Lemmas about units in a MonoidWithZero or a GroupWithZero.

We also define Ring.inverse, a globally defined function on any ring (in fact any MonoidWithZero), which inverts units and sends non-units to zero.

@[simp]

theorem Units.ne_zero {M₀ : Type u_2} [MonoidWithZero M₀] [Nontrivial M₀] (u : M₀ˣ) : ↑u ≠ 0

An element of the unit group of a nonzero monoid with zero represented as an element of the monoid is nonzero.

@[simp]

theorem Units.mul_left_eq_zero {M₀ : Type u_2} [MonoidWithZero M₀] (u : M₀ˣ) {a : M₀} : a * ↑u = 0 ↔ a = 0

@[simp]

theorem Units.mul_right_eq_zero {M₀ : Type u_2} [MonoidWithZero M₀] (u : M₀ˣ) {a : M₀} : ↑u * a = 0 ↔ a = 0

theorem IsUnit.ne_zero {M₀ : Type u_2} [MonoidWithZero M₀] [Nontrivial M₀] {a : M₀} (ha : IsUnit a) : a ≠ 0

theorem IsUnit.mul_right_eq_zero {M₀ : Type u_2} [MonoidWithZero M₀] {a b : M₀} (ha : IsUnit a) : a * b = 0 ↔ b = 0

theorem IsUnit.mul_left_eq_zero {M₀ : Type u_2} [MonoidWithZero M₀] {a b : M₀} (hb : IsUnit b) : a * b = 0 ↔ a = 0

@[simp]

theorem isUnit_zero_iff {M₀ : Type u_2} [MonoidWithZero M₀] : IsUnit 0 ↔ 0 = 1

theorem not_isUnit_zero {M₀ : Type u_2} [MonoidWithZero M₀] [Nontrivial M₀] : ¬IsUnit 0

noncomputable def Ring.inverse {M₀ : Type u_2} [MonoidWithZero M₀] : M₀ → M₀

Introduce a function inverse on a monoid with zero M₀, which sends x to x⁻¹ if x is invertible and to 0 otherwise. This definition is somewhat ad hoc, but one needs a fully (rather than partially) defined inverse function for some purposes, including for calculus.

Note that while this is in the Ring namespace for brevity, it requires the weaker assumption MonoidWithZero M₀ instead of Ring M₀.

Equations

• Ring.inverse x = if h : IsUnit x then ↑h.unit⁻¹ else 0

@[simp]

theorem Ring.inverse_unit {M₀ : Type u_2} [MonoidWithZero M₀] (u : M₀ˣ) : inverse ↑u = ↑u⁻¹

By definition, if x is invertible then inverse x = x⁻¹.

theorem Ring.IsUnit.ringInverse {M₀ : Type u_2} [MonoidWithZero M₀] {x : M₀} (h : IsUnit x) : IsUnit (inverse x)

theorem Ring.inverse_of_isUnit {M₀ : Type u_2} [MonoidWithZero M₀] {x : M₀} (h : IsUnit x) : inverse x = ↑h.unit⁻¹

@[simp]

theorem Ring.inverse_non_unit {M₀ : Type u_2} [MonoidWithZero M₀] (x : M₀) (h : ¬IsUnit x) : inverse x = 0

By definition, if x is not invertible then inverse x = 0.

theorem Ring.mul_inverse_cancel {M₀ : Type u_2} [MonoidWithZero M₀] (x : M₀) (h : IsUnit x) : x * inverse x = 1

theorem Ring.inverse_mul_cancel {M₀ : Type u_2} [MonoidWithZero M₀] (x : M₀) (h : IsUnit x) : inverse x * x = 1

theorem Ring.mul_inverse_cancel_right {M₀ : Type u_2} [MonoidWithZero M₀] (x y : M₀) (h : IsUnit x) : y * x * inverse x = y

theorem Ring.inverse_mul_cancel_right {M₀ : Type u_2} [MonoidWithZero M₀] (x y : M₀) (h : IsUnit x) : y * inverse x * x = y

theorem Ring.mul_inverse_cancel_left {M₀ : Type u_2} [MonoidWithZero M₀] (x y : M₀) (h : IsUnit x) : x * (inverse x * y) = y

theorem Ring.inverse_mul_cancel_left {M₀ : Type u_2} [MonoidWithZero M₀] (x y : M₀) (h : IsUnit x) : inverse x * (x * y) = y

theorem Ring.inverse_mul_eq_iff_eq_mul {M₀ : Type u_2} [MonoidWithZero M₀] (x y z : M₀) (h : IsUnit x) : inverse x * y = z ↔ y = x * z

theorem Ring.eq_mul_inverse_iff_mul_eq {M₀ : Type u_2} [MonoidWithZero M₀] (x y z : M₀) (h : IsUnit z) : x = y * inverse z ↔ x * z = y

