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Mathlib.Algebra.GroupWithZero.Units.Lemmas

Further lemmas about units in a MonoidWithZero or a GroupWithZero. #

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@[simp]
theorem div_self {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (h : a ≠ 0) :
a / a = 1
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theorem eq_mul_inv_iff_mul_eq₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} (hc : c ≠ 0) :
a = b * c⁻¹ ↔ a * c = b
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theorem eq_inv_mul_iff_mul_eq₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} (hb : b ≠ 0) :
a = b⁻¹ * c ↔ b * a = c
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theorem inv_mul_eq_iff_eq_mul₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} (ha : a ≠ 0) :
a⁻¹ * b = c ↔ b = a * c
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theorem mul_inv_eq_iff_eq_mul₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} (hb : b ≠ 0) :
a * b⁻¹ = c ↔ a = c * b
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theorem mul_inv_eq_one₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} (hb : b ≠ 0) :
a * b⁻¹ = 1 ↔ a = b
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theorem inv_mul_eq_one₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} (ha : a ≠ 0) :
a⁻¹ * b = 1 ↔ a = b
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theorem mul_eq_one_iff_eq_inv₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} (hb : b ≠ 0) :
a * b = 1 ↔ a = b⁻¹
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theorem mul_eq_one_iff_inv_eq₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} (ha : a ≠ 0) :
a * b = 1 ↔ a⁻¹ = b
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theorem mul_eq_of_eq_mul_inv₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {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.

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theorem mul_eq_of_eq_inv_mul₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {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.

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theorem eq_mul_of_inv_mul_eq₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {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.

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theorem eq_mul_of_mul_inv_eq₀ {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {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.

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@[simp]
theorem div_mul_cancel {G₀ : Type u_3} [GroupWithZero G₀] {b : G₀} (a : G₀) (h : b ≠ 0) :
a / b * b = a
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@[simp]
theorem mul_div_cancel {G₀ : Type u_3} [GroupWithZero G₀] {b : G₀} (a : G₀) (h : b ≠ 0) :
a * b / b = a
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theorem mul_one_div_cancel {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (h : a ≠ 0) :
a * (1 / a) = 1
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theorem one_div_mul_cancel {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} (h : a ≠ 0) :
1 / a * a = 1
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theorem div_left_inj' {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} (hc : c ≠ 0) :
a / c = b / c ↔ a = b
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theorem div_eq_iff {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} (hb : b ≠ 0) :
a / b = c ↔ a = c * b
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theorem eq_div_iff {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} (hb : b ≠ 0) :
c = a / b ↔ c * b = a
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theorem div_eq_iff_mul_eq {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} (hb : b ≠ 0) :
a / b = c ↔ c * b = a
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theorem eq_div_iff_mul_eq {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} (hc : c ≠ 0) :
a = b / c ↔ a * c = b
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theorem div_eq_of_eq_mul {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} (hb : b ≠ 0) :
a = c * b → a / b = c
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theorem eq_div_of_mul_eq {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} (hc : c ≠ 0) :
a * c = b → a = b / c
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theorem div_eq_one_iff_eq {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} (hb : b ≠ 0) :
a / b = 1 ↔ a = b
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theorem div_mul_left {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} (hb : b ≠ 0) :
b / (a * b) = 1 / a
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theorem mul_div_mul_right {G₀ : Type u_3} [GroupWithZero G₀] {c : G₀} (a : G₀) (b : G₀) (hc : c ≠ 0) :
a * c / (b * c) = a / b
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theorem mul_mul_div {G₀ : Type u_3} [GroupWithZero G₀] {b : G₀} (a : G₀) (hb : b ≠ 0) :
a = a * b * (1 / b)
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theorem div_div_div_cancel_right {G₀ : Type u_3} [GroupWithZero G₀] {b : G₀} {c : G₀} (a : G₀) (hc : c ≠ 0) :
a / c / (b / c) = a / b
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theorem div_mul_div_cancel {G₀ : Type u_3} [GroupWithZero G₀] {b : G₀} {c : G₀} (a : G₀) (hc : c ≠ 0) :
a / c * (c / b) = a / b
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theorem div_mul_cancel_of_imp {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} (h : b = 0 → a = 0) :
a / b * b = a
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theorem mul_div_cancel_of_imp {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} (h : b = 0 → a = 0) :
a * b / b = a
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@[simp]
theorem divp_mk0 {G₀ : Type u_3} [GroupWithZero G₀] (a : G₀) {b : G₀} (hb : b ≠ 0) :
a /ₚ Units.mk0 b hb = a / b
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theorem Commute.div_eq_div_iff {G₀ : Type u_3} [GroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} {d : G₀} (hbd : Commute b d) (hb : b ≠ 0) (hd : d ≠ 0) :
a / b = c / d ↔ a * d = c * b

The MonoidWithZero version of div_eq_div_iff_mul_eq_mul.

