mathlib documentation

algebra.group.basic

Basic lemmas about semigroups, monoids, and groups #

This file lists various basic lemmas about semigroups, monoids, and groups. Most proofs are one-liners from the corresponding axioms. For the definitions of semigroups, monoids and groups, see algebra/group/defs.lean.

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theorem comp_assoc_left {α : Type u} (f : α → α → α) [is_associative α f] (x y : α) :
f x ∘ f y = f (f x y)

Composing two associative operations of f : α → α → α on the left is equal to an associative operation on the left.

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theorem comp_assoc_right {α : Type u} (f : α → α → α) [is_associative α f] (x y : α) :
((λ (z : α), f z x) ∘ λ (z : α), f z y) = λ (z : α), f z (f y x)

Composing two associative operations of f : α → α → α on the right is equal to an associative operation on the right.

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@[simp]
theorem comp_add_left {α : Type u_1} [add_semigroup α] (x y : α) :
has_add.add x ∘ has_add.add y = has_add.add (x + y)

Composing two additions on the left by y then x is equal to a addition on the left by x + y.

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@[simp]
theorem comp_mul_left {α : Type u_1} [semigroup α] (x y : α) :
has_mul.mul x ∘ has_mul.mul y = has_mul.mul (x * y)

Composing two multiplications on the left by y then x is equal to a multiplication on the left by x * y.

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@[simp]
theorem comp_add_right {α : Type u_1} [add_semigroup α] (x y : α) :
((λ (_x : α), _x + x) ∘ λ (_x : α), _x + y) = λ (_x : α), _x + (y + x)

Composing two additions on the right by y and x is equal to a addition on the right by y + x.

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@[simp]
theorem comp_mul_right {α : Type u_1} [semigroup α] (x y : α) :
((λ (_x : α), _x * x) ∘ λ (_x : α), _x * y) = λ (_x : α), _x * y * x

Composing two multiplications on the right by y and x is equal to a multiplication on the right by y * x.

