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.

@[simp]

theorem pow_dite {α : Type u_1} {β : Type u_2} [Pow α β] (p : Prop) [Decidable p] (a : α) (b : p → β) (c : ¬p → β) : (a ^ if h : p then b h else c h) = if h : p then a ^ b h else a ^ c h

@[simp]

theorem dite_smul {α : Type u_1} {β : Type u_2} [SMul β α] (p : Prop) [Decidable p] (a : α) (b : p → β) (c : ¬p → β) : (if h : p then b h else c h) • a = if h : p then b h • a else c h • a

@[simp]

theorem dite_pow {α : Type u_1} {β : Type u_2} [Pow α β] (p : Prop) [Decidable p] (a : p → α) (b : ¬p → α) (c : β) : (if h : p then a h else b h) ^ c = if h : p then a h ^ c else b h ^ c

@[simp]

theorem smul_dite {α : Type u_1} {β : Type u_2} [SMul β α] (p : Prop) [Decidable p] (a : p → α) (b : ¬p → α) (c : β) : (c • if h : p then a h else b h) = if h : p then c • a h else c • b h

@[simp]

theorem pow_ite {α : Type u_1} {β : Type u_2} [Pow α β] (p : Prop) [Decidable p] (a : α) (b c : β) : (a ^ if p then b else c) = if p then a ^ b else a ^ c

@[simp]

theorem ite_smul {α : Type u_1} {β : Type u_2} [SMul β α] (p : Prop) [Decidable p] (a : α) (b c : β) : (if p then b else c) • a = if p then b • a else c • a

@[simp]

theorem ite_pow {α : Type u_1} {β : Type u_2} [Pow α β] (p : Prop) [Decidable p] (a b : α) (c : β) : (if p then a else b) ^ c = if p then a ^ c else b ^ c

@[simp]

theorem smul_ite {α : Type u_1} {β : Type u_2} [SMul β α] (p : Prop) [Decidable p] (a b : α) (c : β) : (c • if p then a else b) = if p then c • a else c • b

@[simp]

theorem vadd_ite {α : Type u_1} {β : Type u_2} [VAdd β α] (p : Prop) [Decidable p] (a b : α) (c : β) : (c +ᵥ if p then a else b) = if p then c +ᵥ a else c +ᵥ b

@[simp]

theorem ite_vadd {α : Type u_1} {β : Type u_2} [VAdd β α] (p : Prop) [Decidable p] (a : α) (b c : β) : (if p then b else c) +ᵥ a = if p then b +ᵥ a else c +ᵥ a

@[simp]

theorem dite_vadd {α : Type u_1} {β : Type u_2} [VAdd β α] (p : Prop) [Decidable p] (a : α) (b : p → β) (c : ¬p → β) : (if h : p then b h else c h) +ᵥ a = if h : p then b h +ᵥ a else c h +ᵥ a

@[simp]

theorem vadd_dite {α : Type u_1} {β : Type u_2} [VAdd β α] (p : Prop) [Decidable p] (a : p → α) (b : ¬p → α) (c : β) : (c +ᵥ if h : p then a h else b h) = if h : p then c +ᵥ a h else c +ᵥ b h

theorem mul_right_injective {G : Type u_3} [Mul G] [IsLeftCancelMul G] (a : G) : Function.Injective fun (x : G) => a * x

theorem add_right_injective {G : Type u_3} [Add G] [IsLeftCancelAdd G] (a : G) : Function.Injective fun (x : G) => a + x

@[simp]

theorem mul_right_inj {G : Type u_3} [Mul G] [IsLeftCancelMul G] (a : G) {b c : G} : a * b = a * c ↔ b = c

@[simp]

theorem add_right_inj {G : Type u_3} [Add G] [IsLeftCancelAdd G] (a : G) {b c : G} : a + b = a + c ↔ b = c

theorem mul_ne_mul_right {G : Type u_3} [Mul G] [IsLeftCancelMul G] (a : G) {b c : G} : a * b ≠ a * c ↔ b ≠ c

theorem add_ne_add_right {G : Type u_3} [Add G] [IsLeftCancelAdd G] (a : G) {b c : G} : a + b ≠ a + c ↔ b ≠ c

theorem mul_left_injective {G : Type u_3} [Mul G] [IsRightCancelMul G] (a : G) : Function.Injective fun (x : G) => x * a

theorem add_left_injective {G : Type u_3} [Add G] [IsRightCancelAdd G] (a : G) : Function.Injective fun (x : G) => x + a

@[simp]

theorem mul_left_inj {G : Type u_3} [Mul G] [IsRightCancelMul G] (a : G) {b c : G} : b * a = c * a ↔ b = c

