₚ u⁻¹ = a * u
@[simp]
theorem divp_mul_cancel {α : Type u} [] (a : α) (u : αˣ) :
a /ₚ u * u = a
@[simp]
theorem mul_divp_cancel {α : Type u} [] (a : α) (u : αˣ) :
a * u /ₚ u = a
@[simp]
theorem divp_left_inj {α : Type u} [] (u : αˣ) {a : α} {b : α} :
a /ₚ u = b /ₚ u a = b
theorem divp_divp_eq_divp_mul {α : Type u} [] (x : α) (u₁ : αˣ) (u₂ : αˣ) :
x /ₚ u₁ /ₚ u₂ = x /ₚ (u₂ * u₁)
theorem divp_eq_iff_mul_eq {α : Type u} [] {x : α} {u : αˣ} {y : α} :
x /ₚ u = y y * u = x
theorem eq_divp_iff_mul_eq {α : Type u} [] {x : α} {u : αˣ} {y : α} :
x = y /ₚ u x * u = y
theorem divp_eq_one_iff_eq {α : Type u} [] {a : α} {u : αˣ} :
a /ₚ u = 1 a = u
@[simp]
theorem one_divp {α : Type u} [] (u : αˣ) :
1 /ₚ u = u⁻¹
theorem inv_eq_one_divp {α : Type u} [] (u : αˣ) :
u⁻¹ = 1 /ₚ u

Used for field_simp to deal with inverses of units.

theorem inv_eq_one_divp' {α : Type u} [] (u : αˣ) :
(1 / u) = 1 /ₚ u

Used for field_simp to deal with inverses of units. This form of the lemma is essential since field_simp likes to use inv_eq_one_div to rewrite ↑u⁻¹ = ↑(1 / u).

theorem val_div_eq_divp {α : Type u} [] (u₁ : αˣ) (u₂ : αˣ) :
(u₁ / u₂) = u₁ /ₚ u₂

field_simp moves division inside αˣ to the right, and this lemma lifts the calculation to α.

