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Mathlib.Algebra.Group.Units

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 #

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

Notation #

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

structure Units (α : Type u) [Monoid α] :
  • val : α

    The underlying value in the base Monoid.

  • inv : α

    The inverse value of val in the base Monoid.

  • val_inv : s * s.inv = 1

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

  • inv_val : s.inv * s = 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.

Instances For

    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) [AddMonoid α] :
      • val : α

        The underlying value in the base AddMonoid.

      • neg : α

        The additive inverse value of val in the base AddMonoid.

      • val_neg : s + s.neg = 0

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

      • neg_val : s.neg + s = 0

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

      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.

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

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

        Equations
        • AddUnits.instCoeHeadAddUnits = { coe := AddUnits.val }
        instance Units.instCoeHeadUnits {α : Type u} [Monoid α] :
        CoeHead αˣ α

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

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

        The additive inverse of an additive unit in an AddMonoid.

        Equations
        • AddUnits.instNeg = { neg := fun u => { val := u.neg, neg := u, val_neg := (_ : u.neg + u = 0), neg_val := (_ : u + u.neg = 0) } }
        instance Units.instInv {α : Type u} [Monoid α] :

        The inverse of a unit in a Monoid.

        Equations
        • Units.instInv = { inv := fun u => { val := u.inv, inv := u, val_inv := (_ : u.inv * u = 1), inv_val := (_ : u * u.inv = 1) } }
        def AddUnits.Simps.val_neg {α : Type u} [AddMonoid α] (u : AddUnits α) :
        α

        See Note [custom simps projection]

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

          See Note [custom simps projection]

          Equations
          Instances For
            theorem AddUnits.val_mk {α : Type u} [AddMonoid α] (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} [Monoid α] (a : α) (b : α) (h₁ : a * b = 1) (h₂ : b * a = 1) :
            { val := a, inv := b, val_inv := h₁, inv_val := h₂ } = a
            abbrev AddUnits.ext.match_1 {α : Type u_1} [AddMonoid α] (motive : (x x_1 : AddUnits α) → x = x_1Prop) :
            (x x_1 : AddUnits α) → (x_2 : x = x_1) → ((v i₁ : α) → (vi₁ : v + i₁ = 0) → (iv₁ : i₁ + v = 0) → (v' i₂ : α) → (vi₂ : v' + i₂ = 0) → (iv₂ : i₂ + v' = 0) → (e : { val := v, neg := i₁, val_neg := vi₁, neg_val := iv₁ } = { val := v', neg := i₂, val_neg := vi₂, neg_val := iv₂ }) → motive { val := v, neg := i₁, val_neg := vi₁, neg_val := iv₁ } { val := v', neg := i₂, val_neg := vi₂, neg_val := iv₂ } e) → motive x x_1 x_2
            Equations
            • One or more equations did not get rendered due to their size.
            Instances For
              theorem AddUnits.ext {α : Type u} [AddMonoid α] :
              Function.Injective AddUnits.val
              theorem Units.ext {α : Type u} [Monoid α] :
              theorem AddUnits.eq_iff {α : Type u} [AddMonoid α] {a : AddUnits α} {b : AddUnits α} :
              a = b a = b
              theorem Units.eq_iff {α : Type u} [Monoid α] {a : αˣ} {b : αˣ} :
              a = b a = b
              theorem AddUnits.ext_iff {α : Type u} [AddMonoid α] {a : AddUnits α} {b : AddUnits α} :
              a = b a = b
              theorem Units.ext_iff {α : Type u} [Monoid α] {a : αˣ} {b : αˣ} :
              a = b a = b

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

              Equations

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

              Equations
              @[simp]
              theorem AddUnits.mk_val {α : Type u} [AddMonoid α] (u : AddUnits α) (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} [Monoid α] (u : αˣ) (y : α) (h₁ : u * y = 1) (h₂ : y * u = 1) :
              { val := u, inv := y, val_inv := h₁, inv_val := h₂ } = u
              def AddUnits.copy {α : Type u} [AddMonoid α] (u : AddUnits α) (val : α) (hv : val = u) (inv : α) (hi : inv = ↑(-u)) :

              Copy an AddUnit, adjusting definitional equalities.