@[simp]

theorem add_left_inj {G : Type u_3} [Add G] [IsRightCancelAdd G] (a : G) {b c : G} : b + a = c + a ↔ b = c

theorem mul_ne_mul_left {G : Type u_3} [Mul G] [IsRightCancelMul G] (a : G) {b c : G} : b * a ≠ c * a ↔ b ≠ c

theorem add_ne_add_left {G : Type u_3} [Add G] [IsRightCancelAdd G] (a : G) {b c : G} : b + a ≠ c + a ↔ b ≠ c

instance Semigroup.to_isAssociative {α : Type u_1} [Semigroup α] : Std.Associative fun (x1 x2 : α) => x1 * x2

instance AddSemigroup.to_isAssociative {α : Type u_1} [AddSemigroup α] : Std.Associative fun (x1 x2 : α) => x1 + x2

@[simp]

theorem comp_mul_left {α : Type u_1} [Semigroup α] (x y : α) : ((fun (x_1 : α) => x * x_1) ∘ fun (x : α) => y * x) = fun (x_1 : α) => x * y * x_1

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_left {α : Type u_1} [AddSemigroup α] (x y : α) : ((fun (x_1 : α) => x + x_1) ∘ fun (x : α) => y + x) = fun (x_1 : α) => x + y + x_1

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

@[simp]

theorem comp_mul_right {α : Type u_1} [Semigroup α] (x y : α) : ((fun (x_1 : α) => x_1 * x) ∘ fun (x : α) => x * y) = fun (x_1 : α) => x_1 * (y * x)

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

@[simp]

theorem comp_add_right {α : Type u_1} [AddSemigroup α] (x y : α) : ((fun (x_1 : α) => x_1 + x) ∘ fun (x : α) => x + y) = fun (x_1 : α) => x_1 + (y + x)

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

instance CommMagma.to_isCommutative {G : Type u_3} [CommMagma G] : Std.Commutative fun (x1 x2 : G) => x1 * x2

instance AddCommMagma.to_isCommutative {G : Type u_3} [AddCommMagma G] : Std.Commutative fun (x1 x2 : G) => x1 + x2

theorem ite_mul_one {M : Type u_4} [MulOneClass M] {P : Prop} [Decidable P] {a b : M} : (if P then a * b else 1) = (if P then a else 1) * if P then b else 1

theorem ite_add_zero {M : Type u_4} [AddZeroClass M] {P : Prop} [Decidable P] {a b : M} : (if P then a + b else 0) = (if P then a else 0) + if P then b else 0

theorem ite_one_mul {M : Type u_4} [MulOneClass M] {P : Prop} [Decidable P] {a b : M} : (if P then 1 else a * b) = (if P then 1 else a) * if P then 1 else b

theorem ite_zero_add {M : Type u_4} [AddZeroClass M] {P : Prop} [Decidable P] {a b : M} : (if P then 0 else a + b) = (if P then 0 else a) + if P then 0 else b

theorem eq_one_iff_eq_one_of_mul_eq_one {M : Type u_4} [MulOneClass M] {a b : M} (h : a * b = 1) : a = 1 ↔ b = 1

theorem eq_zero_iff_eq_zero_of_add_eq_zero {M : Type u_4} [AddZeroClass M] {a b : M} (h : a + b = 0) : a = 0 ↔ b = 0

theorem one_mul_eq_id {M : Type u_4} [MulOneClass M] : (fun (x : M) => 1 * x) = id

theorem zero_add_eq_id {M : Type u_4} [AddZeroClass M] : (fun (x : M) => 0 + x) = id

theorem mul_one_eq_id {M : Type u_4} [MulOneClass M] : (fun (x : M) => x * 1) = id

theorem add_zero_eq_id {M : Type u_4} [AddZeroClass M] : (fun (x : M) => x + 0) = id

theorem mul_left_comm {G : Type u_3} [CommSemigroup G] (a b c : G) : a * (b * c) = b * (a * c)

theorem add_left_comm {G : Type u_3} [AddCommSemigroup G] (a b c : G) : a + (b + c) = b + (a + c)

theorem mul_right_comm {G : Type u_3} [CommSemigroup G] (a b c : G) : a * b * c = a * c * b

theorem add_right_comm {G : Type u_3} [AddCommSemigroup G] (a b c : G) : a + b + c = a + c + b

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

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

theorem mul_rotate {G : Type u_3} [CommSemigroup G] (a b c : G) : a * b * c = b * c * a

theorem add_rotate {G : Type u_3} [AddCommSemigroup G] (a b c : G) : a + b + c = b + c + a