theorem AddLeftCancelMonoid.eq_zero_of_add_right {α : Type u} [] {a : α} {b : α} (h : a + b = 0) :
a = 0
theorem LeftCancelMonoid.eq_one_of_mul_right {α : Type u} [] [] {a : α} {b : α} (h : a * b = 1) :
a = 1
theorem AddLeftCancelMonoid.eq_zero_of_add_left {α : Type u} [] {a : α} {b : α} (h : a + b = 0) :
b = 0
theorem LeftCancelMonoid.eq_one_of_mul_left {α : Type u} [] [] {a : α} {b : α} (h : a * b = 1) :
b = 1
@[simp]
theorem AddLeftCancelMonoid.add_eq_zero {α : Type u} [] {a : α} {b : α} :
a + b = 0 a = 0 b = 0
@[simp]
theorem LeftCancelMonoid.mul_eq_one {α : Type u} [] [] {a : α} {b : α} :
a * b = 1 a = 1 b = 1
theorem AddLeftCancelMonoid.add_ne_zero {α : Type u} [] {a : α} {b : α} :
a + b 0 a 0 b 0
theorem LeftCancelMonoid.mul_ne_one {α : Type u} [] [] {a : α} {b : α} :
a * b 1 a 1 b 1
theorem AddRightCancelMonoid.eq_zero_of_add_right {α : Type u} [] {a : α} {b : α} (h : a + b = 0) :
a = 0
theorem RightCancelMonoid.eq_one_of_mul_right {α : Type u} [] {a : α} {b : α} (h : a * b = 1) :
a = 1
theorem AddRightCancelMonoid.eq_zero_of_add_left {α : Type u} [] {a : α} {b : α} (h : a + b = 0) :
b = 0
theorem RightCancelMonoid.eq_one_of_mul_left {α : Type u} [] {a : α} {b : α} (h : a * b = 1) :
b = 1
@[simp]
theorem AddRightCancelMonoid.add_eq_zero {α : Type u} [] {a : α} {b : α} :
a + b = 0 a = 0 b = 0
@[simp]
theorem RightCancelMonoid.mul_eq_one {α : Type u} [] {a : α} {b : α} :
a * b = 1 a = 1 b = 1
theorem AddRightCancelMonoid.add_ne_zero {α : Type u} [] {a : α} {b : α} :
a + b 0 a 0 b 0
theorem RightCancelMonoid.mul_ne_one {α : Type u} [] {a : α} {b : α} :
a * b 1 a 1 b 1
theorem eq_zero_of_add_right' {α : Type u} [] [] {a : α} {b : α} (h : a + b = 0) :
a = 0
theorem eq_one_of_mul_right' {α : Type u} [] [] {a : α} {b : α} (h : a * b = 1) :
a = 1
theorem eq_zero_of_add_left' {α : Type u} [] [] {a : α} {b : α} (h : a + b = 0) :
b = 0
theorem eq_one_of_mul_left' {α : Type u} [] [] {a : α} {b : α} (h : a * b = 1) :
b = 1
theorem add_eq_zero' {α : Type u} [] [] {a : α} {b : α} :
a + b = 0 a = 0 b = 0
theorem mul_eq_one' {α : Type u} [] [] {a : α} {b : α} :
a * b = 1 a = 1 b = 1
theorem add_ne_zero' {α : Type u} [] [] {a : α} {b : α} :
a + b 0 a 0 b 0
theorem mul_ne_one' {α : Type u} [] [] {a : α} {b : α} :
a * b 1 a 1 b 1
theorem divp_mul_eq_mul_divp {α : Type u} [] (x : α) (y : α) (u : αˣ) :
x /ₚ u * y = x * y /ₚ u
theorem divp_eq_divp_iff {α : Type u} [] {x : α} {y : α} {ux : αˣ} {uy : αˣ} :
x /ₚ ux = y /ₚ uy x * uy = y * ux
theorem divp_mul_divp {α : Type u} [] (x : α) (y : α) (ux : αˣ) (uy : αˣ) :
x /ₚ ux * (y /ₚ uy) = x * y /ₚ (ux * uy)
theorem eq_zero_of_add_right {α : Type u} [] [] {a : α} {b : α} (h : a + b = 0) :
a = 0
theorem eq_one_of_mul_right {α : Type u} [] [] {a : α} {b : α} (h : a * b = 1) :
a = 1
theorem eq_zero_of_add_left {α : Type u} [] [] {a : α} {b : α} (h : a + b = 0) :
b = 0
theorem eq_one_of_mul_left {α : Type u} [] [] {a : α} {b : α} (h : a * b = 1) :
b = 1
@[simp]
theorem add_eq_zero {α : Type u} [] [] {a : α} {b : α} :
a + b = 0 a = 0 b = 0
@[simp]
theorem mul_eq_one {α : Type u} [] [] {a : α} {b : α} :
a * b = 1 a = 1 b = 1
theorem add_ne_zero {α : Type u} [] [] {a : α} {b : α} :
a + b 0 a 0 b 0
theorem mul_ne_one {α : Type u} [] [] {a : α} {b : α} :
a * b 1 a 1 b 1

# IsUnit predicate #

def IsAddUnit {M : Type u_1} [] (a : M) :

An element a : M of an AddMonoid is an AddUnit if it has a two-sided additive inverse. The actual definition says that a is equal to some u : AddUnits M, where AddUnits M is a bundled version of IsAddUnit.

Equations
• = ∃ (u : ), u = a
Instances For
def IsUnit {M : Type u_1} [] (a : M) :

An element a : M of a Monoid is a unit if it has a two-sided inverse. The actual definition says that a is equal to some u : Mˣ, where Mˣ is a bundled version of IsUnit.