              Equations
              • AddUnits.copy u val hv inv hi = { val := val, neg := inv, val_neg := (_ : val + inv = 0), neg_val := (_ : inv + val = 0) }
              Instances For
                theorem AddUnits.copy.proof_2 {α : Type u_1} [AddMonoid α] (u : AddUnits α) (val : α) (hv : val = u) (inv : α) (hi : inv = ↑(-u)) :
                inv + val = 0
                theorem AddUnits.copy.proof_1 {α : Type u_1} [AddMonoid α] (u : AddUnits α) (val : α) (hv : val = u) (inv : α) (hi : inv = ↑(-u)) :
                val + inv = 0
                @[simp]
                theorem Units.val_inv_copy {α : Type u} [Monoid α] (u : αˣ) (val : α) (hv : val = u) (inv : α) (hi : inv = u⁻¹) :
                (Units.copy u val hv inv hi)⁻¹ = inv
                @[simp]
                theorem Units.val_copy {α : Type u} [Monoid α] (u : αˣ) (val : α) (hv : val = u) (inv : α) (hi : inv = u⁻¹) :
                ↑(Units.copy u val hv inv hi) = val
                @[simp]
                theorem AddUnits.val_neg_copy {α : Type u} [AddMonoid α] (u : AddUnits α) (val : α) (hv : val = u) (inv : α) (hi : inv = ↑(-u)) :
                ↑(-AddUnits.copy u val hv inv hi) = inv
                @[simp]
                theorem AddUnits.val_copy {α : Type u} [AddMonoid α] (u : AddUnits α) (val : α) (hv : val = u) (inv : α) (hi : inv = ↑(-u)) :
                ↑(AddUnits.copy u val hv inv hi) = val
                def Units.copy {α : Type u} [Monoid α] (u : αˣ) (val : α) (hv : val = u) (inv : α) (hi : inv = u⁻¹) :
                αˣ

                Copy a unit, adjusting definition equalities.

                Equations
                • Units.copy u val hv inv hi = { val := val, inv := inv, val_inv := (_ : val * inv = 1), inv_val := (_ : inv * val = 1) }
                Instances For
                  theorem AddUnits.copy_eq {α : Type u} [AddMonoid α] (u : AddUnits α) (val : α) (hv : val = u) (inv : α) (hi : inv = ↑(-u)) :
                  AddUnits.copy u val hv inv hi = u
                  theorem Units.copy_eq {α : Type u} [Monoid α] (u : αˣ) (val : α) (hv : val = u) (inv : α) (hi : inv = u⁻¹) :
                  Units.copy u val hv inv hi = u
                  theorem AddUnits.instAddZeroClassAddUnits.proof_3 {α : Type u_1} [AddMonoid α] (u₁ : AddUnits α) (u₂ : AddUnits α) :
                  u₂.neg + u₁.neg + (u₁ + u₂) = 0
                  theorem AddUnits.instAddZeroClassAddUnits.proof_2 {α : Type u_1} [AddMonoid α] (u₁ : AddUnits α) (u₂ : AddUnits α) :
                  u₁ + u₂ + (u₂.neg + u₁.neg) = 0

                  Additive units of an additive monoid have an addition and an additive identity.

                  Equations

                  Units of a monoid form have a multiplication and multiplicative identity.

                  Equations
                  theorem AddUnits.instAddGroupAddUnits.proof_2 {α : Type u_1} [AddMonoid α] (a : AddUnits α) :
                  0 + a = a
                  theorem AddUnits.instAddGroupAddUnits.proof_10 {α : Type u_1} [AddMonoid α] :
                  ∀ (n : ) (a : AddUnits α), zsmulRec (Int.negSucc n) a = zsmulRec (Int.negSucc n) a

                  Additive units of an additive monoid form an additive group.

                  Equations
                  theorem AddUnits.instAddGroupAddUnits.proof_5 {α : Type u_1} [AddMonoid α] :
                  ∀ (n : ) (x : AddUnits α), nsmulRec (n + 1) x = nsmulRec (n + 1) x
                  theorem AddUnits.instAddGroupAddUnits.proof_6 {α : Type u_1} [AddMonoid α] :
                  ∀ (x x_1 x_2 : AddUnits α), x + x_1 + x_2 = x + (x_1 + x_2)
                  theorem AddUnits.instAddGroupAddUnits.proof_7 {α : Type u_1} [AddMonoid α] :
                  ∀ (a b : AddUnits α), a - b = a - b
                  theorem AddUnits.instAddGroupAddUnits.proof_4 {α : Type u_1} [AddMonoid α] :
                  ∀ (x : AddUnits α), nsmulRec 0 x = nsmulRec 0 x
                  theorem AddUnits.instAddGroupAddUnits.proof_8 {α : Type u_1} [AddMonoid α] :
                  ∀ (a : AddUnits α), zsmulRec 0 a = zsmulRec 0 a
                  theorem AddUnits.instAddGroupAddUnits.proof_3 {α : Type u_1} [AddMonoid α] (a : AddUnits α) :
                  a + 0 = a
                  theorem AddUnits.instAddGroupAddUnits.proof_9 {α : Type u_1} [AddMonoid α] :
                  ∀ (n : ) (a : AddUnits α), zsmulRec (Int.ofNat (Nat.succ n)) a = zsmulRec (Int.ofNat (Nat.succ n)) a
                  theorem AddUnits.instAddGroupAddUnits.proof_1 {α : Type u_1} [AddMonoid α] :
                  ∀ (x x_1 x_2 : AddUnits α), x + x_1 + x_2 = x + (x_1 + x_2)
                  instance Units.instGroupUnits {α : Type u} [Monoid α] :

                  Units of a monoid form a group.