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

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

theorem pow_boole {M : Type u_4} [Monoid M] (P : Prop) [Decidable P] (a : M) : (a ^ if P then 1 else 0) = if P then a else 1

theorem boole_nsmul {M : Type u_4} [AddMonoid M] (P : Prop) [Decidable P] (a : M) : (if P then 1 else 0) • a = if P then a else 0

theorem pow_mul_pow_sub {M : Type u_4} [Monoid M] {m n : ℕ} (a : M) (h : m ≤ n) : a ^ m * a ^ (n - m) = a ^ n

theorem nsmul_add_sub_nsmul {M : Type u_4} [AddMonoid M] {m n : ℕ} (a : M) (h : m ≤ n) : m • a + (n - m) • a = n • a

theorem pow_sub_mul_pow {M : Type u_4} [Monoid M] {m n : ℕ} (a : M) (h : m ≤ n) : a ^ (n - m) * a ^ m = a ^ n

theorem sub_nsmul_nsmul_add {M : Type u_4} [AddMonoid M] {m n : ℕ} (a : M) (h : m ≤ n) : (n - m) • a + m • a = n • a

theorem mul_pow_sub_one {M : Type u_4} [Monoid M] {n : ℕ} (hn : n ≠ 0) (a : M) : a * a ^ (n - 1) = a ^ n

theorem sub_one_nsmul_add {M : Type u_4} [AddMonoid M] {n : ℕ} (hn : n ≠ 0) (a : M) : a + (n - 1) • a = n • a

theorem pow_sub_one_mul {M : Type u_4} [Monoid M] {n : ℕ} (hn : n ≠ 0) (a : M) : a ^ (n - 1) * a = a ^ n

theorem add_sub_one_nsmul {M : Type u_4} [AddMonoid M] {n : ℕ} (hn : n ≠ 0) (a : M) : (n - 1) • a + a = n • a

theorem pow_eq_pow_mod {M : Type u_4} [Monoid M] {a : M} {n : ℕ} (m : ℕ) (ha : a ^ n = 1) : a ^ m = a ^ (m % n)

If x ^ n = 1, then x ^ m is the same as x ^ (m % n)

theorem nsmul_eq_mod_nsmul {M : Type u_4} [AddMonoid M] {a : M} {n : ℕ} (m : ℕ) (ha : n • a = 0) : m • a = (m % n) • a

If n • x = 0, then m • x is the same as (m % n) • x

theorem pow_mul_pow_eq_one {M : Type u_4} [Monoid M] {a b : M} (n : ℕ) : a * b = 1 → a ^ n * b ^ n = 1

theorem nsmul_add_nsmul_eq_zero {M : Type u_4} [AddMonoid M] {a b : M} (n : ℕ) : a + b = 0 → n • a + n • b = 0

theorem inv_unique {M : Type u_4} [CommMonoid M] {x y z : M} (hy : x * y = 1) (hz : x * z = 1) : y = z

theorem neg_unique {M : Type u_4} [AddCommMonoid M] {x y z : M} (hy : x + y = 0) (hz : x + z = 0) : y = z

theorem mul_pow {M : Type u_4} [CommMonoid M] (a b : M) (n : ℕ) : (a * b) ^ n = a ^ n * b ^ n

theorem nsmul_add {M : Type u_4} [AddCommMonoid M] (a b : M) (n : ℕ) : n • (a + b) = n • a + n • b

@[simp]

theorem mul_right_eq_self {M : Type u_4} [LeftCancelMonoid M] {a b : M} : a * b = a ↔ b = 1

@[simp]

theorem add_right_eq_self {M : Type u_4} [AddLeftCancelMonoid M] {a b : M} : a + b = a ↔ b = 0

@[simp]

theorem self_eq_mul_right {M : Type u_4} [LeftCancelMonoid M] {a b : M} : a = a * b ↔ b = 1

@[simp]

theorem self_eq_add_right {M : Type u_4} [AddLeftCancelMonoid M] {a b : M} : a = a + b ↔ b = 0

theorem mul_right_ne_self {M : Type u_4} [LeftCancelMonoid M] {a b : M} : a * b ≠ a ↔ b ≠ 1

theorem add_right_ne_self {M : Type u_4} [AddLeftCancelMonoid M] {a b : M} : a + b ≠ a ↔ b ≠ 0

theorem self_ne_mul_right {M : Type u_4} [LeftCancelMonoid M] {a b : M} : a ≠ a * b ↔ b ≠ 1

theorem self_ne_add_right {M : Type u_4} [AddLeftCancelMonoid M] {a b : M} : a ≠ a + b ↔ b ≠ 0

@[simp]

theorem mul_left_eq_self {M : Type u_4} [RightCancelMonoid M] {a b : M} : a * b = b ↔ a = 1