Equations
• = ∃ (u : Mˣ), u = a
Instances For
theorem isAddUnit_iff_exists {M : Type u_1} [] {x : M} :
∃ (b : M), x + b = 0 b + x = 0

See isAddUnit_iff_exists_and_exists for a similar lemma with two existentials.

theorem isUnit_iff_exists {M : Type u_1} [] {x : M} :
∃ (b : M), x * b = 1 b * x = 1

See isUnit_iff_exists_and_exists for a similar lemma with two existentials.

theorem isAddUnit_iff_exists_and_exists {M : Type u_1} [] {a : M} :
(∃ (b : M), a + b = 0) ∃ (c : M), c + a = 0

See isAddUnit_iff_exists for a similar lemma with one existential.

theorem isUnit_iff_exists_and_exists {M : Type u_1} [] {a : M} :
(∃ (b : M), a * b = 1) ∃ (c : M), c * a = 1

See isUnit_iff_exists for a similar lemma with one existential.

theorem isAddUnit_of_subsingleton {M : Type u_1} [] [] (a : M) :
theorem isUnit_of_subsingleton {M : Type u_1} [] [] (a : M) :
Equations
• =
instance instCanLiftUnitsValIsUnit {M : Type u_1} [] :
CanLift M Mˣ Units.val IsUnit
Equations
• =

A subsingleton AddMonoid has a unique additive unit.

Equations
theorem instUniqueAddUnitsOfSubsingleton.proof_1 {M : Type u_1} [] [] :
∀ (x : ), x = 0
instance instUniqueUnitsOfSubsingleton {M : Type u_1} [] [] :

A subsingleton Monoid has a unique unit.

Equations
• instUniqueUnitsOfSubsingleton = { toInhabited := Units.instInhabited, uniq := }
@[simp]
@[simp]
theorem Units.isUnit {M : Type u_1} [] (u : Mˣ) :
IsUnit u
@[simp]
theorem isAddUnit_zero {M : Type u_1} [] :
@[simp]
theorem isUnit_one {M : Type u_1} [] :
theorem isAddUnit_of_add_eq_zero {M : Type u_1} [] (a : M) (b : M) (h : a + b = 0) :
theorem isUnit_of_mul_eq_one {M : Type u_1} [] (a : M) (b : M) (h : a * b = 1) :
theorem isAddUnit_of_add_eq_zero_right {M : Type u_1} [] (a : M) (b : M) (h : a + b = 0) :
theorem isUnit_of_mul_eq_one_right {M : Type u_1} [] (a : M) (b : M) (h : a * b = 1) :
theorem IsAddUnit.exists_neg {M : Type u_1} [] {a : M} (h : ) :
∃ (b : M), a + b = 0
theorem IsUnit.exists_right_inv {M : Type u_1} [] {a : M} (h : ) :
∃ (b : M), a * b = 1
theorem IsAddUnit.exists_neg' {M : Type u_1} [] {a : M} (h : ) :
∃ (b : M), b + a = 0
theorem IsUnit.exists_left_inv {M : Type u_1} [] {a : M} (h : ) :
∃ (b : M), b * a = 1
theorem IsAddUnit.add {M : Type u_1} [] {a : M} {b : M} :
theorem IsUnit.mul {M : Type u_1} [] {a : M} {b : M} :
IsUnit (a * b)
theorem IsAddUnit.nsmul {M : Type u_1} [] {a : M} (n : ) :
theorem IsUnit.pow {M : Type u_1} [] {a : M} (n : ) :
IsUnit (a ^ n)
theorem units_eq_one {M : Type u_1} [] [] (u : Mˣ) :
u = 1
theorem isAddUnit_iff_eq_zero {M : Type u_1} [] [Unique (AddUnits M)] {x : M} :
x = 0
theorem isUnit_iff_eq_one {M : Type u_1} [] [] {x : M} :
x = 1
theorem isAddUnit_iff_exists_neg {M : Type u_1} [] {a : M} :
∃ (b : M), a + b = 0
theorem isUnit_iff_exists_inv {M : Type u_1} [] {a : M} :
∃ (b : M), a * b = 1
theorem isAddUnit_iff_exists_neg' {M : Type u_1} [] {a : M} :
∃ (b : M), b + a = 0
theorem isUnit_iff_exists_inv' {M : Type u_1} [] {a : M} :
∃ (b : M), b * a = 1
@[simp]

Addition of a u : AddUnits M on the right doesn't affect IsAddUnit.

