mathlib documentation

algebra.group.basic

Basic lemmas about semigroups, monoids, and groups #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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.

theorem comp_assoc_left {α : Type u_1} (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.

theorem comp_assoc_right {α : Type u_1} (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.

@[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.

@[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.

@[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.

@[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.

theorem ite_mul_one {M : Type u} [mul_one_class M] {P : Prop} [decidable P] {a b : M} :

ite P (a * b) 1 = ite P a 1 * ite P b 1

theorem ite_add_zero {M : Type u} [add_zero_class M] {P : Prop} [decidable P] {a b : M} :

ite P (a + b) 0 = ite P a 0 + ite P b 0

theorem ite_zero_add {M : Type u} [add_zero_class M] {P : Prop} [decidable P] {a b : M} :

ite P 0 (a + b) = ite P 0 a + ite P 0 b

theorem ite_one_mul {M : Type u} [mul_one_class M] {P : Prop} [decidable P] {a b : M} :

ite P 1 (a * b) = ite P 1 a * ite P 1 b

theorem eq_zero_iff_eq_zero_of_add_eq_zero {M : Type u} [add_zero_class M] {a b : M} (h : a + b = 0) :

a = 0 ↔ b = 0

theorem eq_one_iff_eq_one_of_mul_eq_one {M : Type u} [mul_one_class M] {a b : M} (h : a * b = 1) :

a = 1 ↔ b = 1

theorem one_mul_eq_id {M : Type u} [mul_one_class M] :

has_mul.mul 1 = id

theorem zero_add_eq_id {M : Type u} [add_zero_class M] :

has_add.add 0 = id

theorem add_zero_eq_id {M : Type u} [add_zero_class M] :

(λ (_x : M), _x + 0) = id

theorem mul_one_eq_id {M : Type u} [mul_one_class M] :

(λ (_x : M), _x * 1) = id

theorem mul_left_comm {G : Type u_3} [comm_semigroup G] (a b c : G) :

a * (b * c) = b * (a * c)

theorem add_left_comm {G : Type u_3} [add_comm_semigroup G] (a b c : G) :

a + (b + c) = b + (a + c)

theorem mul_right_comm {G : Type u_3} [comm_semigroup G] (a b c : G) :

a * b * c = a * c * b

theorem add_right_comm {G : Type u_3} [add_comm_semigroup G] (a b c : G) :

a + b + c = a + c + b

theorem add_add_add_comm {G : Type u_3} [add_comm_semigroup G] (a b c d : G) :

a + b + (c + d) = a + c + (b + d)

theorem mul_mul_mul_comm {G : Type u_3} [comm_semigroup G] (a b c d : G) :

a * b * (c * d) = a * c * (b * d)

theorem add_rotate {G : Type u_3} [add_comm_semigroup G] (a b c : G) :

a + b + c = b + c + a

theorem mul_rotate {G : Type u_3} [comm_semigroup G] (a b c : G) :

a * b * c = b * c * a

theorem add_rotate' {G : Type u_3} [add_comm_semigroup G] (a b c : G) :

a + (b + c) = b + (c + a)

theorem mul_rotate' {G : Type u_3} [comm_semigroup G] (a b c : G) :

a * (b * c) = b * (c * a)

theorem bit0_add {M : Type u} [add_comm_semigroup M] (a b : M) :

bit0 (a + b) = bit0 a + bit0 b

theorem bit1_add {M : Type u} [add_comm_semigroup M] [has_one M] (a b : M) :

bit1 (a + b) = bit0 a + bit1 b

theorem bit1_add' {M : Type u} [add_comm_semigroup M] [has_one M] (a b : M) :

bit1 (a + b) = bit1 a + bit0 b

@[simp]

theorem bit0_zero {M : Type u} [add_monoid M] :

bit0 0 = 0

@[simp]

theorem bit1_zero {M : Type u} [add_monoid M] [has_one M] :

bit1 0 = 1

theorem inv_unique {M : Type u} [comm_monoid M] {x y z : M} (hy : x * y = 1) (hz : x * z = 1) :

y = z

theorem neg_unique {M : Type u} [add_comm_monoid M] {x y z : M} (hy : x + y = 0) (hz : x + z = 0) :

y = z

@[simp]

theorem add_right_eq_self {M : Type u} [add_left_cancel_monoid M] {a b : M} :

a + b = a ↔ b = 0

@[simp]

theorem mul_right_eq_self {M : Type u} [left_cancel_monoid M] {a b : M} :

a * b = a ↔ b = 1

@[simp]

theorem self_eq_mul_right {M : Type u} [left_cancel_monoid M] {a b : M} :

a = a * b ↔ b = 1

@[simp]

theorem self_eq_add_right {M : Type u} [add_left_cancel_monoid M] {a b : M} :

a = a + b ↔ b = 0

@[simp]

theorem add_left_eq_self {M : Type u} [add_right_cancel_monoid M] {a b : M} :

a + b = b ↔ a = 0

@[simp]

theorem mul_left_eq_self {M : Type u} [right_cancel_monoid M] {a b : M} :

a * b = b ↔ a = 1

@[simp]

theorem self_eq_add_left {M : Type u} [add_right_cancel_monoid M] {a b : M} :

b = a + b ↔ a = 0

@[simp]

theorem self_eq_mul_left {M : Type u} [right_cancel_monoid M] {a b : M} :

b = a * b ↔ a = 1

@[simp]

theorem inv_involutive {G : Type u_3} [has_involutive_inv G] :

function.involutive has_inv.inv

@[simp]

theorem neg_involutive {G : Type u_3} [has_involutive_neg G] :

function.involutive has_neg.neg

@[simp]

theorem neg_surjective {G : Type u_3} [has_involutive_neg G] :

function.surjective has_neg.neg

@[simp]

theorem inv_surjective {G : Type u_3} [has_involutive_inv G] :

function.surjective has_inv.inv

theorem neg_injective {G : Type u_3} [has_involutive_neg G] :

function.injective has_neg.neg

theorem inv_injective {G : Type u_3} [has_involutive_inv G] :

function.injective has_inv.inv

@[simp]

theorem inv_inj {G : Type u_3} [has_involutive_inv G] {a b : G} :

a⁻¹ = b⁻¹ ↔ a = b

@[simp]

theorem neg_inj {G : Type u_3} [has_involutive_neg G] {a b : G} :