@[simp]

theorem add_left_eq_self {M : Type u_4} [AddRightCancelMonoid M] {a b : M} : a + b = b ↔ a = 0

@[simp]

theorem self_eq_mul_left {M : Type u_4} [RightCancelMonoid M] {a b : M} : b = a * b ↔ a = 1

@[simp]

theorem self_eq_add_left {M : Type u_4} [AddRightCancelMonoid M] {a b : M} : b = a + b ↔ a = 0

theorem mul_left_ne_self {M : Type u_4} [RightCancelMonoid M] {a b : M} : a * b ≠ b ↔ a ≠ 1

theorem add_left_ne_self {M : Type u_4} [AddRightCancelMonoid M] {a b : M} : a + b ≠ b ↔ a ≠ 0

theorem self_ne_mul_left {M : Type u_4} [RightCancelMonoid M] {a b : M} : b ≠ a * b ↔ a ≠ 1

theorem self_ne_add_left {M : Type u_4} [AddRightCancelMonoid M] {a b : M} : b ≠ a + b ↔ a ≠ 0

theorem eq_iff_eq_of_mul_eq_mul {α : Type u_1} [CancelCommMonoid α] {a b c d : α} (h : a * b = c * d) : a = c ↔ b = d

theorem eq_iff_eq_of_add_eq_add {α : Type u_1} [AddCancelCommMonoid α] {a b c d : α} (h : a + b = c + d) : a = c ↔ b = d

theorem ne_iff_ne_of_mul_eq_mul {α : Type u_1} [CancelCommMonoid α] {a b c d : α} (h : a * b = c * d) : a ≠ c ↔ b ≠ d

theorem ne_iff_ne_of_add_eq_add {α : Type u_1} [AddCancelCommMonoid α] {a b c d : α} (h : a + b = c + d) : a ≠ c ↔ b ≠ d

@[simp]

theorem inv_involutive {G : Type u_3} [InvolutiveInv G] : Function.Involutive Inv.inv

@[simp]

theorem neg_involutive {G : Type u_3} [InvolutiveNeg G] : Function.Involutive Neg.neg

@[simp]

theorem inv_surjective {G : Type u_3} [InvolutiveInv G] : Function.Surjective Inv.inv

@[simp]

theorem neg_surjective {G : Type u_3} [InvolutiveNeg G] : Function.Surjective Neg.neg

theorem inv_injective {G : Type u_3} [InvolutiveInv G] : Function.Injective Inv.inv

theorem neg_injective {G : Type u_3} [InvolutiveNeg G] : Function.Injective Neg.neg

@[simp]

theorem inv_inj {G : Type u_3} [InvolutiveInv G] {a b : G} : a⁻¹ = b⁻¹ ↔ a = b

@[simp]

theorem neg_inj {G : Type u_3} [InvolutiveNeg G] {a b : G} : -a = -b ↔ a = b

theorem inv_eq_iff_eq_inv {G : Type u_3} [InvolutiveInv G] {a b : G} : a⁻¹ = b ↔ a = b⁻¹

theorem neg_eq_iff_eq_neg {G : Type u_3} [InvolutiveNeg G] {a b : G} : -a = b ↔ a = -b

theorem inv_comp_inv (G : Type u_3) [InvolutiveInv G] : Inv.inv ∘ Inv.inv = id

theorem neg_comp_neg (G : Type u_3) [InvolutiveNeg G] : Neg.neg ∘ Neg.neg = id

theorem leftInverse_inv (G : Type u_3) [InvolutiveInv G] : Function.LeftInverse (fun (a : G) => a⁻¹) fun (a : G) => a⁻¹

theorem leftInverse_neg (G : Type u_3) [InvolutiveNeg G] : Function.LeftInverse (fun (a : G) => -a) fun (a : G) => -a

theorem rightInverse_inv (G : Type u_3) [InvolutiveInv G] : Function.RightInverse (fun (a : G) => a⁻¹) fun (a : G) => a⁻¹

theorem rightInverse_neg (G : Type u_3) [InvolutiveNeg G] : Function.RightInverse (fun (a : G) => -a) fun (a : G) => -a

theorem inv_eq_one_div {G : Type u_3} [DivInvMonoid G] (x : G) : x⁻¹ = 1 / x

theorem neg_eq_zero_sub {G : Type u_3} [SubNegMonoid G] (x : G) : -x = 0 - x

theorem mul_one_div {G : Type u_3} [DivInvMonoid G] (x y : G) : x * (1 / y) = x / y

theorem add_zero_sub {G : Type u_3} [SubNegMonoid G] (x y : G) : x + (0 - y) = x - y

theorem mul_div_assoc {G : Type u_3} [DivInvMonoid G] (a b c : G) : a * b / c = a * (b / c)

theorem add_sub_assoc {G : Type u_3} [SubNegMonoid G] (a b c : G) : a + b - c = a + (b - c)

theorem mul_div_assoc' {G : Type u_3} [DivInvMonoid G] (a b c : G) : a * (b / c) = a * b / c

theorem add_sub_assoc' {G : Type u_3} [SubNegMonoid G] (a b c : G) : a + (b - c) = a + b - c