@[simp]

theorem Ring.inverse_one (M₀ : Type u_2) [MonoidWithZero M₀] : inverse 1 = 1

@[simp]

theorem Ring.inverse_zero (M₀ : Type u_2) [MonoidWithZero M₀] : inverse 0 = 0

theorem IsUnit.ring_inverse {M₀ : Type u_2} [MonoidWithZero M₀] {a : M₀} : IsUnit a → IsUnit (Ring.inverse a)

@[simp]

theorem isUnit_ring_inverse {M₀ : Type u_2} [MonoidWithZero M₀] {a : M₀} : IsUnit (Ring.inverse a) ↔ IsUnit a

def Units.mk0 {G₀ : Type u_3} [GroupWithZero G₀] (a : G₀) (ha : a ≠ 0) : G₀ˣ

Embed a non-zero element of a GroupWithZero into the unit group. By combining this function with the operations on units, or the /ₚ operation, it is possible to write a division as a partial function with three arguments.

Equations

• Units.mk0 a ha = { val := a, inv := a⁻¹, val_inv := ⋯, inv_val := ⋯ }

@[simp]

theorem Units.mk0_one {G₀ : Type u_3} [GroupWithZero G₀] (h : 1 ≠ 0 := ⋯) : mk0 1 h = 1

@[simp]

theorem Units.val_mk0 {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (h : a ≠ 0) : ↑(mk0 a h) = a

@[simp]

theorem Units.mk0_val {G₀ : Type u_3} [GroupWithZero G₀] (u : G₀ˣ) (h : ↑u ≠ 0) : mk0 (↑u) h = u

theorem Units.mul_inv' {G₀ : Type u_3} [GroupWithZero G₀] (u : G₀ˣ) : ↑u * (↑u)⁻¹ = 1

theorem Units.inv_mul' {G₀ : Type u_3} [GroupWithZero G₀] (u : G₀ˣ) : (↑u)⁻¹ * ↑u = 1

@[simp]

theorem Units.mk0_inj {G₀ : Type u_3} [GroupWithZero G₀] {a b : G₀} (ha : a ≠ 0) (hb : b ≠ 0) : mk0 a ha = mk0 b hb ↔ a = b

theorem Units.exists0 {G₀ : Type u_3} [GroupWithZero G₀] {p : G₀ˣ → Prop} : (∃ (g : G₀ˣ), p g) ↔ ∃ (g : G₀), ∃ (hg : g ≠ 0), p (mk0 g hg)

In a group with zero, an existential over a unit can be rewritten in terms of Units.mk0.

theorem Units.exists0' {G₀ : Type u_3} [GroupWithZero G₀] {p : (g : G₀) → g ≠ 0 → Prop} : (∃ (g : G₀), ∃ (hg : g ≠ 0), p g hg) ↔ ∃ (g : G₀ˣ), p ↑g ⋯

An alternative version of Units.exists0. This one is useful if Lean cannot figure out p when using Units.exists0 from right to left.

@[simp]

theorem Units.exists_iff_ne_zero {G₀ : Type u_3} [GroupWithZero G₀] {p : G₀ → Prop} : (∃ (u : G₀ˣ), p ↑u) ↔ ∃ (x : G₀), x ≠ 0 ∧ p x

theorem GroupWithZero.eq_zero_or_unit {G₀ : Type u_3} [GroupWithZero G₀] (a : G₀) : a = 0 ∨ ∃ (u : G₀ˣ), a = ↑u

theorem IsUnit.mk0 {G₀ : Type u_3} [GroupWithZero G₀] (x : G₀) (hx : x ≠ 0) : IsUnit x

@[simp]

theorem isUnit_iff_ne_zero {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} : IsUnit a ↔ a ≠ 0

theorem Ne.isUnit {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} : a ≠ 0 → IsUnit a

Alias of the reverse direction of isUnit_iff_ne_zero.