                  Equations
                  • Units.instGroupUnits = let src := inferInstance; Group.mk (_ : ∀ (u : αˣ), u⁻¹ * u = 1)
                  theorem AddUnits.instAddCommGroupAddUnits.proof_1 {α : Type u_1} [AddCommMonoid α] :
                  ∀ (x x_1 : AddUnits α), x + x_1 = x_1 + x

                  Additive units of an additive commutative monoid form an additive commutative group.

                  Equations

                  Units of a commutative monoid form a commutative group.

                  Equations

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

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

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

                  Equations
                  • Units.instInhabitedUnits = { default := 1 }
                  instance AddUnits.instReprAddUnits {α : Type u} [AddMonoid α] [Repr α] :

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

                  Equations
                  • AddUnits.instReprAddUnits = { reprPrec := reprPrec AddUnits.val }
                  instance Units.instReprUnits {α : Type u} [Monoid α] [Repr α] :

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

                  Equations
                  • Units.instReprUnits = { reprPrec := reprPrec Units.val }
                  @[simp]
                  theorem AddUnits.val_add {α : Type u} [AddMonoid α] (a : AddUnits α) (b : AddUnits α) :
                  ↑(a + b) = a + b
                  @[simp]
                  theorem Units.val_mul {α : Type u} [Monoid α] (a : αˣ) (b : αˣ) :
                  ↑(a * b) = a * b
                  @[simp]
                  theorem AddUnits.val_zero {α : Type u} [AddMonoid α] :
                  0 = 0
                  @[simp]
                  theorem Units.val_one {α : Type u} [Monoid α] :
                  1 = 1
                  @[simp]
                  theorem AddUnits.val_eq_zero {α : Type u} [AddMonoid α] {a : AddUnits α} :
                  a = 0 a = 0
                  @[simp]
                  theorem Units.val_eq_one {α : Type u} [Monoid α] {a : αˣ} :
                  a = 1 a = 1
                  @[simp]
                  theorem AddUnits.neg_mk {α : Type u} [AddMonoid α] (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} [Monoid α] (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} [AddMonoid α] (a : AddUnits α) :
                  a.neg = ↑(-a)
                  @[simp]
                  theorem Units.inv_eq_val_inv {α : Type u} [Monoid α] (a : αˣ) :
                  a.inv = a⁻¹
                  @[simp]
                  theorem AddUnits.neg_add {α : Type u} [AddMonoid α] (a : AddUnits α) :
                  ↑(-a) + a = 0
                  @[simp]
                  theorem Units.inv_mul {α : Type u} [Monoid α] (a : αˣ) :
                  a⁻¹ * a = 1
                  @[simp]
                  theorem AddUnits.add_neg {α : Type u} [AddMonoid α] (a : AddUnits α) :
                  a + ↑(-a) = 0
                  @[simp]
                  theorem Units.mul_inv {α : Type u} [Monoid α] (a : αˣ) :
                  a * a⁻¹ = 1
                  theorem AddUnits.neg_add_of_eq {α : Type u} [AddMonoid α] {u : AddUnits α} {a : α} (h : u = a) :
                  ↑(-u) + a = 0
                  theorem Units.inv_mul_of_eq {α : Type u} [Monoid α] {u : αˣ} {a : α} (h : u = a) :
                  u⁻¹ * a = 1
                  theorem AddUnits.add_neg_of_eq {α : Type u} [AddMonoid α] {u : AddUnits α} {a : α} (h : u = a) :
                  a + ↑(-u) = 0