@[simp]

theorem one_div {G : Type u_3} [DivInvMonoid G] (a : G) : 1 / a = a⁻¹

@[simp]

theorem zero_sub {G : Type u_3} [SubNegMonoid G] (a : G) : 0 - a = -a

theorem mul_div {G : Type u_3} [DivInvMonoid G] (a b c : G) : a * (b / c) = a * b / c

theorem add_sub {G : Type u_3} [SubNegMonoid G] (a b c : G) : a + (b - c) = a + b - c

theorem div_eq_mul_one_div {G : Type u_3} [DivInvMonoid G] (a b : G) : a / b = a * (1 / b)

theorem sub_eq_add_zero_sub {G : Type u_3} [SubNegMonoid G] (a b : G) : a - b = a + (0 - b)

@[simp]

theorem div_one {G : Type u_3} [DivInvOneMonoid G] (a : G) : a / 1 = a

@[simp]

theorem sub_zero {G : Type u_3} [SubNegZeroMonoid G] (a : G) : a - 0 = a

theorem one_div_one {G : Type u_3} [DivInvOneMonoid G] : 1 / 1 = 1

theorem zero_sub_zero {G : Type u_3} [SubNegZeroMonoid G] : 0 - 0 = 0

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

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

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

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

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

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

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

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

theorem eq_of_inv_mul_eq_one {α : Type u_1} [DivisionMonoid α] {a b : α} (h : a⁻¹ * b = 1) : a = b

theorem eq_of_neg_add_eq_zero {α : Type u_1} [SubtractionMonoid α] {a b : α} (h : -a + b = 0) : a = b

theorem eq_of_mul_inv_eq_one {α : Type u_1} [DivisionMonoid α] {a b : α} (h : a * b⁻¹ = 1) : a = b

theorem eq_of_add_neg_eq_zero {α : Type u_1} [SubtractionMonoid α] {a b : α} (h : a + -b = 0) : a = b

theorem div_ne_one_of_ne {α : Type u_1} [DivisionMonoid α] {a b : α} : a ≠ b → a / b ≠ 1

theorem sub_ne_zero_of_ne {α : Type u_1} [SubtractionMonoid α] {a b : α} : a ≠ b → a - b ≠ 0

theorem one_div_mul_one_div_rev {α : Type u_1} [DivisionMonoid α] (a b : α) : 1 / a * (1 / b) = 1 / (b * a)

theorem zero_sub_add_zero_sub_rev {α : Type u_1} [SubtractionMonoid α] (a b : α) : 0 - a + (0 - b) = 0 - (b + a)

theorem inv_div_left {α : Type u_1} [DivisionMonoid α] (a b : α) : a⁻¹ / b = (b * a)⁻¹

theorem neg_sub_left {α : Type u_1} [SubtractionMonoid α] (a b : α) : -a - b = -(b + a)

@[simp]

theorem inv_div {α : Type u_1} [DivisionMonoid α] (a b : α) : (a / b)⁻¹ = b / a

@[simp]

theorem neg_sub {α : Type u_1} [SubtractionMonoid α] (a b : α) : -(a - b) = b - a

theorem one_div_div {α : Type u_1} [DivisionMonoid α] (a b : α) : 1 / (a / b) = b / a

theorem zero_sub_sub {α : Type u_1} [SubtractionMonoid α] (a b : α) : 0 - (a - b) = b - a

theorem one_div_one_div {α : Type u_1} [DivisionMonoid α] (a : α) : 1 / (1 / a) = a

theorem zero_sub_zero_sub {α : Type u_1} [SubtractionMonoid α] (a : α) : 0 - (0 - a) = a

theorem div_eq_div_iff_comm {α : Type u_1} [DivisionMonoid α] (a b c : α) {d : α} : a / b = c / d ↔ b / a = d / c

theorem sub_eq_sub_iff_comm {α : Type u_1} [SubtractionMonoid α] (a b c : α) {d : α} : a - b = c - d ↔ b - a = d - c

@[instance 100]

instance DivisionMonoid.toDivInvOneMonoid {α : Type u_1} [DivisionMonoid α] : DivInvOneMonoid α