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theorem ite_mul_one {M : Type u} [monoid M] {P : Prop} [decidable P] {a b : M} :
ite P (a * b) 1 = (ite P a 1) * ite P b 1
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theorem ite_add_zero {M : Type u} [add_monoid M] {P : Prop} [decidable P] {a b : M} :
ite P (a + b) 0 = ite P a 0 + ite P b 0
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theorem eq_zero_iff_eq_zero_of_add_eq_zero {M : Type u} [add_monoid M] {a b : M} (h : a + b = 0) :
a = 0 ↔ b = 0
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theorem eq_one_iff_eq_one_of_mul_eq_one {M : Type u} [monoid M] {a b : M} (h : a * b = 1) :
a = 1 ↔ b = 1
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theorem mul_left_comm {G : Type u} [comm_semigroup G] (a b c : G) :
a * b * c = b * a * c
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theorem add_left_comm {G : Type u} [add_comm_semigroup G] (a b c : G) :
a + (b + c) = b + (a + c)
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theorem mul_right_comm {G : Type u} [comm_semigroup G] (a b c : G) :
(a * b) * c = (a * c) * b
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theorem add_right_comm {G : Type u} [add_comm_semigroup G] (a b c : G) :
a + b + c = a + c + b
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theorem add_add_add_comm {G : Type u} [add_comm_semigroup G] (a b c d : G) :
a + b + (c + d) = a + c + (b + d)
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theorem mul_mul_mul_comm {G : Type u} [comm_semigroup G] (a b c d : G) :
(a * b) * c * d = (a * c) * b * d
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@[simp]
theorem bit0_zero {M : Type u} [add_monoid M] :
bit0 0 = 0
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@[simp]
theorem bit1_zero {M : Type u} [add_monoid M] [has_one M] :
bit1 0 = 1
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theorem inv_unique {M : Type u} [comm_monoid M] {x y z : M} (hy : x * y = 1) (hz : x * z = 1) :
y = z
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theorem neg_unique {M : Type u} [add_comm_monoid M] {x y z : M} (hy : x + y = 0) (hz : x + z = 0) :
y = z
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@[simp]
theorem add_right_eq_self {M : Type u} [add_left_cancel_monoid M] {a b : M} :
a + b = a ↔ b = 0
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@[simp]
theorem mul_right_eq_self {M : Type u} [left_cancel_monoid M] {a b : M} :
a * b = a ↔ b = 1
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@[simp]
theorem self_eq_mul_right {M : Type u} [left_cancel_monoid M] {a b : M} :
a = a * b ↔ b = 1
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@[simp]
theorem self_eq_add_right {M : Type u} [add_left_cancel_monoid M] {a b : M} :
a = a + b ↔ b = 0
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@[simp]
theorem add_left_eq_self {M : Type u} [add_right_cancel_monoid M] {a b : M} :
a + b = b ↔ a = 0
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@[simp]
theorem mul_left_eq_self {M : Type u} [right_cancel_monoid M] {a b : M} :
a * b = b ↔ a = 1
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@[simp]
theorem self_eq_add_left {M : Type u} [add_right_cancel_monoid M] {a b : M} :
b = a + b ↔ a = 0
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@[simp]
theorem self_eq_mul_left {M : Type u} [right_cancel_monoid M] {a b : M} :
b = a * b ↔ a = 1
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theorem neg_eq_zero_sub {G : Type u} [sub_neg_monoid G] (x : G) :
-x = 0 - x
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theorem inv_eq_one_div {G : Type u} [div_inv_monoid G] (x : G) :
x⁻¹ = 1 / x
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theorem add_zero_sub {G : Type u} [sub_neg_monoid G] (x y : G) :
x + (0 - y) = x - y
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theorem mul_one_div {G : Type u} [div_inv_monoid G] (x y : G) :
x * (1 / y) = x / y
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theorem mul_div_assoc {G : Type u} [div_inv_monoid G] {a b c : G} :
a * b / c = a * (b / c)
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theorem mul_div_assoc' {G : Type u} [div_inv_monoid G] (a b c : G) :
a * (b / c) = a * b / c
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@[simp]
theorem one_div {G : Type u} [div_inv_monoid G] (a : G) :
1 / a = a⁻¹
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@[simp]
theorem zero_sub {G : Type u} [sub_neg_monoid G] (a : G) :
0 - a = -a
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@[simp]