                  theorem Units.mul_inv_of_eq {α : Type u} [Monoid α] {u : αˣ} {a : α} (h : u = a) :
                  a * u⁻¹ = 1
                  @[simp]
                  theorem AddUnits.add_neg_cancel_left {α : Type u} [AddMonoid α] (a : AddUnits α) (b : α) :
                  a + (↑(-a) + b) = b
                  @[simp]
                  theorem Units.mul_inv_cancel_left {α : Type u} [Monoid α] (a : αˣ) (b : α) :
                  a * (a⁻¹ * b) = b
                  @[simp]
                  theorem AddUnits.neg_add_cancel_left {α : Type u} [AddMonoid α] (a : AddUnits α) (b : α) :
                  ↑(-a) + (a + b) = b
                  @[simp]
                  theorem Units.inv_mul_cancel_left {α : Type u} [Monoid α] (a : αˣ) (b : α) :
                  a⁻¹ * (a * b) = b
                  @[simp]
                  theorem AddUnits.add_neg_cancel_right {α : Type u} [AddMonoid α] (a : α) (b : AddUnits α) :
                  a + b + ↑(-b) = a
                  @[simp]
                  theorem Units.mul_inv_cancel_right {α : Type u} [Monoid α] (a : α) (b : αˣ) :
                  a * b * b⁻¹ = a
                  @[simp]
                  theorem AddUnits.neg_add_cancel_right {α : Type u} [AddMonoid α] (a : α) (b : AddUnits α) :
                  a + ↑(-b) + b = a
                  @[simp]
                  theorem Units.inv_mul_cancel_right {α : Type u} [Monoid α] (a : α) (b : αˣ) :
                  a * b⁻¹ * b = a
                  @[simp]
                  theorem AddUnits.add_right_inj {α : Type u} [AddMonoid α] (a : AddUnits α) {b : α} {c : α} :
                  a + b = a + c b = c
                  @[simp]
                  theorem Units.mul_right_inj {α : Type u} [Monoid α] (a : αˣ) {b : α} {c : α} :
                  a * b = a * c b = c
                  @[simp]
                  theorem AddUnits.add_left_inj {α : Type u} [AddMonoid α] (a : AddUnits α) {b : α} {c : α} :
                  b + a = c + a b = c
                  @[simp]
                  theorem Units.mul_left_inj {α : Type u} [Monoid α] (a : αˣ) {b : α} {c : α} :
                  b * a = c * a b = c
                  theorem AddUnits.eq_add_neg_iff_add_eq {α : Type u} [AddMonoid α] (c : AddUnits α) {a : α} {b : α} :
                  a = b + ↑(-c) a + c = b
                  theorem Units.eq_mul_inv_iff_mul_eq {α : Type u} [Monoid α] (c : αˣ) {a : α} {b : α} :
                  a = b * c⁻¹ a * c = b
                  theorem AddUnits.eq_neg_add_iff_add_eq {α : Type u} [AddMonoid α] (b : AddUnits α) {a : α} {c : α} :
                  a = ↑(-b) + c b + a = c
                  theorem Units.eq_inv_mul_iff_mul_eq {α : Type u} [Monoid α] (b : αˣ) {a : α} {c : α} :
                  a = b⁻¹ * c b * a = c
                  theorem AddUnits.neg_add_eq_iff_eq_add {α : Type u} [AddMonoid α] (a : AddUnits α) {b : α} {c : α} :
                  ↑(-a) + b = c b = a + c
                  theorem Units.inv_mul_eq_iff_eq_mul {α : Type u} [Monoid α] (a : αˣ) {b : α} {c : α} :
                  a⁻¹ * b = c b = a * c
                  theorem AddUnits.add_neg_eq_iff_eq_add {α : Type u} [AddMonoid α] (b : AddUnits α) {a : α} {c : α} :
                  a + ↑(-b) = c a = c + b
                  theorem Units.mul_inv_eq_iff_eq_mul {α : Type u} [Monoid α] (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} [AddMonoid α] {u : AddUnits α} {a : α} (h : a + u = 0) :
                  ↑(-u) = a
                  theorem Units.inv_eq_of_mul_eq_one_left {α : Type u} [Monoid α] {u : αˣ} {a : α} (h : a * u = 1) :
                  u⁻¹ = a