Equations

• DivisionMonoid.toDivInvOneMonoid = DivInvOneMonoid.mk ⋯

@[instance 100]

instance SubtractionMonoid.toSubNegZeroMonoid {α : Type u_1} [SubtractionMonoid α] : SubNegZeroMonoid α

Equations

• SubtractionMonoid.toSubNegZeroMonoid = SubNegZeroMonoid.mk ⋯

@[simp]

theorem inv_pow {α : Type u_1} [DivisionMonoid α] (a : α) (n : ℕ) : a⁻¹ ^ n = (a ^ n)⁻¹

@[simp]

theorem neg_nsmul {α : Type u_1} [SubtractionMonoid α] (a : α) (n : ℕ) : n • -a = -(n • a)

@[simp]

theorem one_zpow {α : Type u_1} [DivisionMonoid α] (n : ℤ) : 1 ^ n = 1

theorem zsmul_zero {α : Type u_1} [SubtractionMonoid α] (n : ℤ) : n • 0 = 0

@[simp]

theorem zpow_neg {α : Type u_1} [DivisionMonoid α] (a : α) (n : ℤ) : a ^ (-n) = (a ^ n)⁻¹

@[simp]

theorem neg_zsmul {α : Type u_1} [SubtractionMonoid α] (a : α) (n : ℤ) : -n • a = -(n • a)

theorem mul_zpow_neg_one {α : Type u_1} [DivisionMonoid α] (a b : α) : (a * b) ^ (-1) = b ^ (-1) * a ^ (-1)

theorem neg_one_zsmul_add {α : Type u_1} [SubtractionMonoid α] (a b : α) : -1 • (a + b) = -1 • b + -1 • a

theorem inv_zpow {α : Type u_1} [DivisionMonoid α] (a : α) (n : ℤ) : a⁻¹ ^ n = (a ^ n)⁻¹

theorem zsmul_neg {α : Type u_1} [SubtractionMonoid α] (a : α) (n : ℤ) : n • -a = -(n • a)

@[simp]

theorem inv_zpow' {α : Type u_1} [DivisionMonoid α] (a : α) (n : ℤ) : a⁻¹ ^ n = a ^ (-n)

@[simp]

theorem zsmul_neg' {α : Type u_1} [SubtractionMonoid α] (a : α) (n : ℤ) : n • -a = -n • a

theorem one_div_pow {α : Type u_1} [DivisionMonoid α] (a : α) (n : ℕ) : (1 / a) ^ n = 1 / a ^ n

theorem nsmul_zero_sub {α : Type u_1} [SubtractionMonoid α] (a : α) (n : ℕ) : n • (0 - a) = 0 - n • a

theorem one_div_zpow {α : Type u_1} [DivisionMonoid α] (a : α) (n : ℤ) : (1 / a) ^ n = 1 / a ^ n

theorem zsmul_zero_sub {α : Type u_1} [SubtractionMonoid α] (a : α) (n : ℤ) : n • (0 - a) = 0 - n • a

@[simp]

theorem inv_eq_one {α : Type u_1} [DivisionMonoid α] {a : α} : a⁻¹ = 1 ↔ a = 1

@[simp]

theorem neg_eq_zero {α : Type u_1} [SubtractionMonoid α] {a : α} : -a = 0 ↔ a = 0

@[simp]

theorem one_eq_inv {α : Type u_1} [DivisionMonoid α] {a : α} : 1 = a⁻¹ ↔ a = 1

@[simp]

theorem zero_eq_neg {α : Type u_1} [SubtractionMonoid α] {a : α} : 0 = -a ↔ a = 0

theorem inv_ne_one {α : Type u_1} [DivisionMonoid α] {a : α} : a⁻¹ ≠ 1 ↔ a ≠ 1

theorem neg_ne_zero {α : Type u_1} [SubtractionMonoid α] {a : α} : -a ≠ 0 ↔ a ≠ 0

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

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

theorem zpow_mul {α : Type u_1} [DivisionMonoid α] (a : α) (m n : ℤ) : a ^ (m * n) = (a ^ m) ^ n

theorem mul_zsmul' {α : Type u_1} [SubtractionMonoid α] (a : α) (m n : ℤ) : (m * n) • a = n • m • a

theorem zpow_mul' {α : Type u_1} [DivisionMonoid α] (a : α) (m n : ℤ) : a ^ (m * n) = (a ^ n) ^ m

theorem mul_zsmul {α : Type u_1} [SubtractionMonoid α] (a : α) (m n : ℤ) : (m * n) • a = m • n • a

theorem div_div_eq_mul_div {α : Type u_1} [DivisionMonoid α] (a b c : α) : a / (b / c) = a * c / b

theorem sub_sub_eq_add_sub {α : Type u_1} [SubtractionMonoid α] (a b c : α) : a - (b - c) = a + c - b