theorem neg_add_cancel_right {G : Type u} [add_group G] (a b : G) :
a + -b + b = a
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@[simp]
theorem inv_mul_cancel_right {G : Type u} [group G] (a b : G) :
(a * b⁻¹) * b = a
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@[simp]
theorem one_inv {G : Type u} [group G] :
1⁻¹ = 1
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@[simp]
theorem neg_zero {G : Type u} [add_group G] :
-0 = 0
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theorem left_inverse_neg (G : Type u_1) [add_group G] :
function.left_inverse (λ (a : G), -a) (λ (a : G), -a)
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theorem left_inverse_inv (G : Type u_1) [group G] :
function.left_inverse (λ (a : G), a⁻¹) (λ (a : G), a⁻¹)
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@[simp]
theorem inv_involutive {G : Type u} [group G] :
function.involutive has_inv.inv
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@[simp]
theorem neg_involutive {G : Type u} [add_group G] :
function.involutive has_neg.neg
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theorem neg_injective {G : Type u} [add_group G] :
function.injective has_neg.neg
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theorem inv_injective {G : Type u} [group G] :
function.injective has_inv.inv
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@[simp]
theorem inv_inj {G : Type u} [group G] {a b : G} :
a⁻¹ = b⁻¹ ↔ a = b
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@[simp]
theorem neg_inj {G : Type u} [add_group G] {a b : G} :
-a = -b ↔ a = b
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@[simp]
theorem mul_inv_cancel_left {G : Type u} [group G] (a b : G) :
a * a⁻¹ * b = b
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@[simp]
theorem add_neg_cancel_left {G : Type u} [add_group G] (a b : G) :
a + (-a + b) = b
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theorem mul_left_surjective {G : Type u} [group G] (a : G) :
function.surjective (has_mul.mul a)
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theorem add_left_surjective {G : Type u} [add_group G] (a : G) :
function.surjective (has_add.add a)
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theorem add_right_surjective {G : Type u} [add_group G] (a : G) :
function.surjective (λ (x : G), x + a)
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theorem mul_right_surjective {G : Type u} [group G] (a : G) :
function.surjective (λ (x : G), x * a)
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@[simp]
theorem mul_inv_rev {G : Type u} [group G] (a b : G) :
(a * b)⁻¹ = b⁻¹ * a⁻¹
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@[simp]
theorem neg_add_rev {G : Type u} [add_group G] (a b : G) :
-(a + b) = -b + -a
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theorem eq_inv_of_eq_inv {G : Type u} [group G] {a b : G} (h : a = b⁻¹) :
b = a⁻¹
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theorem eq_neg_of_eq_neg {G : Type u} [add_group G] {a b : G} (h : a = -b) :
b = -a
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theorem eq_inv_of_mul_eq_one {G : Type u} [group G] {a b : G} (h : a * b = 1) :
a = b⁻¹
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theorem eq_neg_of_add_eq_zero {G : Type u} [add_group G] {a b : G} (h : a + b = 0) :
a = -b
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theorem eq_add_neg_of_add_eq {G : Type u} [add_group G] {a b c : G} (h : a + c = b) :
a = b + -c
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theorem eq_mul_inv_of_mul_eq {G : Type u} [group G] {a b c : G} (h : a * c = b) :
a = b * c⁻¹
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theorem eq_neg_add_of_add_eq {G : Type u} [add_group G] {a b c : G} (h : b + a = c) :
a = -b + c
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theorem eq_inv_mul_of_mul_eq {G : Type u} [group G] {a b c : G} (h : b * a = c) :
a = b⁻¹ * c
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theorem inv_mul_eq_of_eq_mul {G : Type u} [group G] {a b c : G} (h : b = a * c) :
a⁻¹ * b = c
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theorem neg_add_eq_of_eq_add {G : Type u} [add_group G] {a b c : G} (h : b = a + c) :
-a + b = c
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theorem add_neg_eq_of_eq_add {G : Type u} [add_group G] {a b c : G} (h : a = c + b) :
a + -b = c
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theorem mul_inv_eq_of_eq_mul {G : Type u} [group G] {a b c : G} (h : a = c * b) :
a * b⁻¹ = c