                  theorem AddUnits.neg_eq_of_add_eq_zero_right {α : Type u} [AddMonoid α] {u : AddUnits α} {a : α} (h : u + a = 0) :
                  ↑(-u) = a
                  theorem Units.inv_eq_of_mul_eq_one_right {α : Type u} [Monoid α] {u : αˣ} {a : α} (h : u * a = 1) :
                  u⁻¹ = a
                  theorem AddUnits.eq_neg_of_add_eq_zero_left {α : Type u} [AddMonoid α] {u : AddUnits α} {a : α} (h : u + a = 0) :
                  a = ↑(-u)
                  theorem Units.eq_inv_of_mul_eq_one_left {α : Type u} [Monoid α] {u : αˣ} {a : α} (h : u * a = 1) :
                  a = u⁻¹
                  theorem AddUnits.eq_neg_of_add_eq_zero_right {α : Type u} [AddMonoid α] {u : AddUnits α} {a : α} (h : a + u = 0) :
                  a = ↑(-u)
                  theorem Units.eq_inv_of_mul_eq_one_right {α : Type u} [Monoid α] {u : αˣ} {a : α} (h : a * u = 1) :
                  a = u⁻¹
                  @[simp]
                  theorem AddUnits.add_neg_eq_zero {α : Type u} [AddMonoid α] {u : AddUnits α} {a : α} :
                  a + ↑(-u) = 0 a = u
                  @[simp]
                  theorem Units.mul_inv_eq_one {α : Type u} [Monoid α] {u : αˣ} {a : α} :
                  a * u⁻¹ = 1 a = u
                  @[simp]
                  theorem AddUnits.neg_add_eq_zero {α : Type u} [AddMonoid α] {u : AddUnits α} {a : α} :
                  ↑(-u) + a = 0 u = a
                  @[simp]
                  theorem Units.inv_mul_eq_one {α : Type u} [Monoid α] {u : αˣ} {a : α} :
                  u⁻¹ * a = 1 u = a
                  theorem AddUnits.add_eq_zero_iff_eq_neg {α : Type u} [AddMonoid α] {u : AddUnits α} {a : α} :
                  a + u = 0 a = ↑(-u)
                  theorem Units.mul_eq_one_iff_eq_inv {α : Type u} [Monoid α] {u : αˣ} {a : α} :
                  a * u = 1 a = u⁻¹
                  theorem AddUnits.add_eq_zero_iff_neg_eq {α : Type u} [AddMonoid α] {u : AddUnits α} {a : α} :
                  u + a = 0 ↑(-u) = a
                  theorem Units.mul_eq_one_iff_inv_eq {α : Type u} [Monoid α] {u : αˣ} {a : α} :
                  u * a = 1 u⁻¹ = a
                  theorem AddUnits.neg_unique {α : Type u} [AddMonoid α] {u₁ : AddUnits α} {u₂ : AddUnits α} (h : u₁ = u₂) :
                  ↑(-u₁) = ↑(-u₂)
                  theorem Units.inv_unique {α : Type u} [Monoid α] {u₁ : αˣ} {u₂ : αˣ} (h : u₁ = u₂) :
                  u₁⁻¹ = u₂⁻¹
                  @[simp]
                  theorem AddUnits.val_neg_eq_neg_val {M : Type u_1} [SubtractionMonoid M] (u : AddUnits M) :
                  ↑(-u) = -u
                  @[simp]
                  theorem Units.val_inv_eq_inv_val {M : Type u_1} [DivisionMonoid M] (u : Mˣ) :
                  u⁻¹ = (u)⁻¹
                  theorem AddUnits.mkOfAddEqZero.proof_1 {α : Type u_1} [AddCommMonoid α] (a : α) (b : α) (hab : a + b = 0) :
                  b + a = 0
                  def AddUnits.mkOfAddEqZero {α : Type u} [AddCommMonoid α] (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
                  Instances For
                    def Units.mkOfMulEqOne {α : Type u} [CommMonoid α] (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} [AddCommMonoid α] {a : α} {b : α} (h : a + b = 0) :
                      @[simp]
                      theorem Units.val_mkOfMulEqOne {α : Type u} [CommMonoid α] {a : α} {b : α} (h : a * b = 1) :
                      ↑(Units.mkOfMulEqOne a b h) = a
                      def divp {α : Type u} [Monoid α] (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.