@[simp]

theorem div_inv_eq_mul {α : Type u_1} [DivisionMonoid α] (a b : α) : a / b⁻¹ = a * b

@[simp]

theorem sub_neg_eq_add {α : Type u_1} [SubtractionMonoid α] (a b : α) : a - -b = a + b

theorem div_mul_eq_div_div_swap {α : Type u_1} [DivisionMonoid α] (a b c : α) : a / (b * c) = a / c / b

theorem sub_add_eq_sub_sub_swap {α : Type u_1} [SubtractionMonoid α] (a b c : α) : a - (b + c) = a - c - b

theorem mul_inv {α : Type u_1} [DivisionCommMonoid α] (a b : α) : (a * b)⁻¹ = a⁻¹ * b⁻¹

theorem neg_add {α : Type u_1} [SubtractionCommMonoid α] (a b : α) : -(a + b) = -a + -b

theorem inv_div' {α : Type u_1} [DivisionCommMonoid α] (a b : α) : (a / b)⁻¹ = a⁻¹ / b⁻¹

theorem neg_sub' {α : Type u_1} [SubtractionCommMonoid α] (a b : α) : -(a - b) = -a - -b

theorem div_eq_inv_mul {α : Type u_1} [DivisionCommMonoid α] (a b : α) : a / b = b⁻¹ * a

theorem sub_eq_neg_add {α : Type u_1} [SubtractionCommMonoid α] (a b : α) : a - b = -b + a

theorem inv_mul_eq_div {α : Type u_1} [DivisionCommMonoid α] (a b : α) : a⁻¹ * b = b / a

theorem neg_add_eq_sub {α : Type u_1} [SubtractionCommMonoid α] (a b : α) : -a + b = b - a

theorem inv_div_comm {α : Type u_1} [DivisionCommMonoid α] (a b : α) : a⁻¹ / b = b⁻¹ / a

theorem neg_sub_comm {α : Type u_1} [SubtractionCommMonoid α] (a b : α) : -a - b = -b - a

theorem inv_mul' {α : Type u_1} [DivisionCommMonoid α] (a b : α) : (a * b)⁻¹ = a⁻¹ / b

theorem neg_add' {α : Type u_1} [SubtractionCommMonoid α] (a b : α) : -(a + b) = -a - b

theorem inv_div_inv {α : Type u_1} [DivisionCommMonoid α] (a b : α) : a⁻¹ / b⁻¹ = b / a

theorem neg_sub_neg {α : Type u_1} [SubtractionCommMonoid α] (a b : α) : -a - -b = b - a

theorem inv_inv_div_inv {α : Type u_1} [DivisionCommMonoid α] (a b : α) : (a⁻¹ / b⁻¹)⁻¹ = a / b

theorem neg_neg_sub_neg {α : Type u_1} [SubtractionCommMonoid α] (a b : α) : -(-a - -b) = a - b

theorem one_div_mul_one_div {α : Type u_1} [DivisionCommMonoid α] (a b : α) : 1 / a * (1 / b) = 1 / (a * b)

theorem zero_sub_add_zero_sub {α : Type u_1} [SubtractionCommMonoid α] (a b : α) : 0 - a + (0 - b) = 0 - (a + b)

theorem div_right_comm {α : Type u_1} [DivisionCommMonoid α] (a b c : α) : a / b / c = a / c / b

theorem sub_right_comm {α : Type u_1} [SubtractionCommMonoid α] (a b c : α) : a - b - c = a - c - b

theorem div_div {α : Type u_1} [DivisionCommMonoid α] (a b c : α) : a / b / c = a / (b * c)

theorem sub_sub {α : Type u_1} [SubtractionCommMonoid α] (a b c : α) : a - b - c = a - (b + c)

theorem div_mul {α : Type u_1} [DivisionCommMonoid α] (a b c : α) : a / b * c = a / (b / c)

theorem sub_add {α : Type u_1} [SubtractionCommMonoid α] (a b c : α) : a - b + c = a - (b - c)

theorem mul_div_left_comm {α : Type u_1} [DivisionCommMonoid α] (a b c : α) : a * (b / c) = b * (a / c)

theorem add_sub_left_comm {α : Type u_1} [SubtractionCommMonoid α] (a b c : α) : a + (b - c) = b + (a - c)

theorem mul_div_right_comm {α : Type u_1} [DivisionCommMonoid α] (a b c : α) : a * b / c = a / c * b

theorem add_sub_right_comm {α : Type u_1} [SubtractionCommMonoid α] (a b c : α) : a + b - c = a - c + b

theorem div_mul_eq_div_div {α : Type u_1} [DivisionCommMonoid α] (a b c : α) : a / (b * c) = a / b / c