@[instance 10]

instance GroupWithZero.noZeroDivisors {G₀ : Type u_3} [GroupWithZero G₀] : NoZeroDivisors G₀

@[simp]

theorem Units.mk0_mul {G₀ : Type u_3} [GroupWithZero G₀] (x y : G₀) (hxy : x * y ≠ 0) : mk0 (x * y) hxy = mk0 x ⋯ * mk0 y ⋯

theorem div_ne_zero {G₀ : Type u_3} [GroupWithZero G₀] {a b : G₀} (ha : a ≠ 0) (hb : b ≠ 0) : a / b ≠ 0

@[simp]

theorem div_eq_zero_iff {G₀ : Type u_3} [GroupWithZero G₀] {a b : G₀} : a / b = 0 ↔ a = 0 ∨ b = 0

theorem div_ne_zero_iff {G₀ : Type u_3} [GroupWithZero G₀] {a b : G₀} : a / b ≠ 0 ↔ a ≠ 0 ∧ b ≠ 0

@[simp]

theorem div_self {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (h : a ≠ 0) : a / a = 1

theorem eq_mul_inv_iff_mul_eq₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hc : c ≠ 0) : a = b * c⁻¹ ↔ a * c = b

theorem eq_inv_mul_iff_mul_eq₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hb : b ≠ 0) : a = b⁻¹ * c ↔ b * a = c

theorem inv_mul_eq_iff_eq_mul₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (ha : a ≠ 0) : a⁻¹ * b = c ↔ b = a * c

theorem mul_inv_eq_iff_eq_mul₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hb : b ≠ 0) : a * b⁻¹ = c ↔ a = c * b

theorem mul_inv_eq_one₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b : G₀} (hb : b ≠ 0) : a * b⁻¹ = 1 ↔ a = b

theorem inv_mul_eq_one₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b : G₀} (ha : a ≠ 0) : a⁻¹ * b = 1 ↔ a = b

theorem mul_eq_one_iff_eq_inv₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b : G₀} (hb : b ≠ 0) : a * b = 1 ↔ a = b⁻¹

theorem mul_eq_one_iff_inv_eq₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b : G₀} (ha : a ≠ 0) : a * b = 1 ↔ a⁻¹ = b

theorem mul_eq_of_eq_mul_inv₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (ha : a ≠ 0) (h : a = c * b⁻¹) : a * b = c

A variant of eq_mul_inv_iff_mul_eq₀ that moves the nonzero hypothesis to another variable.

theorem mul_eq_of_eq_inv_mul₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hb : b ≠ 0) (h : b = a⁻¹ * c) : a * b = c

A variant of eq_inv_mul_iff_mul_eq₀ that moves the nonzero hypothesis to another variable.

theorem eq_mul_of_inv_mul_eq₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hc : c ≠ 0) (h : b⁻¹ * a = c) : a = b * c

A variant of inv_mul_eq_iff_eq_mul₀ that moves the nonzero hypothesis to another variable.

theorem eq_mul_of_mul_inv_eq₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hb : b ≠ 0) (h : a * c⁻¹ = b) : a = b * c

A variant of mul_inv_eq_iff_eq_mul₀ that moves the nonzero hypothesis to another variable.

@[simp]

theorem div_mul_cancel₀ {G₀ : Type u_3} [GroupWithZero G₀] {b : G₀} (a : G₀) (h : b ≠ 0) : a / b * b = a

theorem mul_one_div_cancel {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (h : a ≠ 0) : a * (1 / a) = 1

theorem one_div_mul_cancel {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (h : a ≠ 0) : 1 / a * a = 1

theorem div_left_inj' {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hc : c ≠ 0) : a / c = b / c ↔ a = b

theorem div_eq_iff {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hb : b ≠ 0) : a / b = c ↔ a = c * b

theorem eq_div_iff {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hb : b ≠ 0) : c = a / b ↔ c * b = a

theorem div_eq_iff_mul_eq {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hb : b ≠ 0) : a / b = c ↔ c * b = a

theorem eq_div_iff_mul_eq {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hc : c ≠ 0) : a = b / c ↔ a * c = b

theorem div_eq_of_eq_mul {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hb : b ≠ 0) : a = c * b → a / b = c