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

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                          @[simp]
                          theorem divp_self {α : Type u} [Monoid α] (u : αˣ) :
                          u /ₚ u = 1
                          @[simp]
                          theorem divp_one {α : Type u} [Monoid α] (a : α) :
                          a /ₚ 1 = a
                          theorem divp_assoc {α : Type u} [Monoid α] (a : α) (b : α) (u : αˣ) :
                          a * b /ₚ u = a * (b /ₚ u)
                          theorem divp_assoc' {α : Type u} [Monoid α] (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} [Monoid α] {a : α} (u : αˣ) :
                          a /ₚ u⁻¹ = a * u
                          @[simp]
                          theorem divp_mul_cancel {α : Type u} [Monoid α] (a : α) (u : αˣ) :
                          a /ₚ u * u = a
                          @[simp]
                          theorem mul_divp_cancel {α : Type u} [Monoid α] (a : α) (u : αˣ) :
                          a * u /ₚ u = a
                          @[simp]
                          theorem divp_left_inj {α : Type u} [Monoid α] (u : αˣ) {a : α} {b : α} :
                          a /ₚ u = b /ₚ u a = b
                          theorem divp_divp_eq_divp_mul {α : Type u} [Monoid α] (x : α) (u₁ : αˣ) (u₂ : αˣ) :
                          x /ₚ u₁ /ₚ u₂ = x /ₚ (u₂ * u₁)
                          theorem divp_eq_iff_mul_eq {α : Type u} [Monoid α] {x : α} {u : αˣ} {y : α} :
                          x /ₚ u = y y * u = x
                          theorem eq_divp_iff_mul_eq {α : Type u} [Monoid α] {x : α} {u : αˣ} {y : α} :
                          x = y /ₚ u x * u = y
                          theorem divp_eq_one_iff_eq {α : Type u} [Monoid α] {a : α} {u : αˣ} :
                          a /ₚ u = 1 a = u
                          @[simp]
                          theorem one_divp {α : Type u} [Monoid α] (u : αˣ) :
                          1 /ₚ u = u⁻¹
                          theorem inv_eq_one_divp {α : Type u} [Monoid α] (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} [Monoid α] (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} [Monoid α] (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 divp_mul_eq_mul_divp {α : Type u} [CommMonoid α] (x : α) (y : α) (u : αˣ) :
                          x /ₚ u * y = x * y /ₚ u
                          theorem divp_eq_divp_iff {α : Type u} [CommMonoid α] {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} [CommMonoid α] (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} [AddCommMonoid α] [Subsingleton (AddUnits α)] {a : α} {b : α} (h : a + b = 0) :
                          a = 0
                          theorem eq_one_of_mul_right {α : Type u} [CommMonoid α] [Subsingleton αˣ] {a : α} {b : α} (h : a * b = 1) :
                          a = 1
                          theorem eq_zero_of_add_left {α : Type u} [AddCommMonoid α] [Subsingleton (AddUnits α)] {a : α} {b : α} (h : a + b = 0) :
                          b = 0
                          theorem eq_one_of_mul_left {α : Type u} [CommMonoid α] [Subsingleton αˣ] {a : α} {b : α} (h : a * b = 1) :
                          b = 1
                          @[simp]
                          theorem add_eq_zero {α : Type u} [AddCommMonoid α] [Subsingleton (AddUnits α)] {a : α} {b : α} :
                          a + b = 0 a = 0 b = 0
                          @[simp]
                          theorem mul_eq_one {α : Type u} [CommMonoid α] [Subsingleton αˣ] {a : α} {b : α} :
                          a * b = 1 a = 1 b = 1

                          IsUnit predicate #

                          def IsAddUnit {M : Type u_1} [AddMonoid M] (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.

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                            def IsUnit {M : Type u_1} [Monoid M] (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 is a bundled version of IsUnit.

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                              theorem isUnit_of_subsingleton {M : Type u_1} [Monoid M] [Subsingleton M] (a : M) :
                              theorem instCanLiftAddUnitsValIsAddUnit.proof_1 {M : Type u_1} [AddMonoid M] :
                              CanLift M (AddUnits M) AddUnits.val IsAddUnit

                              A subsingleton AddMonoid has a unique additive unit.

                              Equations
                              • instUniqueAddUnits = { toInhabited := { default := 0 }, uniq := (_ : ∀ (a : AddUnits M), a = 0) }
                              theorem instUniqueAddUnits.proof_1 {M : Type u_1} [AddMonoid M] [Subsingleton M] (a : AddUnits M) :
                              a = 0
                              instance instUniqueUnits {M : Type u_1} [Monoid M] [Subsingleton M] :

                              A subsingleton Monoid has a unique unit.