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theorem eq_add_of_add_neg_eq {G : Type u} [add_group G] {a b c : G} (h : a + -c = b) :
a = b + c
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theorem eq_mul_of_mul_inv_eq {G : Type u} [group G] {a b c : G} (h : a * c⁻¹ = b) :
a = b * c
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theorem eq_mul_of_inv_mul_eq {G : Type u} [group G] {a b c : G} (h : b⁻¹ * a = c) :
a = b * c
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theorem eq_add_of_neg_add_eq {G : Type u} [add_group G] {a b c : G} (h : -b + a = c) :
a = b + c
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theorem mul_eq_of_eq_inv_mul {G : Type u} [group G] {a b c : G} (h : b = a⁻¹ * c) :
a * b = c
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theorem add_eq_of_eq_neg_add {G : Type u} [add_group G] {a b c : G} (h : b = -a + c) :
a + b = c
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theorem mul_eq_of_eq_mul_inv {G : Type u} [group G] {a b c : G} (h : a = c * b⁻¹) :
a * b = c
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theorem add_eq_of_eq_add_neg {G : Type u} [add_group G] {a b c : G} (h : a = c + -b) :
a + b = c
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@[simp]
theorem inv_eq_one {G : Type u} [group G] {a : G} :
a⁻¹ = 1 ↔ a = 1
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@[simp]
theorem neg_eq_zero {G : Type u} [add_group G] {a : G} :
-a = 0 ↔ a = 0
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@[simp]
theorem one_eq_inv {G : Type u} [group G] {a : G} :
1 = a⁻¹ ↔ a = 1
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@[simp]
theorem zero_eq_neg {G : Type u} [add_group G] {a : G} :
0 = -a ↔ a = 0
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theorem inv_ne_one {G : Type u} [group G] {a : G} :
a⁻¹ ≠ 1 ↔ a ≠ 1
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theorem neg_ne_zero {G : Type u} [add_group G] {a : G} :
-a ≠ 0 ↔ a ≠ 0
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theorem eq_inv_iff_eq_inv {G : Type u} [group G] {a b : G} :
a = b⁻¹ ↔ b = a⁻¹
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theorem eq_neg_iff_eq_neg {G : Type u} [add_group G] {a b : G} :
a = -b ↔ b = -a
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theorem neg_eq_iff_neg_eq {G : Type u} [add_group G] {a b : G} :
-a = b ↔ -b = a
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theorem inv_eq_iff_inv_eq {G : Type u} [group G] {a b : G} :
a⁻¹ = b ↔ b⁻¹ = a
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theorem mul_eq_one_iff_eq_inv {G : Type u} [group G] {a b : G} :
a * b = 1 ↔ a = b⁻¹
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theorem add_eq_zero_iff_eq_neg {G : Type u} [add_group G] {a b : G} :
a + b = 0 ↔ a = -b
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theorem mul_eq_one_iff_inv_eq {G : Type u} [group G] {a b : G} :
a * b = 1 ↔ a⁻¹ = b
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theorem add_eq_zero_iff_neg_eq {G : Type u} [add_group G] {a b : G} :
a + b = 0 ↔ -a = b
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theorem eq_inv_iff_mul_eq_one {G : Type u} [group G] {a b : G} :
a = b⁻¹ ↔ a * b = 1
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theorem eq_neg_iff_add_eq_zero {G : Type u} [add_group G] {a b : G} :
a = -b ↔ a + b = 0
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theorem neg_eq_iff_add_eq_zero {G : Type u} [add_group G] {a b : G} :
-a = b ↔ a + b = 0
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theorem inv_eq_iff_mul_eq_one {G : Type u} [group G] {a b : G} :
a⁻¹ = b ↔ a * b = 1
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theorem eq_add_neg_iff_add_eq {G : Type u} [add_group G] {a b c : G} :
a = b + -c ↔ a + c = b
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theorem eq_mul_inv_iff_mul_eq {G : Type u} [group G] {a b c : G} :
a = b * c⁻¹ ↔ a * c = b
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theorem eq_inv_mul_iff_mul_eq {G : Type u} [group G] {a b c : G} :
a = b⁻¹ * c ↔ b * a = c
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theorem eq_neg_add_iff_add_eq {G : Type u} [add_group G] {a b c : G} :
a = -b + c ↔ b + a = c
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theorem neg_add_eq_iff_eq_add {G : Type u} [add_group G] {a b c : G} :
-a + b = c ↔ b = a + c
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theorem inv_mul_eq_iff_eq_mul {G : Type u} [group G] {a b c : G} :
a⁻¹ * b = c ↔ b = a * c
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theorem add_neg_eq_iff_eq_add {G : Type u} [add_group G] {a b c : G} :