-a = -b ↔ a = b

theorem eq_inv_of_eq_inv {G : Type u_3} [has_involutive_inv G] {a b : G} (h : a = b⁻¹) :

theorem eq_neg_of_eq_neg {G : Type u_3} [has_involutive_neg G] {a b : G} (h : a = -b) :

b = -a

theorem eq_inv_iff_eq_inv {G : Type u_3} [has_involutive_inv G] {a b : G} :

a = b⁻¹ ↔ b = a⁻¹

theorem eq_neg_iff_eq_neg {G : Type u_3} [has_involutive_neg G] {a b : G} :

a = -b ↔ b = -a

theorem neg_eq_iff_neg_eq {G : Type u_3} [has_involutive_neg G] {a b : G} :

-a = b ↔ -b = a

theorem inv_eq_iff_inv_eq {G : Type u_3} [has_involutive_inv G] {a b : G} :

a⁻¹ = b ↔ b⁻¹ = a

@[simp]

theorem inv_comp_inv (G : Type u_3) [has_involutive_inv G] :

has_inv.inv ∘ has_inv.inv = id

@[simp]

theorem neg_comp_neg (G : Type u_3) [has_involutive_neg G] :

has_neg.neg ∘ has_neg.neg = id

theorem left_inverse_neg (G : Type u_3) [has_involutive_neg G] :

function.left_inverse (λ (a : G), -a) (λ (a : G), -a)

theorem left_inverse_inv (G : Type u_3) [has_involutive_inv G] :

function.left_inverse (λ (a : G), a⁻¹) (λ (a : G), a⁻¹)

theorem right_inverse_neg (G : Type u_3) [has_involutive_neg G] :

function.left_inverse (λ (a : G), -a) (λ (a : G), -a)

theorem right_inverse_inv (G : Type u_3) [has_involutive_inv G] :

function.left_inverse (λ (a : G), a⁻¹) (λ (a : G), a⁻¹)

theorem neg_eq_zero_sub {G : Type u_3} [sub_neg_monoid G] (x : G) :

-x = 0 - x

theorem inv_eq_one_div {G : Type u_3} [div_inv_monoid G] (x : G) :

x⁻¹ = 1 / x

theorem add_zero_sub {G : Type u_3} [sub_neg_monoid G] (x y : G) :

x + (0 - y) = x - y

theorem mul_one_div {G : Type u_3} [div_inv_monoid G] (x y : G) :

x * (1 / y) = x / y

theorem mul_div_assoc {G : Type u_3} [div_inv_monoid G] (a b c : G) :

a * b / c = a * (b / c)

theorem add_sub_assoc {G : Type u_3} [sub_neg_monoid G] (a b c : G) :

a + b - c = a + (b - c)

theorem add_sub_assoc' {G : Type u_3} [sub_neg_monoid G] (a b c : G) :

a + (b - c) = a + b - c

theorem mul_div_assoc' {G : Type u_3} [div_inv_monoid G] (a b c : G) :

a * (b / c) = a * b / c

@[simp]

theorem one_div {G : Type u_3} [div_inv_monoid G] (a : G) :

1 / a = a⁻¹

@[simp]

theorem zero_sub {G : Type u_3} [sub_neg_monoid G] (a : G) :

0 - a = -a

theorem mul_div {G : Type u_3} [div_inv_monoid G] (a b c : G) :

a * (b / c) = a * b / c

theorem add_sub {G : Type u_3} [sub_neg_monoid G] (a b c : G) :

a + (b - c) = a + b - c

theorem sub_eq_add_zero_sub {G : Type u_3} [sub_neg_monoid G] (a b : G) :

a - b = a + (0 - b)

theorem div_eq_mul_one_div {G : Type u_3} [div_inv_monoid G] (a b : G) :

a / b = a * (1 / b)

@[simp]

theorem sub_zero {G : Type u_3} [sub_neg_zero_monoid G] (a : G) :

a - 0 = a

@[simp]

theorem div_one {G : Type u_3} [div_inv_one_monoid G] (a : G) :

a / 1 = a

theorem one_div_one {G : Type u_3} [div_inv_one_monoid G] :

1 / 1 = 1

theorem zero_sub_zero {G : Type u_3} [sub_neg_zero_monoid G] :

0 - 0 = 0

theorem inv_eq_of_mul_eq_one_left {α : Type u_1} [division_monoid α] {a b : α} (h : a * b = 1) :

theorem neg_eq_of_add_eq_zero_left {α : Type u_1} [subtraction_monoid α] {a b : α} (h : a + b = 0) :

-b = a

theorem eq_neg_of_add_eq_zero_left {α : Type u_1} [subtraction_monoid α] {a b : α} (h : a + b = 0) :

a = -b

theorem eq_inv_of_mul_eq_one_left {α : Type u_1} [division_monoid α] {a b : α} (h : a * b = 1) :

theorem eq_neg_of_add_eq_zero_right {α : Type u_1} [subtraction_monoid α] {a b : α} (h : a + b = 0) :

b = -a

theorem eq_inv_of_mul_eq_one_right {α : Type u_1} [division_monoid α] {a b : α} (h : a * b = 1) :

theorem eq_zero_sub_of_add_eq_zero_left {α : Type u_1} [subtraction_monoid α] {a b : α} (h : b + a = 0) :

b = 0 - a

theorem eq_one_div_of_mul_eq_one_left {α : Type u_1} [division_monoid α] {a b : α} (h : b * a = 1) :

b = 1 / a

theorem eq_zero_sub_of_add_eq_zero_right {α : Type u_1} [subtraction_monoid α] {a b : α} (h : a + b = 0) :