theorem eq_div_of_mul_eq {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hc : c ≠ 0) : a * c = b → a = b / c

theorem div_eq_one_iff_eq {G₀ : Type u_3} [GroupWithZero G₀] {a b : G₀} (hb : b ≠ 0) : a / b = 1 ↔ a = b

theorem div_mul_cancel_right₀ {G₀ : Type u_3} [GroupWithZero G₀] {b : G₀} (hb : b ≠ 0) (a : G₀) : b / (a * b) = a⁻¹

theorem mul_div_mul_right {G₀ : Type u_3} [GroupWithZero G₀] {c : G₀} (a b : G₀) (hc : c ≠ 0) : a * c / (b * c) = a / b

theorem mul_mul_div {G₀ : Type u_3} [GroupWithZero G₀] {b : G₀} (a : G₀) (hb : b ≠ 0) : a = a * b * (1 / b)

theorem div_div_div_cancel_right₀ {G₀ : Type u_3} [GroupWithZero G₀] {c : G₀} (hc : c ≠ 0) (a b : G₀) : a / c / (b / c) = a / b

theorem div_mul_div_cancel₀ {G₀ : Type u_3} [GroupWithZero G₀] {a b c : G₀} (hb : b ≠ 0) : a / b * (b / c) = a / c

theorem div_mul_cancel_of_imp {G₀ : Type u_3} [GroupWithZero G₀] {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a

theorem mul_div_cancel_of_imp {G₀ : Type u_3} [GroupWithZero G₀] {a b : G₀} (h : b = 0 → a = 0) : a * b / b = a

@[simp]

theorem divp_mk0 {G₀ : Type u_3} [GroupWithZero G₀] {b : G₀} (a : G₀) (hb : b ≠ 0) : a /ₚ Units.mk0 b hb = a / b

theorem pow_sub₀ {G₀ : Type u_3} [GroupWithZero G₀] {m n : ℕ} (a : G₀) (ha : a ≠ 0) (h : n ≤ m) : a ^ (m - n) = a ^ m * (a ^ n)⁻¹

theorem pow_sub_of_lt {G₀ : Type u_3} [GroupWithZero G₀] {m n : ℕ} (a : G₀) (h : n < m) : a ^ (m - n) = a ^ m * (a ^ n)⁻¹

theorem inv_pow_sub₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {m n : ℕ} (ha : a ≠ 0) (h : n ≤ m) : a⁻¹ ^ (m - n) = (a ^ m)⁻¹ * a ^ n

theorem inv_pow_sub_of_lt {G₀ : Type u_3} [GroupWithZero G₀] {m n : ℕ} (a : G₀) (h : n < m) : a⁻¹ ^ (m - n) = (a ^ m)⁻¹ * a ^ n

theorem zpow_sub₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (ha : a ≠ 0) (m n : ℤ) : a ^ (m - n) = a ^ m / a ^ n

theorem zpow_natCast_sub_natCast₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (ha : a ≠ 0) (m n : ℕ) : a ^ (↑m - ↑n) = a ^ m / a ^ n

theorem zpow_natCast_sub_one₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (ha : a ≠ 0) (n : ℕ) : a ^ (↑n - 1) = a ^ n / a

theorem zpow_one_sub_natCast₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (ha : a ≠ 0) (n : ℕ) : a ^ (1 - ↑n) = a / a ^ n

theorem zpow_ne_zero {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (n : ℤ) : a ≠ 0 → a ^ n ≠ 0

theorem eq_zero_of_zpow_eq_zero {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {n : ℤ} : a ^ n = 0 → a = 0

theorem zpow_eq_zero_iff {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {n : ℤ} (hn : n ≠ 0) : a ^ n = 0 ↔ a = 0

theorem zpow_ne_zero_iff {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {n : ℤ} (hn : n ≠ 0) : a ^ n ≠ 0 ↔ a ≠ 0

theorem zpow_neg_mul_zpow_self {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (n : ℤ) (ha : a ≠ 0) : a ^ (-n) * a ^ n = 1

theorem Ring.inverse_eq_inv {G₀ : Type u_3} [GroupWithZero G₀] (a : G₀) : inverse a = a⁻¹

@[simp]

theorem Ring.inverse_eq_inv' {G₀ : Type u_3} [GroupWithZero G₀] : inverse = Inv.inv