@[simp]
theorem Units.isUnit_mul_units {M : Type u_1} [] (a : M) (u : Mˣ) :
IsUnit (a * u)

Multiplication by a u : Mˣ on the right doesn't affect IsUnit.

@[simp]

Addition of a u : AddUnits M on the left doesn't affect IsAddUnit.

@[simp]
theorem Units.isUnit_units_mul {M : Type u_3} [] (u : Mˣ) (a : M) :
IsUnit (u * a)

Multiplication by a u : Mˣ on the left doesn't affect IsUnit.

theorem isUnit_of_mul_isUnit_left {M : Type u_1} [] {x : M} {y : M} (hu : IsUnit (x * y)) :
theorem isUnit_of_mul_isUnit_right {M : Type u_1} [] {x : M} {y : M} (hu : IsUnit (x * y)) :
@[simp]
theorem IsAddUnit.add_iff {M : Type u_1} [] {x : M} {y : M} :
@[simp]
theorem IsUnit.mul_iff {M : Type u_1} [] {x : M} {y : M} :
IsUnit (x * y)
noncomputable def IsUnit.unit {M : Type u_1} [] {a : M} (h : ) :

The element of the group of units, corresponding to an element of a monoid which is a unit. When α is a DivisionMonoid, use IsUnit.unit' instead.

Equations
• h.unit = .copy a
Instances For
noncomputable def IsAddUnit.addUnit {N : Type u_2} [] {a : N} (h : ) :

"The element of the additive group of additive units, corresponding to an element of an additive monoid which is an additive unit. When α is a SubtractionMonoid, use IsAddUnit.addUnit' instead.

Equations
Instances For
@[simp]
@[simp]
theorem IsUnit.unit_of_val_units {M : Type u_1} [] {a : Mˣ} (h : IsUnit a) :
h.unit = a
@[simp]
theorem IsAddUnit.addUnit_spec {M : Type u_1} [] {a : M} (h : ) :
@[simp]
theorem IsUnit.unit_spec {M : Type u_1} [] {a : M} (h : ) :
h.unit = a
@[simp]
@[simp]
theorem IsUnit.unit_one {M : Type u_1} [] (h : ) :
h.unit = 1
@[simp]
theorem IsAddUnit.val_neg_add {M : Type u_1} [] {a : M} (h : ) :
@[simp]
theorem IsUnit.val_inv_mul {M : Type u_1} [] {a : M} (h : ) :
h.unit⁻¹ * a = 1
@[simp]
theorem IsAddUnit.add_val_neg {M : Type u_1} [] {a : M} (h : ) :
@[simp]
theorem IsUnit.mul_val_inv {M : Type u_1} [] {a : M} (h : ) :
a * h.unit⁻¹ = 1
instance IsAddUnit.instDecidableOfExistsAddUnitsEqVal {M : Type u_1} [] (x : M) [h : Decidable (∃ (u : ), u = x)] :

IsAddUnit x is decidable if we can decide if x comes from AddUnits M.

Equations
instance IsUnit.instDecidableOfExistsUnitsEqVal {M : Type u_1} [] (x : M) [h : Decidable (∃ (u : Mˣ), u = x)] :

IsUnit x is decidable if we can decide if x comes from Mˣ.