b = 0 - a

theorem eq_one_div_of_mul_eq_one_right {α : Type u_1} [division_monoid α] {a b : α} (h : a * b = 1) :

b = 1 / a

theorem eq_of_sub_eq_zero {α : Type u_1} [subtraction_monoid α] {a b : α} (h : a - b = 0) :

a = b

theorem eq_of_div_eq_one {α : Type u_1} [division_monoid α] {a b : α} (h : a / b = 1) :

a = b

theorem div_ne_one_of_ne {α : Type u_1} [division_monoid α] {a b : α} :

a ≠ b → a / b ≠ 1

theorem sub_ne_zero_of_ne {α : Type u_1} [subtraction_monoid α] {a b : α} :

a ≠ b → a - b ≠ 0

theorem zero_sub_add_zero_sub_rev {α : Type u_1} [subtraction_monoid α] (a b : α) :

0 - a + (0 - b) = 0 - (b + a)

theorem one_div_mul_one_div_rev {α : Type u_1} [division_monoid α] (a b : α) :

1 / a * (1 / b) = 1 / (b * a)

theorem neg_sub_left {α : Type u_1} [subtraction_monoid α] (a b : α) :

-a - b = -(b + a)

theorem inv_div_left {α : Type u_1} [division_monoid α] (a b : α) :

a⁻¹ / b = (b * a)⁻¹

@[simp]

theorem neg_sub {α : Type u_1} [subtraction_monoid α] (a b : α) :

-(a - b) = b - a

@[simp]

theorem inv_div {α : Type u_1} [division_monoid α] (a b : α) :

(a / b)⁻¹ = b / a

@[simp]

theorem one_div_div {α : Type u_1} [division_monoid α] (a b : α) :

1 / (a / b) = b / a

@[simp]

theorem zero_sub_sub {α : Type u_1} [subtraction_monoid α] (a b : α) :

0 - (a - b) = b - a

theorem one_div_one_div {α : Type u_1} [division_monoid α] (a : α) :

1 / (1 / a) = a

theorem zero_sub_zero_sub {α : Type u_1} [subtraction_monoid α] (a : α) :

0 - (0 - a) = a

@[protected, instance]

def division_monoid.to_div_inv_one_monoid {α : Type u_1} [division_monoid α] :

div_inv_one_monoid α

Equations

division_monoid.to_div_inv_one_monoid = {mul := div_inv_monoid.mul (division_monoid.to_div_inv_monoid α), mul_assoc := _, one := div_inv_monoid.one (division_monoid.to_div_inv_monoid α), one_mul := _, mul_one := _, npow := div_inv_monoid.npow (division_monoid.to_div_inv_monoid α), npow_zero' := _, npow_succ' := _, inv := div_inv_monoid.inv (division_monoid.to_div_inv_monoid α), div := div_inv_monoid.div (division_monoid.to_div_inv_monoid α), div_eq_mul_inv := _, zpow := div_inv_monoid.zpow (division_monoid.to_div_inv_monoid α), zpow_zero' := _, zpow_succ' := _, zpow_neg' := _, inv_one := _}

@[protected, instance]

def subtraction_monoid.to_sub_neg_zero_monoid {α : Type u_1} [subtraction_monoid α] :

sub_neg_zero_monoid α

Equations

subtraction_monoid.to_sub_neg_zero_monoid = {add := sub_neg_monoid.add (subtraction_monoid.to_sub_neg_monoid α), add_assoc := _, zero := sub_neg_monoid.zero (subtraction_monoid.to_sub_neg_monoid α), zero_add := _, add_zero := _, nsmul := sub_neg_monoid.nsmul (subtraction_monoid.to_sub_neg_monoid α), nsmul_zero' := _, nsmul_succ' := _, neg := sub_neg_monoid.neg (subtraction_monoid.to_sub_neg_monoid α), sub := sub_neg_monoid.sub (subtraction_monoid.to_sub_neg_monoid α), sub_eq_add_neg := _, zsmul := sub_neg_monoid.zsmul (subtraction_monoid.to_sub_neg_monoid α), zsmul_zero' := _, zsmul_succ' := _, zsmul_neg' := _, neg_zero := _}

@[simp]

theorem inv_eq_one {α : Type u_1} [division_monoid α] {a : α} :

a⁻¹ = 1 ↔ a = 1

@[simp]

theorem neg_eq_zero {α : Type u_1} [subtraction_monoid α] {a : α} :

-a = 0 ↔ a = 0

@[simp]

theorem one_eq_inv {α : Type u_1} [division_monoid α] {a : α} :

1 = a⁻¹ ↔ a = 1

@[simp]

theorem zero_eq_neg {α : Type u_1} [subtraction_monoid α] {a : α} :

0 = -a ↔ a = 0

theorem inv_ne_one {α : Type u_1} [division_monoid α] {a : α} :

a⁻¹ ≠ 1 ↔ a ≠ 1

theorem neg_ne_zero {α : Type u_1} [subtraction_monoid α] {a : α} :

-a ≠ 0 ↔ a ≠ 0

theorem eq_of_zero_sub_eq_zero_sub {α : Type u_1} [subtraction_monoid α] {a b : α} (h : 0 - a = 0 - b) :

a = b

theorem eq_of_one_div_eq_one_div {α : Type u_1} [division_monoid α] {a b : α} (h : 1 / a = 1 / b) :

a = b

theorem div_div_eq_mul_div {α : Type u_1} [division_monoid α] (a b c : α) :

a / (b / c) = a * c / b

theorem sub_sub_eq_add_sub {α : Type u_1} [subtraction_monoid α] (a b c : α) :

a - (b - c) = a + c - b

@[simp]

theorem sub_neg_eq_add {α : Type u_1} [subtraction_monoid α] (a b : α) :

a - -b = a + b

@[simp]

theorem div_inv_eq_mul {α : Type u_1} [division_monoid α] (a b : α) :

a / b⁻¹ = a * b

theorem div_mul_eq_div_div_swap {α : Type u_1} [division_monoid α] (a b c : α) :

a / (b * c) = a / c / b

theorem sub_add_eq_sub_sub_swap {α : Type u_1} [subtraction_monoid α] (a b c : α) :

a - (b + c) = a - c - b

theorem bit0_neg {α : Type u_1} [subtraction_monoid α] (a : α) :

bit0 (-a) = -bit0 a

theorem mul_inv {α : Type u_1} [division_comm_monoid α] (a b : α) :