@[instance 10]

instance CommGroupWithZero.toCancelCommMonoidWithZero {G₀ : Type u_3} [CommGroupWithZero G₀] : CancelCommMonoidWithZero G₀

Equations

• CommGroupWithZero.toCancelCommMonoidWithZero = { toMonoid := MonoidWithZero.toMonoid, mul_comm := ⋯, toZero := MonoidWithZero.toZero, zero_mul := ⋯, mul_zero := ⋯, toIsLeftCancelMulZero := ⋯ }

@[instance 100]

instance CommGroupWithZero.toDivisionCommMonoid {G₀ : Type u_3} [CommGroupWithZero G₀] : DivisionCommMonoid G₀

Equations

• One or more equations did not get rendered due to their size.

theorem div_mul_cancel_left₀ {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} (ha : a ≠ 0) (b : G₀) : a / (a * b) = b⁻¹

theorem mul_div_cancel_left_of_imp {G₀ : Type u_3} [CommGroupWithZero G₀] {a b : G₀} (h : a = 0 → b = 0) : a * b / a = b

theorem mul_div_cancel_of_imp' {G₀ : Type u_3} [CommGroupWithZero G₀] {a b : G₀} (h : b = 0 → a = 0) : b * (a / b) = a

theorem mul_div_cancel₀ {G₀ : Type u_3} [CommGroupWithZero G₀] {b : G₀} (a : G₀) (hb : b ≠ 0) : b * (a / b) = a

theorem mul_div_mul_left {G₀ : Type u_3} [CommGroupWithZero G₀] {c : G₀} (a b : G₀) (hc : c ≠ 0) : c * a / (c * b) = a / b

theorem mul_eq_mul_of_div_eq_div {G₀ : Type u_3} [CommGroupWithZero G₀] {b d : G₀} (a c : G₀) (hb : b ≠ 0) (hd : d ≠ 0) (h : a / b = c / d) : a * d = c * b

theorem div_eq_div_iff {G₀ : Type u_3} [CommGroupWithZero G₀] {a b c d : G₀} (hb : b ≠ 0) (hd : d ≠ 0) : a / b = c / d ↔ a * d = c * b

theorem div_eq_div_iff_div_eq_div' {G₀ : Type u_3} [CommGroupWithZero G₀] {a b c d : G₀} (hb : b ≠ 0) (hc : c ≠ 0) : a / b = c / d ↔ a / c = b / d

The CommGroupWithZero version of div_eq_div_iff_div_eq_div.

@[simp]

theorem div_div_cancel₀ {G₀ : Type u_3} [CommGroupWithZero G₀] {a b : G₀} (ha : a ≠ 0) : a / (a / b) = b

@[deprecated div_div_cancel₀ (since := "2024-11-25")]

theorem div_div_cancel' {G₀ : Type u_3} [CommGroupWithZero G₀] {a b : G₀} (ha : a ≠ 0) : a / (a / b) = b

Alias of div_div_cancel₀.

theorem div_div_cancel_left' {G₀ : Type u_3} [CommGroupWithZero G₀] {a b : G₀} (ha : a ≠ 0) : a / b / a = b⁻¹

theorem div_helper {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} (b : G₀) (h : a ≠ 0) : 1 / (a * b) * a = 1 / b

theorem div_div_div_cancel_left' {G₀ : Type u_3} [CommGroupWithZero G₀] {c : G₀} (a b : G₀) (hc : c ≠ 0) : c / a / (c / b) = b / a

@[simp]

theorem div_mul_div_cancel₀' {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} (ha : a ≠ 0) (b c : G₀) : a / b * (c / a) = c / b

noncomputable def groupWithZeroOfIsUnitOrEqZero {M : Type u_4} [Nontrivial M] [hM : MonoidWithZero M] (h : ∀ (a : M), IsUnit a ∨ a = 0) : GroupWithZero M

Constructs a GroupWithZero structure on a MonoidWithZero consisting only of units and 0.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def commGroupWithZeroOfIsUnitOrEqZero {M : Type u_4} [Nontrivial M] [hM : CommMonoidWithZero M] (h : ∀ (a : M), IsUnit a ∨ a = 0) : CommGroupWithZero M

Constructs a CommGroupWithZero structure on a CommMonoidWithZero consisting only of units and 0.

Equations

• One or more equations did not get rendered due to their size.