Equations
theorem IsAddUnit.add_left_inj {M : Type u_1} [] {a : M} {b : M} {c : M} (h : ) :
b + a = c + a b = c
theorem IsUnit.mul_left_inj {M : Type u_1} [] {a : M} {b : M} {c : M} (h : ) :
b * a = c * a b = c
theorem IsAddUnit.add_right_inj {M : Type u_1} [] {a : M} {b : M} {c : M} (h : ) :
a + b = a + c b = c
theorem IsUnit.mul_right_inj {M : Type u_1} [] {a : M} {b : M} {c : M} (h : ) :
a * b = a * c b = c
theorem IsAddUnit.add_left_cancel {M : Type u_1} [] {a : M} {b : M} {c : M} (h : ) :
a + b = a + cb = c
theorem IsUnit.mul_left_cancel {M : Type u_1} [] {a : M} {b : M} {c : M} (h : ) :
a * b = a * cb = c
theorem IsAddUnit.add_right_cancel {M : Type u_1} [] {a : M} {b : M} {c : M} (h : ) :
a + b = c + ba = c
theorem IsUnit.mul_right_cancel {M : Type u_1} [] {a : M} {b : M} {c : M} (h : ) :
a * b = c * ba = c
theorem IsAddUnit.add_right_injective {M : Type u_1} [] {a : M} (h : ) :
Function.Injective fun (x : M) => a + x
theorem IsUnit.mul_right_injective {M : Type u_1} [] {a : M} (h : ) :
Function.Injective fun (x : M) => a * x
theorem IsAddUnit.add_left_injective {M : Type u_1} [] {b : M} (h : ) :
Function.Injective fun (x : M) => x + b
theorem IsUnit.mul_left_injective {M : Type u_1} [] {b : M} (h : ) :
Function.Injective fun (x : M) => x * b
Function.Bijective fun (x : M) => a + x
theorem IsUnit.isUnit_iff_mulLeft_bijective {M : Type u_1} [] {a : M} :
Function.Bijective fun (x : M) => a * x
Function.Bijective fun (x : M) => x + a
theorem IsUnit.isUnit_iff_mulRight_bijective {M : Type u_1} [] {a : M} :
Function.Bijective fun (x : M) => x * a
@[simp]
-a + a = 0
@[simp]
theorem IsUnit.inv_mul_cancel {α : Type u} [] {a : α} :
a⁻¹ * a = 1
@[simp]
a + -a = 0
@[simp]
theorem IsUnit.mul_inv_cancel {α : Type u} [] {a : α} :
a * a⁻¹ = 1
def IsAddUnit.addUnit' {α : Type u} {a : α} (h : ) :

The element of the additive group of additive units, corresponding to an element of an additive monoid which is an additive unit. As opposed to IsAddUnit.addUnit, the negation is computable and comes from the negation on α. This is useful to transfer properties of negation in AddUnits α to α. See also toAddUnits.

Equations
• h.addUnit' = { val := a, neg := -a, val_neg := , neg_val := }
Instances For
@[simp]
theorem IsUnit.val_unit' {α : Type u} [] {a : α} (h : ) :
h.unit' = a
@[simp]
theorem IsAddUnit.val_addUnit' {α : Type u} {a : α} (h : ) :
def IsUnit.unit' {α : Type u} [] {a : α} (h : ) :
αˣ

The element of the group of units, corresponding to an element of a monoid which is a unit. As opposed to IsUnit.unit, the inverse is computable and comes from the inversion on α. This is useful to transfer properties of inversion in Units α to α. See also toUnits.