(a * b)⁻¹ = a⁻¹ * b⁻¹

theorem neg_add {α : Type u_1} [subtraction_comm_monoid α] (a b : α) :

-(a + b) = -a + -b

theorem inv_div' {α : Type u_1} [division_comm_monoid α] (a b : α) :

(a / b)⁻¹ = a⁻¹ / b⁻¹

theorem neg_sub' {α : Type u_1} [subtraction_comm_monoid α] (a b : α) :

-(a - b) = -a - -b

theorem sub_eq_neg_add {α : Type u_1} [subtraction_comm_monoid α] (a b : α) :

a - b = -b + a

theorem div_eq_inv_mul {α : Type u_1} [division_comm_monoid α] (a b : α) :

a / b = b⁻¹ * a

theorem neg_add_eq_sub {α : Type u_1} [subtraction_comm_monoid α] (a b : α) :

-a + b = b - a

theorem inv_mul_eq_div {α : Type u_1} [division_comm_monoid α] (a b : α) :

a⁻¹ * b = b / a

theorem inv_mul' {α : Type u_1} [division_comm_monoid α] (a b : α) :

(a * b)⁻¹ = a⁻¹ / b

theorem neg_add' {α : Type u_1} [subtraction_comm_monoid α] (a b : α) :

-(a + b) = -a - b

@[simp]

theorem inv_div_inv {α : Type u_1} [division_comm_monoid α] (a b : α) :

a⁻¹ / b⁻¹ = b / a

@[simp]

theorem neg_sub_neg {α : Type u_1} [subtraction_comm_monoid α] (a b : α) :

-a - -b = b - a

theorem neg_neg_sub_neg {α : Type u_1} [subtraction_comm_monoid α] (a b : α) :

-(-a - -b) = a - b

theorem inv_inv_div_inv {α : Type u_1} [division_comm_monoid α] (a b : α) :

(a⁻¹ / b⁻¹)⁻¹ = a / b

theorem one_div_mul_one_div {α : Type u_1} [division_comm_monoid α] (a b : α) :

1 / a * (1 / b) = 1 / (a * b)

theorem zero_sub_add_zero_sub {α : Type u_1} [subtraction_comm_monoid α] (a b : α) :

0 - a + (0 - b) = 0 - (a + b)

theorem sub_right_comm {α : Type u_1} [subtraction_comm_monoid α] (a b c : α) :

a - b - c = a - c - b

theorem div_right_comm {α : Type u_1} [division_comm_monoid α] (a b c : α) :

a / b / c = a / c / b

theorem sub_sub {α : Type u_1} [subtraction_comm_monoid α] (a b c : α) :

a - b - c = a - (b + c)

theorem div_div {α : Type u_1} [division_comm_monoid α] (a b c : α) :

a / b / c = a / (b * c)

theorem div_mul {α : Type u_1} [division_comm_monoid α] (a b c : α) :

a / b * c = a / (b / c)

theorem sub_add {α : Type u_1} [subtraction_comm_monoid α] (a b c : α) :

a - b + c = a - (b - c)

theorem add_sub_left_comm {α : Type u_1} [subtraction_comm_monoid α] (a b c : α) :

a + (b - c) = b + (a - c)

theorem mul_div_left_comm {α : Type u_1} [division_comm_monoid α] (a b c : α) :

a * (b / c) = b * (a / c)

theorem mul_div_right_comm {α : Type u_1} [division_comm_monoid α] (a b c : α) :

a * b / c = a / c * b

theorem add_sub_right_comm {α : Type u_1} [subtraction_comm_monoid α] (a b c : α) :

a + b - c = a - c + b

theorem div_mul_eq_div_div {α : Type u_1} [division_comm_monoid α] (a b c : α) :

a / (b * c) = a / b / c

theorem sub_add_eq_sub_sub {α : Type u_1} [subtraction_comm_monoid α] (a b c : α) :

a - (b + c) = a - b - c

theorem div_mul_eq_mul_div {α : Type u_1} [division_comm_monoid α] (a b c : α) :

a / b * c = a * c / b

theorem sub_add_eq_add_sub {α : Type u_1} [subtraction_comm_monoid α] (a b c : α) :

a - b + c = a + c - b

theorem mul_comm_div {α : Type u_1} [division_comm_monoid α] (a b c : α) :

a / b * c = a * (c / b)

theorem add_comm_sub {α : Type u_1} [subtraction_comm_monoid α] (a b c : α) :

a - b + c = a + (c - b)

theorem sub_add_comm {α : Type u_1} [subtraction_comm_monoid α] (a b c : α) :

a - b + c = c - b + a

theorem div_mul_comm {α : Type u_1} [division_comm_monoid α] (a b c : α) :

a / b * c = c / b * a

theorem sub_add_eq_sub_add_zero_sub {α : Type u_1} [subtraction_comm_monoid α] (a b c : α) :

a - (b + c) = a - b + (0 - c)

theorem div_mul_eq_div_mul_one_div {α : Type u_1} [division_comm_monoid α] (a b c : α) :

a / (b * c) = a / b * (1 / c)

theorem sub_sub_sub_eq {α : Type u_1} [subtraction_comm_monoid α] (a b c d : α) :

a - b - (c - d) = a + d - (b + c)

theorem div_div_div_eq {α : Type u_1} [division_comm_monoid α] (a b c d : α) :

a / b / (c / d) = a * d / (b * c)