a + -b = c ↔ a = c + b
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theorem mul_inv_eq_iff_eq_mul {G : Type u} [group G] {a b c : G} :
a * b⁻¹ = c ↔ a = c * b
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theorem mul_inv_eq_one {G : Type u} [group G] {a b : G} :
a * b⁻¹ = 1 ↔ a = b
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theorem add_neg_eq_zero {G : Type u} [add_group G] {a b : G} :
a + -b = 0 ↔ a = b
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theorem inv_mul_eq_one {G : Type u} [group G] {a b : G} :
a⁻¹ * b = 1 ↔ a = b
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theorem neg_add_eq_zero {G : Type u} [add_group G] {a b : G} :
-a + b = 0 ↔ a = b
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theorem sub_left_injective {G : Type u} [add_group G] {b : G} :
function.injective (λ (a : G), a - b)
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theorem div_left_injective {G : Type u} [group G] {b : G} :
function.injective (λ (a : G), a / b)
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theorem div_right_injective {G : Type u} [group G] {b : G} :
function.injective (λ (a : G), b / a)
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theorem sub_right_injective {G : Type u} [add_group G] {b : G} :
function.injective (λ (a : G), b - a)
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@[simp]
theorem sub_self {G : Type u} [add_group G] (a : G) :
a - a = 0
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@[simp]
theorem sub_add_cancel {G : Type u} [add_group G] (a b : G) :
a - b + b = a
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@[simp]
theorem add_sub_cancel {G : Type u} [add_group G] (a b : G) :
a + b - b = a
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theorem add_sub_assoc {G : Type u} [add_group G] (a b c : G) :
a + b - c = a + (b - c)
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theorem eq_of_sub_eq_zero {G : Type u} [add_group G] {a b : G} (h : a - b = 0) :
a = b
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@[simp]
theorem sub_zero {G : Type u} [add_group G] (a : G) :
a - 0 = a
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theorem sub_ne_zero_of_ne {G : Type u} [add_group G] {a b : G} (h : a ≠ b) :
a - b ≠ 0
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@[simp]
theorem sub_neg_eq_add {G : Type u} [add_group G] (a b : G) :
a - -b = a + b
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@[simp]
theorem neg_sub {G : Type u} [add_group G] (a b : G) :
-(a - b) = b - a
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theorem add_sub {G : Type u} [add_group G] (a b c : G) :
a + (b - c) = a + b - c
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theorem sub_add_eq_sub_sub_swap {G : Type u} [add_group G] (a b c : G) :
a - (b + c) = a - c - b
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@[simp]
theorem add_sub_add_right_eq_sub {G : Type u} [add_group G] (a b c : G) :
a + c - (b + c) = a - b
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theorem eq_sub_of_add_eq {G : Type u} [add_group G] {a b c : G} (h : a + c = b) :
a = b - c
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theorem sub_eq_of_eq_add {G : Type u} [add_group G] {a b c : G} (h : a = c + b) :
a - b = c
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theorem eq_add_of_sub_eq {G : Type u} [add_group G] {a b c : G} (h : a - c = b) :
a = b + c
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theorem add_eq_of_eq_sub {G : Type u} [add_group G] {a b c : G} (h : a = c - b) :
a + b = c
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@[simp]
theorem sub_right_inj {G : Type u} [add_group G] {a b c : G} :
a - b = a - c ↔ b = c
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@[simp]
theorem sub_left_inj {G : Type u} [add_group G] {a b c : G} :
b - a = c - a ↔ b = c
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theorem sub_add_sub_cancel {G : Type u} [add_group G] (a b c : G) :
a - b + (b - c) = a - c
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theorem sub_sub_sub_cancel_right {G : Type u} [add_group G] (a b c : G) :
a - c - (b - c) = a - b
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theorem sub_sub_assoc_swap {G : Type u} [add_group G] {a b c : G} :
a - (b - c) = a + c - b
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theorem sub_eq_zero {G : Type u} [add_group G] {a b : G} :
a - b = 0 ↔ a = b
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theorem sub_eq_zero_of_eq {G : Type u} [add_group G] {a b : G} :
a = b → a - b = 0