                              Equations
                              • instUniqueUnits = { toInhabited := { default := 1 }, uniq := (_ : ∀ (a : Mˣ), a = 1) }
                              @[simp]
                              theorem AddUnits.isAddUnit {M : Type u_1} [AddMonoid M] (u : AddUnits M) :
                              @[simp]
                              theorem Units.isUnit {M : Type u_1} [Monoid M] (u : Mˣ) :
                              IsUnit u
                              @[simp]
                              theorem isAddUnit_zero {M : Type u_1} [AddMonoid M] :
                              @[simp]
                              theorem isUnit_one {M : Type u_1} [Monoid M] :
                              theorem isAddUnit_of_add_eq_zero {M : Type u_1} [AddCommMonoid M] (a : M) (b : M) (h : a + b = 0) :
                              theorem isUnit_of_mul_eq_one {M : Type u_1} [CommMonoid M] (a : M) (b : M) (h : a * b = 1) :
                              theorem IsAddUnit.exists_neg {M : Type u_1} [AddMonoid M] {a : M} (h : IsAddUnit a) :
                              b, a + b = 0
                              theorem IsUnit.exists_right_inv {M : Type u_1} [Monoid M] {a : M} (h : IsUnit a) :
                              b, a * b = 1
                              theorem IsAddUnit.exists_neg' {M : Type u_1} [AddMonoid M] {a : M} (h : IsAddUnit a) :
                              b, b + a = 0
                              theorem IsUnit.exists_left_inv {M : Type u_1} [Monoid M] {a : M} (h : IsUnit a) :
                              b, b * a = 1
                              abbrev isAddUnit_iff_exists_neg.match_1 {M : Type u_1} [AddCommMonoid M] {a : M} (motive : (b, a + b = 0) → Prop) :
                              (x : b, a + b = 0) → ((b : M) → (hab : a + b = 0) → motive (_ : b, a + b = 0)) → motive x
                              Equations
                              Instances For
                                theorem isAddUnit_iff_exists_neg {M : Type u_1} [AddCommMonoid M] {a : M} :
                                IsAddUnit a b, a + b = 0
                                theorem isUnit_iff_exists_inv {M : Type u_1} [CommMonoid M] {a : M} :
                                IsUnit a b, a * b = 1
                                theorem isAddUnit_iff_exists_neg' {M : Type u_1} [AddCommMonoid M] {a : M} :
                                IsAddUnit a b, b + a = 0
                                theorem isUnit_iff_exists_inv' {M : Type u_1} [CommMonoid M] {a : M} :
                                IsUnit a b, b * a = 1
                                theorem IsAddUnit.add {M : Type u_1} [AddMonoid M] {x : M} {y : M} :
                                IsAddUnit xIsAddUnit yIsAddUnit (x + y)
                                theorem IsUnit.mul {M : Type u_1} [Monoid M] {x : M} {y : M} :
                                IsUnit xIsUnit yIsUnit (x * y)
                                abbrev AddUnits.isAddUnit_add_addUnits.match_1 {M : Type u_1} [AddMonoid M] (a : M) (u : AddUnits M) (motive : IsAddUnit (a + u)Prop) :
                                (x : IsAddUnit (a + u)) → ((v : AddUnits M) → (hv : v = a + u) → motive (_ : u, u = a + u)) → motive x
                                Equations
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                                  @[simp]
                                  theorem AddUnits.isAddUnit_add_addUnits {M : Type u_1} [AddMonoid M] (a : M) (u : AddUnits M) :

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

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

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

                                  abbrev AddUnits.isAddUnit_addUnits_add.match_1 {M : Type u_1} [AddMonoid M] (u : AddUnits M) (a : M) (motive : IsAddUnit (u + a)Prop) :
                                  (x : IsAddUnit (u + a)) → ((v : AddUnits M) → (hv : v = u + a) → motive (_ : u, u = u + a)) → motive x
                                  Equations
                                  Instances For
                                    @[simp]
                                    theorem AddUnits.isAddUnit_addUnits_add {M : Type u_3} [AddMonoid M] (u : AddUnits M) (a : M) :

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

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

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

                                    theorem isAddUnit_of_add_isAddUnit_left {M : Type u_1} [AddCommMonoid M] {x : M} {y : M} (hu : IsAddUnit (x + y)) :
                                    abbrev isAddUnit_of_add_isAddUnit_left.match_1 {M : Type u_1} [AddCommMonoid M] {x : M} {y : M} (motive : (b, x + y + b = 0) → Prop) :
                                    (x : b, x + y + b = 0) → ((z : M) → (hz : x + y + z = 0) → motive (_ : b, x + y + b = 0)) → motive x
                                    Equations
                                    Instances For
                                      theorem isUnit_of_mul_isUnit_left {M : Type u_1} [CommMonoid M] {x : M} {y : M} (hu : IsUnit (x * y)) :
                                      theorem isAddUnit_of_add_isAddUnit_right {M : Type u_1} [AddCommMonoid M] {x : M} {y : M} (hu : IsAddUnit (x + y)) :
                                      theorem isUnit_of_mul_isUnit_right {M : Type u_1} [CommMonoid M] {x : M} {y : M} (hu : IsUnit (x * y)) :
                                      @[simp]
                                      theorem IsAddUnit.add_iff {M : Type u_1} [AddCommMonoid M] {x : M} {y : M} :
                                      @[simp]
                                      theorem IsUnit.mul_iff {M : Type u_1} [CommMonoid M] {x : M} {y : M} :
                                      noncomputable def IsUnit.unit {M : Type u_1} [Monoid M] {a : M} (h : IsUnit a) :

                                      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.

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                                        noncomputable def IsAddUnit.addUnit {N : Type u_2} [AddMonoid N] {a : N} (h : IsAddUnit a) :