Equations
• h.unit' = { val := a, inv := a⁻¹, val_inv := , inv_val := }
Instances For
theorem IsAddUnit.val_neg_addUnit' {α : Type u} {a : α} (h : ) :
theorem IsUnit.val_inv_unit' {α : Type u} [] {a : α} (h : ) :
h.unit'⁻¹ = a⁻¹
@[simp]
theorem IsAddUnit.add_neg_cancel_left {α : Type u} {a : α} (h : ) (b : α) :
a + (-a + b) = b
@[simp]
theorem IsUnit.mul_inv_cancel_left {α : Type u} [] {a : α} (h : ) (b : α) :
a * (a⁻¹ * b) = b
@[simp]
theorem IsAddUnit.neg_add_cancel_left {α : Type u} {a : α} (h : ) (b : α) :
-a + (a + b) = b
@[simp]
theorem IsUnit.inv_mul_cancel_left {α : Type u} [] {a : α} (h : ) (b : α) :
a⁻¹ * (a * b) = b
@[simp]
theorem IsAddUnit.add_neg_cancel_right {α : Type u} {b : α} (h : ) (a : α) :
a + b + -b = a
@[simp]
theorem IsUnit.mul_inv_cancel_right {α : Type u} [] {b : α} (h : ) (a : α) :
a * b * b⁻¹ = a
@[simp]
theorem IsAddUnit.neg_add_cancel_right {α : Type u} {b : α} (h : ) (a : α) :
a + -b + b = a
@[simp]
theorem IsUnit.inv_mul_cancel_right {α : Type u} [] {b : α} (h : ) (a : α) :
a * b⁻¹ * b = a
theorem IsAddUnit.sub_self {α : Type u} {a : α} (h : ) :
a - a = 0
theorem IsUnit.div_self {α : Type u} [] {a : α} (h : ) :
a / a = 1
theorem IsAddUnit.eq_add_neg_iff_add_eq {α : Type u} {a : α} {b : α} {c : α} (h : ) :
a = b + -c a + c = b
theorem IsUnit.eq_mul_inv_iff_mul_eq {α : Type u} [] {a : α} {b : α} {c : α} (h : ) :
a = b * c⁻¹ a * c = b
theorem IsAddUnit.eq_neg_add_iff_add_eq {α : Type u} {a : α} {b : α} {c : α} (h : ) :
a = -b + c b + a = c
theorem IsUnit.eq_inv_mul_iff_mul_eq {α : Type u} [] {a : α} {b : α} {c : α} (h : ) :
a = b⁻¹ * c b * a = c
theorem IsAddUnit.neg_add_eq_iff_eq_add {α : Type u} {a : α} {b : α} {c : α} (h : ) :
-a + b = c b = a + c
theorem IsUnit.inv_mul_eq_iff_eq_mul {α : Type u} [] {a : α} {b : α} {c : α} (h : ) :
a⁻¹ * b = c b = a * c
theorem IsAddUnit.add_neg_eq_iff_eq_add {α : Type u} {a : α} {b : α} {c : α} (h : ) :
a + -b = c a = c + b
theorem IsUnit.mul_inv_eq_iff_eq_mul {α : Type u} [] {a : α} {b : α} {c : α} (h : ) :
a * b⁻¹ = c a = c * b
theorem IsAddUnit.add_neg_eq_zero {α : Type u} {a : α} {b : α} (h : ) :
a + -b = 0 a = b
theorem IsUnit.mul_inv_eq_one {α : Type u} [] {a : α} {b : α} (h : ) :
a * b⁻¹ = 1 a = b
theorem IsAddUnit.neg_add_eq_zero {α : Type u} {a : α} {b : α} (h : ) :
-a + b = 0 a = b
theorem IsUnit.inv_mul_eq_one {α : Type u} [] {a : α} {b : α} (h : ) :
a⁻¹ * b = 1 a = b
theorem IsAddUnit.add_eq_zero_iff_eq_neg {α : Type u} {a : α} {b : α} (h : ) :
a + b = 0 a = -b
theorem IsUnit.mul_eq_one_iff_eq_inv {α : Type u} [] {a : α} {b : α} (h : ) :
a * b = 1 a = b⁻¹
theorem IsAddUnit.add_eq_zero_iff_neg_eq {α : Type u} {a : α} {b : α} (h : ) :
a + b = 0 -a = b
theorem IsUnit.mul_eq_one_iff_inv_eq {α : Type u} [] {a : α} {b : α} (h : ) :
a * b = 1 a⁻¹ = b