theorem div_div_div_comm {α : Type u_1} [division_comm_monoid α] (a b c d : α) :

a / b / (c / d) = a / c / (b / d)

theorem sub_sub_sub_comm {α : Type u_1} [subtraction_comm_monoid α] (a b c d : α) :

a - b - (c - d) = a - c - (b - d)

theorem div_mul_div_comm {α : Type u_1} [division_comm_monoid α] (a b c d : α) :

a / b * (c / d) = a * c / (b * d)

theorem sub_add_sub_comm {α : Type u_1} [subtraction_comm_monoid α] (a b c d : α) :

a - b + (c - d) = a + c - (b + d)

theorem mul_div_mul_comm {α : Type u_1} [division_comm_monoid α] (a b c d : α) :

a * b / (c * d) = a / c * (b / d)

theorem add_sub_add_comm {α : Type u_1} [subtraction_comm_monoid α] (a b c d : α) :

a + b - (c + d) = a - c + (b - d)

@[simp]

theorem sub_eq_neg_self {G : Type u_3} [add_group G] {a b : G} :

a - b = -b ↔ a = 0

@[simp]

theorem div_eq_inv_self {G : Type u_3} [group G] {a b : G} :

a / b = b⁻¹ ↔ a = 1

theorem mul_left_surjective {G : Type u_3} [group G] (a : G) :

function.surjective (has_mul.mul a)

theorem add_left_surjective {G : Type u_3} [add_group G] (a : G) :

function.surjective (has_add.add a)

theorem add_right_surjective {G : Type u_3} [add_group G] (a : G) :

function.surjective (λ (x : G), x + a)

theorem mul_right_surjective {G : Type u_3} [group G] (a : G) :

function.surjective (λ (x : G), x * a)

theorem eq_add_neg_of_add_eq {G : Type u_3} [add_group G] {a b c : G} (h : a + c = b) :

a = b + -c

theorem eq_mul_inv_of_mul_eq {G : Type u_3} [group G] {a b c : G} (h : a * c = b) :

a = b * c⁻¹

theorem eq_neg_add_of_add_eq {G : Type u_3} [add_group G] {a b c : G} (h : b + a = c) :

a = -b + c

theorem eq_inv_mul_of_mul_eq {G : Type u_3} [group G] {a b c : G} (h : b * a = c) :

a = b⁻¹ * c

theorem inv_mul_eq_of_eq_mul {G : Type u_3} [group G] {a b c : G} (h : b = a * c) :

a⁻¹ * b = c

theorem neg_add_eq_of_eq_add {G : Type u_3} [add_group G] {a b c : G} (h : b = a + c) :

-a + b = c

theorem add_neg_eq_of_eq_add {G : Type u_3} [add_group G] {a b c : G} (h : a = c + b) :

a + -b = c

theorem mul_inv_eq_of_eq_mul {G : Type u_3} [group G] {a b c : G} (h : a = c * b) :

a * b⁻¹ = c

theorem eq_add_of_add_neg_eq {G : Type u_3} [add_group G] {a b c : G} (h : a + -c = b) :

a = b + c

theorem eq_mul_of_mul_inv_eq {G : Type u_3} [group G] {a b c : G} (h : a * c⁻¹ = b) :

a = b * c

theorem eq_mul_of_inv_mul_eq {G : Type u_3} [group G] {a b c : G} (h : b⁻¹ * a = c) :

a = b * c

theorem eq_add_of_neg_add_eq {G : Type u_3} [add_group G] {a b c : G} (h : -b + a = c) :

a = b + c

theorem mul_eq_of_eq_inv_mul {G : Type u_3} [group G] {a b c : G} (h : b = a⁻¹ * c) :

a * b = c

theorem add_eq_of_eq_neg_add {G : Type u_3} [add_group G] {a b c : G} (h : b = -a + c) :

a + b = c

theorem mul_eq_of_eq_mul_inv {G : Type u_3} [group G] {a b c : G} (h : a = c * b⁻¹) :

a * b = c

theorem add_eq_of_eq_add_neg {G : Type u_3} [add_group G] {a b c : G} (h : a = c + -b) :

a + b = c

theorem mul_eq_one_iff_eq_inv {G : Type u_3} [group G] {a b : G} :

a * b = 1 ↔ a = b⁻¹

theorem add_eq_zero_iff_eq_neg {G : Type u_3} [add_group G] {a b : G} :

a + b = 0 ↔ a = -b

theorem mul_eq_one_iff_inv_eq {G : Type u_3} [group G] {a b : G} :

a * b = 1 ↔ a⁻¹ = b

theorem add_eq_zero_iff_neg_eq {G : Type u_3} [add_group G] {a b : G} :

a + b = 0 ↔ -a = b

theorem eq_inv_iff_mul_eq_one {G : Type u_3} [group G] {a b : G} :

a = b⁻¹ ↔ a * b = 1

theorem eq_neg_iff_add_eq_zero {G : Type u_3} [add_group G] {a b : G} :

a = -b ↔ a + b = 0

theorem neg_eq_iff_add_eq_zero {G : Type u_3} [add_group G] {a b : G} :

-a = b ↔ a + b = 0

theorem inv_eq_iff_mul_eq_one {G : Type u_3} [group G] {a b : G} :

a⁻¹ = b ↔ a * b = 1

theorem eq_add_neg_iff_add_eq {G : Type u_3} [add_group G] {a b c : G} :

a = b + -c ↔ a + c = b

theorem eq_mul_inv_iff_mul_eq {G : Type u_3} [group G] {a b c : G} :

a = b * c⁻¹ ↔ a * c = b

theorem eq_inv_mul_iff_mul_eq {G : Type u_3} [group G] {a b c : G} :

a = b⁻¹ * c ↔ b * a = c

theorem eq_neg_add_iff_add_eq {G : Type u_3} [add_group G] {a b c : G} :

a = -b + c ↔ b + a = c

theorem neg_add_eq_iff_eq_add {G : Type u_3} [add_group G] {a b c : G} :