Alias of sub_eq_zero.

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theorem sub_ne_zero {G : Type u} [add_group G] {a b : G} :
a - b ≠ 0 ↔ a ≠ b
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@[simp]
theorem sub_eq_self {G : Type u} [add_group G] {a b : G} :
a - b = a ↔ b = 0
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theorem eq_sub_iff_add_eq {G : Type u} [add_group G] {a b c : G} :
a = b - c ↔ a + c = b
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theorem sub_eq_iff_eq_add {G : Type u} [add_group G] {a b c : G} :
a - b = c ↔ a = c + b
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theorem eq_iff_eq_of_sub_eq_sub {G : Type u} [add_group G] {a b c d : G} (H : a - b = c - d) :
a = b ↔ c = d
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theorem left_inverse_sub_add_left {G : Type u} [add_group G] (c : G) :
function.left_inverse (λ (x : G), x - c) (λ (x : G), x + c)
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theorem left_inverse_add_left_sub {G : Type u} [add_group G] (c : G) :
function.left_inverse (λ (x : G), x + c) (λ (x : G), x - c)
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theorem left_inverse_add_right_neg_add {G : Type u} [add_group G] (c : G) :
function.left_inverse (λ (x : G), c + x) (λ (x : G), -c + x)
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theorem left_inverse_neg_add_add_right {G : Type u} [add_group G] (c : G) :
function.left_inverse (λ (x : G), -c + x) (λ (x : G), c + x)
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theorem mul_inv {G : Type u} [comm_group G] (a b : G) :
(a * b)⁻¹ = a⁻¹ * b⁻¹
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theorem neg_add {G : Type u} [add_comm_group G] (a b : G) :
-(a + b) = -a + -b
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theorem sub_add_eq_sub_sub {G : Type u} [add_comm_group G] (a b c : G) :
a - (b + c) = a - b - c
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theorem neg_add_eq_sub {G : Type u} [add_comm_group G] (a b : G) :
-a + b = b - a
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theorem sub_add_eq_add_sub {G : Type u} [add_comm_group G] (a b c : G) :
a - b + c = a + c - b
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theorem sub_sub {G : Type u} [add_comm_group G] (a b c : G) :
a - b - c = a - (b + c)
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theorem sub_add {G : Type u} [add_comm_group G] (a b c : G) :
a - b + c = a - (b - c)
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@[simp]
theorem add_sub_add_left_eq_sub {G : Type u} [add_comm_group G] (a b c : G) :
c + a - (c + b) = a - b
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theorem eq_sub_of_add_eq' {G : Type u} [add_comm_group G] {a b c : G} (h : c + a = b) :
a = b - c
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theorem sub_eq_of_eq_add' {G : Type u} [add_comm_group G] {a b c : G} (h : a = b + c) :
a - b = c
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theorem eq_add_of_sub_eq' {G : Type u} [add_comm_group G] {a b c : G} (h : a - b = c) :
a = b + c
source
theorem add_eq_of_eq_sub' {G : Type u} [add_comm_group G] {a b c : G} (h : b = c - a) :
a + b = c
source
theorem sub_sub_self {G : Type u} [add_comm_group G] (a b : G) :
a - (a - b) = b
source
theorem add_sub_comm {G : Type u} [add_comm_group G] (a b c d : G) :
a + b - (c + d) = a - c + (b - d)
source
theorem sub_eq_sub_add_sub {G : Type u} [add_comm_group G] (a b c : G) :
a - b = c - b + (a - c)
source
theorem neg_neg_sub_neg {G : Type u} [add_comm_group G] (a b : G) :
-(-a - -b) = a - b
source
@[simp]
theorem sub_sub_cancel {G : Type u} [add_comm_group G] (a b : G) :
a - (a - b) = b
source
theorem sub_eq_neg_add {G : Type u} [add_comm_group G] (a b : G) :
a - b = -b + a
source
theorem neg_add' {G : Type u} [add_comm_group G] (a b : G) :
-(a + b) = -a - b
source
@[simp]
theorem neg_sub_neg {G : Type u} [add_comm_group G] (a b : G) :
-a - -b = b - a
source
theorem eq_sub_iff_add_eq' {G : Type u} [add_comm_group G] {a b c : G} :
a = b - c ↔ c + a = b
source
theorem sub_eq_iff_eq_add' {G : Type u} [add_comm_group G] {a b c : G} :
a - b = c ↔ a = b + c
source
@[simp]
theorem add_sub_cancel' {G : Type u} [add_comm_group G] (a b : G) :
a + b - a = b
source
@[simp]
theorem add_sub_cancel'_right {G : Type u} [add_comm_group G] (a b : G) :
a + (b - a) = b
source
theorem add_add_neg_cancel'_right {G : Type u} [add_comm_group G] (a b : G) :
a + (b + -a) = b
source
theorem sub_right_comm {G : Type u} [add_comm_group G] (a b c : G) :
a - b - c = a - c - b
source
@[simp]
theorem add_add_sub_cancel {G : Type u} [add_comm_group G] (a b c : G) :
a + c + (b - c) = a + b
source
@[simp]
theorem sub_add_add_cancel {G : Type u} [add_comm_group G] (a b c : G) :
a - c + (b + c) = a + b
source
@[simp]
theorem sub_add_sub_cancel' {G : Type u} [add_comm_group G] (a b c : G) :
a - b + (c - a) = c - b
source
@[simp]
theorem add_sub_sub_cancel {G : Type u} [add_comm_group G] (a b c : G) :
a + b - (a - c) = b + c
source
@[simp]
theorem sub_sub_sub_cancel_left {G : Type u} [add_comm_group G] (a b c : G) :
c - a - (c - b) = b - a
source
theorem sub_eq_sub_iff_add_eq_add {G : Type u} [add_comm_group G] {a b c d : G} :
a - b = c - d ↔ a + d = c + b
source
theorem sub_eq_sub_iff_sub_eq_sub {G : Type u} [add_comm_group G] {a b c d : G} :
a - b = c - d ↔ a - c = b - d