@[simp]
theorem IsAddUnit.sub_add_cancel {α : Type u} {b : α} (h : ) (a : α) :
a - b + b = a
@[simp]
theorem IsUnit.div_mul_cancel {α : Type u} [] {b : α} (h : ) (a : α) :
a / b * b = a
@[simp]
theorem IsAddUnit.add_sub_cancel_right {α : Type u} {b : α} (h : ) (a : α) :
a + b - b = a
@[simp]
theorem IsUnit.mul_div_cancel_right {α : Type u} [] {b : α} (h : ) (a : α) :
a * b / b = a
theorem IsAddUnit.add_zero_sub_cancel {α : Type u} {a : α} (h : ) :
a + (0 - a) = 0
theorem IsUnit.mul_one_div_cancel {α : Type u} [] {a : α} (h : ) :
a * (1 / a) = 1
theorem IsAddUnit.zero_sub_add_cancel {α : Type u} {a : α} (h : ) :
0 - a + a = 0
theorem IsUnit.one_div_mul_cancel {α : Type u} [] {a : α} (h : ) :
1 / a * a = 1
theorem IsAddUnit.neg {α : Type u} {a : α} (h : ) :
theorem IsUnit.inv {α : Type u} [] {a : α} (h : ) :
theorem IsAddUnit.sub {α : Type u} {a : α} {b : α} (ha : ) (hb : ) :
theorem IsUnit.div {α : Type u} [] {a : α} {b : α} (ha : ) (hb : ) :
IsUnit (a / b)
theorem IsAddUnit.sub_left_inj {α : Type u} {a : α} {b : α} {c : α} (h : ) :
a - c = b - c a = b
theorem IsUnit.div_left_inj {α : Type u} [] {a : α} {b : α} {c : α} (h : ) :
a / c = b / c a = b
theorem IsAddUnit.sub_eq_iff {α : Type u} {a : α} {b : α} {c : α} (h : ) :
a - b = c a = c + b
theorem IsUnit.div_eq_iff {α : Type u} [] {a : α} {b : α} {c : α} (h : ) :
a / b = c a = c * b
theorem IsAddUnit.eq_sub_iff {α : Type u} {a : α} {b : α} {c : α} (h : ) :
a = b - c a + c = b
theorem IsUnit.eq_div_iff {α : Type u} [] {a : α} {b : α} {c : α} (h : ) :
a = b / c a * c = b
theorem IsAddUnit.sub_eq_of_eq_add {α : Type u} {a : α} {b : α} {c : α} (h : ) :
a = c + ba - b = c
theorem IsUnit.div_eq_of_eq_mul {α : Type u} [] {a : α} {b : α} {c : α} (h : ) :
a = c * ba / b = c
theorem IsAddUnit.eq_sub_of_add_eq {α : Type u} {a : α} {b : α} {c : α} (h : ) :
a + c = ba = b - c
theorem IsUnit.eq_div_of_mul_eq {α : Type u} [] {a : α} {b : α} {c : α} (h : ) :
a * c = ba = b / c
theorem IsAddUnit.sub_eq_zero_iff_eq {α : Type u} {a : α} {b : α} (h : ) :
a - b = 0 a = b
theorem IsUnit.div_eq_one_iff_eq {α : Type u} [] {a : α} {b : α} (h : ) :
a / b = 1 a = b
theorem IsAddUnit.sub_add_cancel_right {α : Type u} {b : α} (h : ) (a : α) :
b - (a + b) = -a
theorem IsUnit.div_mul_cancel_right {α : Type u} [] {b : α} (h : ) (a : α) :
b / (a * b) = a⁻¹
theorem IsAddUnit.sub_add_left {α : Type u} {a : α} {b : α} (h : ) :
b - (a + b) = 0 - a
@[deprecated div_mul_cancel_right]
theorem IsUnit.div_mul_left {α : Type u} [] {a : α} {b : α} (h : ) :
b / (a * b) = 1 / a
theorem IsAddUnit.add_sub_add_right {α : Type u} {c : α} (h : ) (a : α) (b : α) :
a + c - (b + c) = a - b
theorem IsUnit.mul_div_mul_right {α : Type u} [] {c : α} (h : ) (a : α) (b : α) :
a * c / (b * c) = a / b
a + b + (0 - b) = a