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theorem div_mul_right {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} (b : G₀) (ha : a ≠ 0) :
a / (a * b) = 1 / b
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theorem mul_div_cancel_left_of_imp {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} {b : G₀} (h : a = 0 → b = 0) :
a * b / a = b
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theorem mul_div_cancel_left {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} (b : G₀) (ha : a ≠ 0) :
a * b / a = b
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theorem mul_div_cancel_of_imp' {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} {b : G₀} (h : b = 0 → a = 0) :
b * (a / b) = a
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theorem mul_div_cancel' {G₀ : Type u_3} [CommGroupWithZero G₀] {b : G₀} (a : G₀) (hb : b ≠ 0) :
b * (a / b) = a
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theorem mul_div_mul_left {G₀ : Type u_3} [CommGroupWithZero G₀] {c : G₀} (a : G₀) (b : G₀) (hc : c ≠ 0) :
c * a / (c * b) = a / b
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theorem mul_eq_mul_of_div_eq_div {G₀ : Type u_3} [CommGroupWithZero G₀] (a : G₀) {b : G₀} (c : G₀) {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0) (h : a / b = c / d) :
a * d = c * b
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theorem div_eq_div_iff {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0) :
a / b = c / d ↔ a * d = c * b
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theorem div_eq_div_iff_div_eq_div' {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} {b : G₀} {c : G₀} {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.

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theorem div_div_cancel' {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} {b : G₀} (ha : a ≠ 0) :
a / (a / b) = b
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theorem div_div_cancel_left' {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} {b : G₀} (ha : a ≠ 0) :
a / b / a = b⁻¹
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theorem div_helper {G₀ : Type u_3} [CommGroupWithZero G₀] {a : G₀} (b : G₀) (h : a ≠ 0) :
1 / (a * b) * a = 1 / b
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theorem div_div_div_cancel_left' {G₀ : Type u_3} [CommGroupWithZero G₀] {c : G₀} (a : G₀) (b : G₀) (hc : c ≠ 0) :
c / a / (c / b) = b / a
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theorem map_ne_zero {M₀ : Type u_2} {G₀ : Type u_3} {F : Type u_6} [MonoidWithZero M₀] [GroupWithZero G₀] [Nontrivial M₀] [FunLike F G₀ M₀] [MonoidWithZeroHomClass F G₀ M₀] (f : F) {a : G₀} :
f a ≠ 0 ↔ a ≠ 0
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@[simp]
theorem map_eq_zero {M₀ : Type u_2} {G₀ : Type u_3} {F : Type u_6} [MonoidWithZero M₀] [GroupWithZero G₀] [Nontrivial M₀] [FunLike F G₀ M₀] [MonoidWithZeroHomClass F G₀ M₀] (f : F) {a : G₀} :
f a = 0 ↔ a = 0
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theorem eq_on_inv₀ {G₀ : Type u_3} {M₀' : Type u_4} {F' : Type u_7} [GroupWithZero G₀] [MonoidWithZero M₀'] [FunLike F' G₀ M₀'] [MonoidWithZeroHomClass F' G₀ M₀'] {a : G₀} (f : F') (g : F') (h : f a = g a) :
f a⁻¹ = g a⁻¹
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@[simp]
theorem map_inv₀ {G₀ : Type u_3} {G₀' : Type u_5} {F : Type u_6} [GroupWithZero G₀] [GroupWithZero G₀'] [FunLike F G₀ G₀'] [MonoidWithZeroHomClass F G₀ G₀'] (f : F) (a : G₀) :
f a⁻¹ = (f a)⁻¹

A monoid homomorphism between groups with zeros sending 0 to 0 sends a⁻¹ to (f a)⁻¹.

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@[simp]
theorem map_div₀ {G₀ : Type u_3} {G₀' : Type u_5} {F : Type u_6} [GroupWithZero G₀] [GroupWithZero G₀'] [FunLike F G₀ G₀'] [MonoidWithZeroHomClass F G₀ G₀'] (f : F) (a : G₀) (b : G₀) :
f (a / b) = f a / f b
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noncomputable def MonoidWithZero.inverse {M : Type u_8} [CommMonoidWithZero M] :
M →*₀ M

We define the inverse as a MonoidWithZeroHom by extending the inverse map by zero on non-units.

Equations
  • MonoidWithZero.inverse = { toZeroHom := { toFun := Ring.inverse, map_zero' := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }
Instances For
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    @[simp]
    theorem MonoidWithZero.coe_inverse {M : Type u_8} [CommMonoidWithZero M] :
    ⇑MonoidWithZero.inverse = Ring.inverse
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    @[simp]
    theorem MonoidWithZero.inverse_apply {M : Type u_8} [CommMonoidWithZero M] (a : M) :
    MonoidWithZero.inverse a = Ring.inverse a
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    def invMonoidWithZeroHom {G₀ : Type u_8} [CommGroupWithZero G₀] :
    G₀ →*₀ G₀

    Inversion on a commutative group with zero, considered as a monoid with zero homomorphism.

    Equations
    • invMonoidWithZeroHom = let __src := invMonoidHom; { toZeroHom := { toFun := __src.toFun, map_zero' := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }
    Instances For
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      @[simp]
      theorem Units.smul_mk0 {G₀ : Type u_3} [GroupWithZero G₀] {α : Type u_8} [SMul G₀ α] {g : G₀} (hg : g ≠ 0) (a : α) :
      Units.mk0 g hg • a = g • a
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      @[simp]
      theorem map_zpow₀ {F : Type u_8} {G₀ : Type u_9} {G₀' : Type u_10} [GroupWithZero G₀] [GroupWithZero G₀'] [FunLike F G₀ G₀'] [MonoidWithZeroHomClass F G₀ G₀'] (f : F) (x : G₀) (n : ℤ) :
      f (x ^ n) = f x ^ n

      If a monoid homomorphism f between two GroupWithZeros maps 0 to 0, then it maps x^n, n : ℤ, to (f x)^n.