-a + b = c ↔ b = a + c

theorem inv_mul_eq_iff_eq_mul {G : Type u_3} [group G] {a b c : G} :

a⁻¹ * b = c ↔ b = a * c

theorem add_neg_eq_iff_eq_add {G : Type u_3} [add_group G] {a b c : G} :

a + -b = c ↔ a = c + b

theorem mul_inv_eq_iff_eq_mul {G : Type u_3} [group G] {a b c : G} :

a * b⁻¹ = c ↔ a = c * b

theorem mul_inv_eq_one {G : Type u_3} [group G] {a b : G} :

a * b⁻¹ = 1 ↔ a = b

theorem add_neg_eq_zero {G : Type u_3} [add_group G] {a b : G} :

a + -b = 0 ↔ a = b

theorem inv_mul_eq_one {G : Type u_3} [group G] {a b : G} :

a⁻¹ * b = 1 ↔ a = b

theorem neg_add_eq_zero {G : Type u_3} [add_group G] {a b : G} :

-a + b = 0 ↔ a = b

theorem sub_left_injective {G : Type u_3} [add_group G] {b : G} :

function.injective (λ (a : G), a - b)

theorem div_left_injective {G : Type u_3} [group G] {b : G} :

function.injective (λ (a : G), a / b)

theorem div_right_injective {G : Type u_3} [group G] {b : G} :

function.injective (λ (a : G), b / a)

theorem sub_right_injective {G : Type u_3} [add_group G] {b : G} :

function.injective (λ (a : G), b - a)

@[simp]

theorem sub_add_cancel {G : Type u_3} [add_group G] (a b : G) :

a - b + b = a

@[simp]

theorem div_mul_cancel' {G : Type u_3} [group G] (a b : G) :

a / b * b = a

@[simp]

theorem div_self' {G : Type u_3} [group G] (a : G) :

a / a = 1

@[simp]

theorem sub_self {G : Type u_3} [add_group G] (a : G) :

a - a = 0

@[simp]

theorem mul_div_cancel'' {G : Type u_3} [group G] (a b : G) :

a * b / b = a

@[simp]

theorem add_sub_cancel {G : Type u_3} [add_group G] (a b : G) :

a + b - b = a

@[simp]

theorem mul_div_mul_right_eq_div {G : Type u_3} [group G] (a b c : G) :

a * c / (b * c) = a / b

@[simp]

theorem add_sub_add_right_eq_sub {G : Type u_3} [add_group G] (a b c : G) :

a + c - (b + c) = a - b

theorem eq_sub_of_add_eq {G : Type u_3} [add_group G] {a b c : G} (h : a + c = b) :

a = b - c

theorem eq_div_of_mul_eq' {G : Type u_3} [group G] {a b c : G} (h : a * c = b) :

a = b / c

theorem div_eq_of_eq_mul'' {G : Type u_3} [group G] {a b c : G} (h : a = c * b) :

a / b = c

theorem sub_eq_of_eq_add {G : Type u_3} [add_group G] {a b c : G} (h : a = c + b) :

a - b = c

theorem eq_mul_of_div_eq {G : Type u_3} [group G] {a b c : G} (h : a / c = b) :

a = b * c

theorem eq_add_of_sub_eq {G : Type u_3} [add_group G] {a b c : G} (h : a - c = b) :

a = b + c

theorem mul_eq_of_eq_div {G : Type u_3} [group G] {a b c : G} (h : a = c / b) :

a * b = c

theorem add_eq_of_eq_sub {G : Type u_3} [add_group G] {a b c : G} (h : a = c - b) :

a + b = c

@[simp]

theorem div_right_inj {G : Type u_3} [group G] {a b c : G} :

a / b = a / c ↔ b = c

@[simp]

theorem sub_right_inj {G : Type u_3} [add_group G] {a b c : G} :

a - b = a - c ↔ b = c

@[simp]

theorem div_left_inj {G : Type u_3} [group G] {a b c : G} :

b / a = c / a ↔ b = c

@[simp]

theorem sub_left_inj {G : Type u_3} [add_group G] {a b c : G} :

b - a = c - a ↔ b = c

@[simp]

theorem div_mul_div_cancel' {G : Type u_3} [group G] (a b c : G) :

a / b * (b / c) = a / c

@[simp]

theorem sub_add_sub_cancel {G : Type u_3} [add_group G] (a b c : G) :

a - b + (b - c) = a - c

@[simp]

theorem div_div_div_cancel_right' {G : Type u_3} [group G] (a b c : G) :

a / c / (b / c) = a / b

@[simp]

theorem sub_sub_sub_cancel_right {G : Type u_3} [add_group G] (a b c : G) :

a - c - (b - c) = a - b

theorem div_eq_one {G : Type u_3} [group G] {a b : G} :

a / b = 1 ↔ a = b

theorem sub_eq_zero {G : Type u_3} [add_group G] {a b : G} :

a - b = 0 ↔ a = b

theorem div_eq_one_of_eq {G : Type u_3} [group G] {a b : G} :

a = b → a / b = 1

Alias of the reverse direction of div_eq_one.

theorem sub_eq_zero_of_eq {G : Type u_3} [add_group G] {a b : G} :

a = b → a - b = 0

Alias of the reverse direction of sub_eq_zero.