theorem IsUnit.mul_mul_div {α : Type u} [] {b : α} (a : α) (h : ) :
a * b * (1 / b) = a
theorem IsAddUnit.sub_add_cancel_left {α : Type u} {a : α} (h : ) (b : α) :
a - (a + b) = -b
theorem IsUnit.div_mul_cancel_left {α : Type u} {a : α} (h : ) (b : α) :
a / (a * b) = b⁻¹
theorem IsAddUnit.sub_add_right {α : Type u} {a : α} (h : ) (b : α) :
a - (a + b) = 0 - b
@[deprecated div_mul_cancel_left]
theorem IsUnit.div_mul_right {α : Type u} {a : α} (h : ) (b : α) :
a / (a * b) = 1 / b
theorem IsAddUnit.add_sub_cancel_left {α : Type u} {a : α} (h : ) (b : α) :
a + b - a = b
theorem IsUnit.mul_div_cancel_left {α : Type u} {a : α} (h : ) (b : α) :
a * b / a = b
theorem IsAddUnit.add_sub_cancel {α : Type u} {a : α} (h : ) (b : α) :
a + (b - a) = b
theorem IsUnit.mul_div_cancel {α : Type u} {a : α} (h : ) (b : α) :
a * (b / a) = b
theorem IsAddUnit.add_sub_add_left {α : Type u} {c : α} (h : ) (a : α) (b : α) :
c + a - (c + b) = a - b
theorem IsUnit.mul_div_mul_left {α : Type u} {c : α} (h : ) (a : α) (b : α) :
c * a / (c * b) = a / b
theorem IsAddUnit.add_eq_add_of_sub_eq_sub {α : Type u} {b : α} {d : α} (hb : ) (hd : ) (a : α) (c : α) (h : a - b = c - d) :
a + d = c + b
theorem IsUnit.mul_eq_mul_of_div_eq_div {α : Type u} {b : α} {d : α} (hb : ) (hd : ) (a : α) (c : α) (h : a / b = c / d) :
a * d = c * b
theorem IsAddUnit.sub_eq_sub_iff {α : Type u} {a : α} {b : α} {c : α} {d : α} (hb : ) (hd : ) :
a - b = c - d a + d = c + b
theorem IsUnit.div_eq_div_iff {α : Type u} {a : α} {b : α} {c : α} {d : α} (hb : ) (hd : ) :
a / b = c / d a * d = c * b
theorem IsAddUnit.sub_sub_cancel {α : Type u} {a : α} {b : α} (h : ) :
a - (a - b) = b
theorem IsUnit.div_div_cancel {α : Type u} {a : α} {b : α} (h : ) :
a / (a / b) = b
theorem IsAddUnit.sub_sub_cancel_left {α : Type u} {a : α} {b : α} (h : ) :
a - b - a = -b
theorem IsUnit.div_div_cancel_left {α : Type u} {a : α} {b : α} (h : ) :
a / b / a = b⁻¹
theorem divp_eq_div {α : Type u} [] (a : α) (u : αˣ) :
a /ₚ u = a / u
theorem Group.isUnit {α : Type u} [] (a : α) :
noncomputable def invOfIsUnit {M : Type u_1} [] (h : ∀ (a : M), ) :
Inv M

Constructs an inv operation for a Monoid consisting only of units.

Equations
• = { inv := fun (a : M) => .unit⁻¹ }
Instances For
noncomputable def groupOfIsUnit {M : Type u_1} [hM : ] (h : ∀ (a : M), ) :

Constructs a Group structure on a Monoid consisting only of units.

Equations
Instances For
noncomputable def commGroupOfIsUnit {M : Type u_1} [hM : ] (h : ∀ (a : M), ) :

Constructs a CommGroup structure on a CommMonoid consisting only of units.

Equations
Instances For
@[deprecated IsUnit.mul_div_cancel]
theorem IsUnit.mul_div_cancel' {α : Type u} {a : α} (h : ) (b : α) :
a * (b / a) = b

Alias of IsUnit.mul_div_cancel.

Alias of IsAddUnit.add_sub_cancel.