mathlib3 documentation

algebra.group_with_zero.units.lemmas

Further lemmas about units in a monoid_with_zero or a group_with_zero. #

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@[simp]
theorem div_self {G₀ : Type u_3} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) :
a / a = 1
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theorem eq_mul_inv_iff_mul_eq₀ {G₀ : Type u_3} [group_with_zero G₀] {a b 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} [group_with_zero G₀] {a b 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} [group_with_zero G₀] {a b 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} [group_with_zero G₀] {a b 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} [group_with_zero G₀] {a b : G₀} (hb : b ≠ 0) :
a * b⁻¹ = 1 ↔ a = b
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theorem inv_mul_eq_one₀ {G₀ : Type u_3} [group_with_zero G₀] {a 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} [group_with_zero G₀] {a 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} [group_with_zero G₀] {a b : G₀} (ha : a ≠ 0) :
a * b = 1 ↔ a⁻¹ = b
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@[simp]
theorem div_mul_cancel {G₀ : Type u_3} [group_with_zero 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} [group_with_zero 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} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) :
a * (1 / a) = 1
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theorem one_div_mul_cancel {G₀ : Type u_3} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) :
1 / a * a = 1
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theorem div_left_inj' {G₀ : Type u_3} [group_with_zero G₀] {a b c : G₀} (hc : c ≠ 0) :
a / c = b / c ↔ a = b
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theorem div_eq_iff {G₀ : Type u_3} [group_with_zero G₀] {a b c : G₀} (hb : b ≠ 0) :
a / b = c ↔ a = c * b
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theorem eq_div_iff {G₀ : Type u_3} [group_with_zero G₀] {a b 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} [group_with_zero G₀] {a b 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} [group_with_zero G₀] {a b 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} [group_with_zero G₀] {a b 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} [group_with_zero G₀] {a b 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} [group_with_zero G₀] {a b : G₀} (hb : b ≠ 0) :
a / b = 1 ↔ a = b
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theorem div_mul_left {G₀ : Type u_3} [group_with_zero G₀] {a b : G₀} (hb : b ≠ 0) :
b / (a * b) = 1 / a
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theorem mul_div_mul_right {G₀ : Type u_3} [group_with_zero G₀] {c : G₀} (a b : G₀) (hc : c ≠ 0) :
a * c / (b * c) = a / b
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theorem mul_mul_div {G₀ : Type u_3} [group_with_zero 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} [group_with_zero G₀] {b 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} [group_with_zero G₀] {b 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} [group_with_zero G₀] {a 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} [group_with_zero G₀] {a b : G₀} (h : b = 0 → a = 0) :
a * b / b = a
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@[simp]
theorem divp_mk0 {G₀ : Type u_3} [group_with_zero G₀] (a : G₀) {b : G₀} (hb : b ≠ 0) :
a /ₚ units.mk0 b hb = a / b
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theorem div_mul_right {G₀ : Type u_3} [comm_group_with_zero 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} [comm_group_with_zero G₀] {a b : G₀} (h : a = 0 → b = 0) :
a * b / a = b
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theorem mul_div_cancel_left {G₀ : Type u_3} [comm_group_with_zero 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} [comm_group_with_zero G₀] {a b : G₀} (h : b = 0 → a = 0) :
b * (a / b) = a
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theorem mul_div_cancel' {G₀ : Type u_3} [comm_group_with_zero 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} [comm_group_with_zero G₀] {c : G₀} (a 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} [comm_group_with_zero 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} [comm_group_with_zero G₀] {a b c d : G₀} (hb : b ≠ 0) (hd : d ≠ 0) :
a / b = c / d ↔ a * d = c * b
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theorem div_div_cancel' {G₀ : Type u_3} [comm_group_with_zero G₀] {a b : G₀} (ha : a ≠ 0) :
a / (a / b) = b
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theorem div_div_cancel_left' {G₀ : Type u_3} [comm_group_with_zero G₀] {a b : G₀} (ha : a ≠ 0) :
a / b / a = b⁻¹
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theorem div_helper {G₀ : Type u_3} [comm_group_with_zero G₀] {a : G₀} (b : G₀) (h : a ≠ 0) :
1 / (a * b) * a = 1 / b
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theorem map_ne_zero {M₀ : Type u_2} {G₀ : Type u_3} {F : Type u_6} [monoid_with_zero M₀] [group_with_zero G₀] [nontrivial M₀] [monoid_with_zero_hom_class 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} [monoid_with_zero M₀] [group_with_zero G₀] [nontrivial M₀] [monoid_with_zero_hom_class 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} [group_with_zero G₀] [monoid_with_zero M₀'] [monoid_with_zero_hom_class F' G₀ M₀'] {a : G₀} (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} [group_with_zero G₀] [group_with_zero G₀'] [monoid_with_zero_hom_class 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} [group_with_zero G₀] [group_with_zero G₀'] [monoid_with_zero_hom_class F G₀ G₀'] (f : F) (a b : G₀) :
⇑f (a / b) = ⇑f a / ⇑f b
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noncomputable def monoid_with_zero.inverse {M : Type u_1} [comm_monoid_with_zero M] :
M →*₀ M

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

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@[simp]
theorem monoid_with_zero.coe_inverse {M : Type u_1} [comm_monoid_with_zero M] :
⇑monoid_with_zero.inverse = ring.inverse
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@[simp]
theorem monoid_with_zero.inverse_apply {M : Type u_1} [comm_monoid_with_zero M] (a : M) :
⇑monoid_with_zero.inverse a = ring.inverse a
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def inv_monoid_with_zero_hom {G₀ : Type u_1} [comm_group_with_zero G₀] :
G₀ →*₀ G₀

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

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@[simp]
theorem units.smul_mk0 {G₀ : Type u_3} [group_with_zero G₀] {α : Type u_1} [has_smul G₀ α] {g : G₀} (hg : g ≠ 0) (a : α) :
units.mk0 g hg • a = g • a