theorem sub_ne_zero {G : Type u_3} [add_group G] {a b : G} :

a - b ≠ 0 ↔ a ≠ b

theorem div_ne_one {G : Type u_3} [group G] {a b : G} :

a / b ≠ 1 ↔ a ≠ b

@[simp]

theorem sub_eq_self {G : Type u_3} [add_group G] {a b : G} :

a - b = a ↔ b = 0

@[simp]

theorem div_eq_self {G : Type u_3} [group G] {a b : G} :

a / b = a ↔ b = 1

theorem eq_div_iff_mul_eq' {G : Type u_3} [group G] {a b c : G} :

a = b / c ↔ a * c = b

theorem eq_sub_iff_add_eq {G : Type u_3} [add_group G] {a b c : G} :

a = b - c ↔ a + c = b

theorem sub_eq_iff_eq_add {G : Type u_3} [add_group G] {a b c : G} :

a - b = c ↔ a = c + b

theorem div_eq_iff_eq_mul {G : Type u_3} [group G] {a b c : G} :

a / b = c ↔ a = c * b

theorem eq_iff_eq_of_sub_eq_sub {G : Type u_3} [add_group G] {a b c d : G} (H : a - b = c - d) :

a = b ↔ c = d

theorem eq_iff_eq_of_div_eq_div {G : Type u_3} [group G] {a b c d : G} (H : a / b = c / d) :

a = b ↔ c = d

theorem left_inverse_sub_add_left {G : Type u_3} [add_group G] (c : G) :

function.left_inverse (λ (x : G), x - c) (λ (x : G), x + c)

theorem left_inverse_div_mul_left {G : Type u_3} [group G] (c : G) :

function.left_inverse (λ (x : G), x / c) (λ (x : G), x * c)

theorem left_inverse_add_left_sub {G : Type u_3} [add_group G] (c : G) :

function.left_inverse (λ (x : G), x + c) (λ (x : G), x - c)

theorem left_inverse_mul_left_div {G : Type u_3} [group G] (c : G) :

function.left_inverse (λ (x : G), x * c) (λ (x : G), x / c)

theorem left_inverse_add_right_neg_add {G : Type u_3} [add_group G] (c : G) :

function.left_inverse (λ (x : G), c + x) (λ (x : G), -c + x)

theorem left_inverse_mul_right_inv_mul {G : Type u_3} [group G] (c : G) :

function.left_inverse (λ (x : G), c * x) (λ (x : G), c⁻¹ * x)

theorem left_inverse_neg_add_add_right {G : Type u_3} [add_group G] (c : G) :

function.left_inverse (λ (x : G), -c + x) (λ (x : G), c + x)

theorem left_inverse_inv_mul_mul_right {G : Type u_3} [group G] (c : G) :

function.left_inverse (λ (x : G), c⁻¹ * x) (λ (x : G), c * x)

theorem exists_npow_eq_one_of_zpow_eq_one {G : Type u_3} [group G] {n : ℤ} (hn : n ≠ 0) {x : G} (h : x ^ n = 1) :

∃ (n : ℕ), 0 < n ∧ x ^ n = 1

theorem exists_nsmul_eq_zero_of_zsmul_eq_zero {G : Type u_3} [add_group G] {n : ℤ} (hn : n ≠ 0) {x : G} (h : n • x = 0) :

∃ (n : ℕ), 0 < n ∧ n • x = 0

theorem div_eq_of_eq_mul' {G : Type u_3} [comm_group G] {a b c : G} (h : a = b * c) :

a / b = c

theorem sub_eq_of_eq_add' {G : Type u_3} [add_comm_group G] {a b c : G} (h : a = b + c) :

a - b = c

@[simp]

theorem add_sub_add_left_eq_sub {G : Type u_3} [add_comm_group G] (a b c : G) :

c + a - (c + b) = a - b

@[simp]

theorem mul_div_mul_left_eq_div {G : Type u_3} [comm_group G] (a b c : G) :

c * a / (c * b) = a / b

theorem eq_div_of_mul_eq'' {G : Type u_3} [comm_group G] {a b c : G} (h : c * a = b) :

a = b / c

theorem eq_sub_of_add_eq' {G : Type u_3} [add_comm_group G] {a b c : G} (h : c + a = b) :

a = b - c

theorem eq_mul_of_div_eq' {G : Type u_3} [comm_group G] {a b c : G} (h : a / b = c) :

a = b * c

theorem eq_add_of_sub_eq' {G : Type u_3} [add_comm_group G] {a b c : G} (h : a - b = c) :

a = b + c

theorem add_eq_of_eq_sub' {G : Type u_3} [add_comm_group G] {a b c : G} (h : b = c - a) :

a + b = c

theorem mul_eq_of_eq_div' {G : Type u_3} [comm_group G] {a b c : G} (h : b = c / a) :

a * b = c

theorem sub_sub_self {G : Type u_3} [add_comm_group G] (a b : G) :

a - (a - b) = b

theorem div_div_self' {G : Type u_3} [comm_group G] (a b : G) :

a / (a / b) = b

theorem div_eq_div_mul_div {G : Type u_3} [comm_group G] (a b c : G) :

a / b = c / b * (a / c)

theorem sub_eq_sub_add_sub {G : Type u_3} [add_comm_group G] (a b c : G) :

a - b = c - b + (a - c)

@[simp]

theorem sub_sub_cancel {G : Type u_3} [add_comm_group G] (a b : G) :

a - (a - b) = b

@[simp]

theorem div_div_cancel {G : Type u_3} [comm_group G] (a b : G) :

a / (a / b) = b

@[simp]

theorem div_div_cancel_left {G : Type u_3} [comm_group G] (a b : G) :

a / b / a = b⁻¹

@[simp]

theorem sub_sub_cancel_left {G : Type u_3} [add_comm_group G] (a b : G) :

a - b - a = -b

theorem eq_div_iff_mul_eq'' {G : Type u_3} [comm_group G] {a b c : G} :

a = b / c ↔ c * a = b

theorem eq_sub_iff_add_eq' {G : Type u_3} [add_comm_group G] {a b c : G} :

a = b - c ↔ c + a = b

theorem div_eq_iff_eq_mul' {G : Type u_3} [comm_group G] {a b c : G} :

a / b = c ↔ a = b * c

theorem sub_eq_iff_eq_add' {G : Type u_3} [add_comm_group G] {a b c : G} :

a - b = c ↔ a = b + c

@[simp]

theorem add_sub_cancel' {G : Type u_3} [add_comm_group G] (a b : G) :

a + b - a = b

@[simp]

theorem mul_div_cancel''' {G : Type u_3} [comm_group G] (a b : G) :