                                        "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
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                                          @[simp]
                                          @[simp]
                                          theorem IsUnit.unit_of_val_units {M : Type u_1} [Monoid M] {a : Mˣ} (h : IsUnit a) :
                                          @[simp]
                                          theorem IsAddUnit.addUnit_spec {M : Type u_1} [AddMonoid M] {a : M} (h : IsAddUnit a) :
                                          @[simp]
                                          theorem IsUnit.unit_spec {M : Type u_1} [Monoid M] {a : M} (h : IsUnit a) :
                                          ↑(IsUnit.unit h) = a
                                          @[simp]
                                          theorem IsAddUnit.val_neg_add {M : Type u_1} [AddMonoid M] {a : M} (h : IsAddUnit a) :
                                          @[simp]
                                          theorem IsUnit.val_inv_mul {M : Type u_1} [Monoid M] {a : M} (h : IsUnit a) :
                                          (IsUnit.unit h)⁻¹ * a = 1
                                          @[simp]
                                          theorem IsAddUnit.add_val_neg {M : Type u_1} [AddMonoid M] {a : M} (h : IsAddUnit a) :
                                          @[simp]
                                          theorem IsUnit.mul_val_inv {M : Type u_1} [Monoid M] {a : M} (h : IsUnit a) :
                                          a * (IsUnit.unit h)⁻¹ = 1
                                          instance IsAddUnit.instDecidableIsAddUnit {M : Type u_1} [AddMonoid M] (x : M) [h : Decidable (u, u = x)] :

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

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

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

                                          Equations
                                          theorem IsAddUnit.add_left_inj {M : Type u_1} [AddMonoid M] {a : M} {b : M} {c : M} (h : IsAddUnit a) :
                                          b + a = c + a b = c
                                          abbrev IsAddUnit.add_left_inj.match_1 {M : Type u_1} [AddMonoid M] {a : M} (motive : IsAddUnit aProp) :
                                          (h : IsAddUnit a) → ((u : AddUnits M) → (hu : u = a) → motive (_ : u, u = a)) → motive h
                                          Equations
                                          Instances For
                                            theorem IsUnit.mul_left_inj {M : Type u_1} [Monoid M] {a : M} {b : M} {c : M} (h : IsUnit a) :
                                            b * a = c * a b = c
                                            theorem IsAddUnit.add_right_inj {M : Type u_1} [AddMonoid M] {a : M} {b : M} {c : M} (h : IsAddUnit a) :
                                            a + b = a + c b = c
                                            theorem IsUnit.mul_right_inj {M : Type u_1} [Monoid M] {a : M} {b : M} {c : M} (h : IsUnit a) :
                                            a * b = a * c b = c
                                            theorem IsAddUnit.add_left_cancel {M : Type u_1} [AddMonoid M] {a : M} {b : M} {c : M} (h : IsAddUnit a) :
                                            a + b = a + cb = c
                                            theorem IsUnit.mul_left_cancel {M : Type u_1} [Monoid M] {a : M} {b : M} {c : M} (h : IsUnit a) :
                                            a * b = a * cb = c
                                            theorem IsAddUnit.add_right_cancel {M : Type u_1} [AddMonoid M] {a : M} {b : M} {c : M} (h : IsAddUnit b) :
                                            a + b = c + ba = c
                                            theorem IsUnit.mul_right_cancel {M : Type u_1} [Monoid M] {a : M} {b : M} {c : M} (h : IsUnit b) :
                                            a * b = c * ba = c
                                            theorem IsAddUnit.add_right_injective {M : Type u_1} [AddMonoid M] {a : M} (h : IsAddUnit a) :
                                            Function.Injective ((fun x x_1 => x + x_1) a)
                                            theorem IsUnit.mul_right_injective {M : Type u_1} [Monoid M] {a : M} (h : IsUnit a) :
                                            Function.Injective ((fun x x_1 => x * x_1) a)
                                            theorem IsAddUnit.add_left_injective {M : Type u_1} [AddMonoid M] {b : M} (h : IsAddUnit b) :
                                            Function.Injective fun x => x + b
                                            theorem IsUnit.mul_left_injective {M : Type u_1} [Monoid M] {b : M} (h : IsUnit b) :
                                            Function.Injective fun x => x * b
                                            @[simp]
                                            theorem IsAddUnit.neg_add_cancel {M : Type u_1} [SubtractionMonoid M] {a : M} :
                                            IsAddUnit a-a + a = 0
                                            @[simp]
                                            theorem IsUnit.inv_mul_cancel {M : Type u_1} [DivisionMonoid M] {a : M} :
                                            IsUnit aa⁻¹ * a = 1
                                            @[simp]
                                            theorem IsAddUnit.add_neg_cancel {M : Type u_1} [SubtractionMonoid M] {a : M} :
                                            IsAddUnit aa + -a = 0
                                            @[simp]
                                            theorem IsUnit.mul_inv_cancel {M : Type u_1} [DivisionMonoid M] {a : M} :
                                            IsUnit aa * a⁻¹ = 1
                                            noncomputable def groupOfIsUnit {M : Type u_1} [hM : Monoid M] (h : ∀ (a : M), IsUnit a) :

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

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                                              noncomputable def commGroupOfIsUnit {M : Type u_1} [hM : CommMonoid M] (h : ∀ (a : M), IsUnit a) :

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

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