a * b / a = b

@[simp]

theorem add_sub_cancel'_right {G : Type u_3} [add_comm_group G] (a b : G) :

a + (b - a) = b

@[simp]

theorem mul_div_cancel'_right {G : Type u_3} [comm_group G] (a b : G) :

a * (b / a) = b

@[simp]

theorem sub_add_cancel' {G : Type u_3} [add_comm_group G] (a b : G) :

a - (a + b) = -b

@[simp]

theorem div_mul_cancel'' {G : Type u_3} [comm_group G] (a b : G) :

a / (a * b) = b⁻¹

theorem mul_mul_inv_cancel'_right {G : Type u_3} [comm_group G] (a b : G) :

a * (b * a⁻¹) = b

theorem add_add_neg_cancel'_right {G : Type u_3} [add_comm_group G] (a b : G) :

a + (b + -a) = b

@[simp]

theorem add_add_sub_cancel {G : Type u_3} [add_comm_group G] (a b c : G) :

a + c + (b - c) = a + b

@[simp]

theorem mul_mul_div_cancel {G : Type u_3} [comm_group G] (a b c : G) :

a * c * (b / c) = a * b

@[simp]

theorem sub_add_add_cancel {G : Type u_3} [add_comm_group G] (a b c : G) :

a - c + (b + c) = a + b

@[simp]

theorem div_mul_mul_cancel {G : Type u_3} [comm_group G] (a b c : G) :

a / c * (b * c) = a * b

@[simp]

theorem div_mul_div_cancel'' {G : Type u_3} [comm_group G] (a b c : G) :

a / b * (c / a) = c / b

@[simp]

theorem sub_add_sub_cancel' {G : Type u_3} [add_comm_group G] (a b c : G) :

a - b + (c - a) = c - b

@[simp]

theorem add_sub_sub_cancel {G : Type u_3} [add_comm_group G] (a b c : G) :

a + b - (a - c) = b + c

@[simp]

theorem mul_div_div_cancel {G : Type u_3} [comm_group G] (a b c : G) :

a * b / (a / c) = b * c

@[simp]

theorem sub_sub_sub_cancel_left {G : Type u_3} [add_comm_group G] (a b c : G) :

c - a - (c - b) = b - a

@[simp]

theorem div_div_div_cancel_left {G : Type u_3} [comm_group G] (a b c : G) :

c / a / (c / b) = b / a

theorem sub_eq_sub_iff_add_eq_add {G : Type u_3} [add_comm_group G] {a b c d : G} :

a - b = c - d ↔ a + d = c + b

theorem div_eq_div_iff_mul_eq_mul {G : Type u_3} [comm_group G] {a b c d : G} :

a / b = c / d ↔ a * d = c * b

theorem sub_eq_sub_iff_sub_eq_sub {G : Type u_3} [add_comm_group G] {a b c d : G} :

a - b = c - d ↔ a - c = b - d

theorem div_eq_div_iff_div_eq_div {G : Type u_3} [comm_group G] {a b c d : G} :

a / b = c / d ↔ a / c = b / d

theorem bit0_sub {M : Type u} [subtraction_comm_monoid M] (a b : M) :

bit0 (a - b) = bit0 a - bit0 b

theorem bit1_sub {M : Type u} [subtraction_comm_monoid M] [has_one M] (a b : M) :

bit1 (a - b) = bit1 a - bit0 b

theorem multiplicative_of_symmetric_of_is_total {α : Type u_1} {β : Type u_2} [monoid β] (p r : α → α → Prop) [is_total α r] (f : α → α → β) (hsymm : symmetric p) (hf_swap : ∀ {a b : α}, p a b → f a b * f b a = 1) (hmul : ∀ {a b c : α}, r a b → r b c → p a b → p b c → p a c → f a c = f a b * f b c) {a b c : α} (pab : p a b) (pbc : p b c) (pac : p a c) :

f a c = f a b * f b c

theorem additive_of_symmetric_of_is_total {α : Type u_1} {β : Type u_2} [add_monoid β] (p r : α → α → Prop) [is_total α r] (f : α → α → β) (hsymm : symmetric p) (hf_swap : ∀ {a b : α}, p a b → f a b + f b a = 0) (hmul : ∀ {a b c : α}, r a b → r b c → p a b → p b c → p a c → f a c = f a b + f b c) {a b c : α} (pab : p a b) (pbc : p b c) (pac : p a c) :

f a c = f a b + f b c

theorem multiplicative_of_is_total {α : Type u_1} {β : Type u_2} [monoid β] (r : α → α → Prop) [is_total α r] (f : α → α → β) (p : α → Prop) (hswap : ∀ {a b : α}, p a → p b → f a b * f b a = 1) (hmul : ∀ {a b c : α}, r a b → r b c → p a → p b → p c → f a c = f a b * f b c) {a b c : α} (pa : p a) (pb : p b) (pc : p c) :

f a c = f a b * f b c

If a binary function from a type equipped with a total relation r to a monoid is anti-symmetric (i.e. satisfies f a b * f b a = 1), in order to show it is multiplicative (i.e. satisfies f a c = f a b * f b c), we may assume r a b and r b c are satisfied. We allow restricting to a subset specified by a predicate p.

theorem additive_of_is_total {α : Type u_1} {β : Type u_2} [add_monoid β] (r : α → α → Prop) [is_total α r] (f : α → α → β) (p : α → Prop) (hswap : ∀ {a b : α}, p a → p b → f a b + f b a = 0) (hmul : ∀ {a b c : α}, r a b → r b c → p a → p b → p c → f a c = f a b + f b c) {a b c : α} (pa : p a) (pb : p b) (pc : p c) :

f a c = f a b + f b c

If a binary function from a type equipped with a total relation r to an additive monoid is anti-symmetric (i.e. satisfies f a b + f b a = 0), in order to show it is additive (i.e. satisfies f a c = f a b + f b c), we may assume r a b and r b c are satisfied. We allow restricting to a subset specified by a predicate p.