theorem sub_add_eq_sub_sub {α : Type u_1} [SubtractionCommMonoid α] (a b c : α) : a - (b + c) = a - b - c

theorem div_mul_eq_mul_div {α : Type u_1} [DivisionCommMonoid α] (a b c : α) : a / b * c = a * c / b

theorem sub_add_eq_add_sub {α : Type u_1} [SubtractionCommMonoid α] (a b c : α) : a - b + c = a + c - b

theorem one_div_mul_eq_div {α : Type u_1} [DivisionCommMonoid α] (a b : α) : 1 / a * b = b / a

theorem zero_sub_add_eq_sub {α : Type u_1} [SubtractionCommMonoid α] (a b : α) : 0 - a + b = b - a

theorem mul_comm_div {α : Type u_1} [DivisionCommMonoid α] (a b c : α) : a / b * c = a * (c / b)

theorem add_comm_sub {α : Type u_1} [SubtractionCommMonoid α] (a b c : α) : a - b + c = a + (c - b)

theorem div_mul_comm {α : Type u_1} [DivisionCommMonoid α] (a b c : α) : a / b * c = c / b * a

theorem sub_add_comm {α : Type u_1} [SubtractionCommMonoid α] (a b c : α) : a - b + c = c - b + a

theorem div_mul_eq_div_mul_one_div {α : Type u_1} [DivisionCommMonoid α] (a b c : α) : a / (b * c) = a / b * (1 / c)

theorem sub_add_eq_sub_add_zero_sub {α : Type u_1} [SubtractionCommMonoid α] (a b c : α) : a - (b + c) = a - b + (0 - c)

theorem div_div_div_eq {α : Type u_1} [DivisionCommMonoid α] (a b c d : α) : a / b / (c / d) = a * d / (b * c)

theorem sub_sub_sub_eq {α : Type u_1} [SubtractionCommMonoid α] (a b c d : α) : a - b - (c - d) = a + d - (b + c)

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

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

theorem div_mul_div_comm {α : Type u_1} [DivisionCommMonoid α] (a b c d : α) : a / b * (c / d) = a * c / (b * d)

theorem sub_add_sub_comm {α : Type u_1} [SubtractionCommMonoid α] (a b c d : α) : a - b + (c - d) = a + c - (b + d)

theorem mul_div_mul_comm {α : Type u_1} [DivisionCommMonoid α] (a b c d : α) : a * b / (c * d) = a / c * (b / d)

theorem add_sub_add_comm {α : Type u_1} [SubtractionCommMonoid α] (a b c d : α) : a + b - (c + d) = a - c + (b - d)

theorem mul_zpow {α : Type u_1} [DivisionCommMonoid α] (a b : α) (n : ℤ) : (a * b) ^ n = a ^ n * b ^ n

theorem zsmul_add {α : Type u_1} [SubtractionCommMonoid α] (a b : α) (n : ℤ) : n • (a + b) = n • a + n • b

theorem div_pow {α : Type u_1} [DivisionCommMonoid α] (a b : α) (n : ℕ) : (a / b) ^ n = a ^ n / b ^ n

theorem nsmul_sub {α : Type u_1} [SubtractionCommMonoid α] (a b : α) (n : ℕ) : n • (a - b) = n • a - n • b

theorem div_zpow {α : Type u_1} [DivisionCommMonoid α] (a b : α) (n : ℤ) : (a / b) ^ n = a ^ n / b ^ n

theorem zsmul_sub {α : Type u_1} [SubtractionCommMonoid α] (a b : α) (n : ℤ) : n • (a - b) = n • a - n • b

@[simp]

theorem div_eq_inv_self {G : Type u_3} [Group G] {a b : G} : a / b = b⁻¹ ↔ a = 1

@[simp]

theorem sub_eq_neg_self {G : Type u_3} [AddGroup G] {a b : G} : a - b = -b ↔ a = 0

theorem mul_left_surjective {G : Type u_3} [Group G] (a : G) : Function.Surjective fun (x : G) => a * x

theorem add_left_surjective {G : Type u_3} [AddGroup G] (a : G) : Function.Surjective fun (x : G) => a + x

theorem mul_right_surjective {G : Type u_3} [Group G] (a : G) : Function.Surjective fun (x : G) => x * a

theorem add_right_surjective {G : Type u_3} [AddGroup G] (a : G) : Function.Surjective fun (x : G) => x + a

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_add_neg_of_add_eq {G : Type u_3} [AddGroup G] {a b c : G} (h : a + c = b) : 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 eq_neg_add_of_add_eq {G : Type u_3} [AddGroup G] {a b c : G} (h : b + a = c) : a = -b + 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} [AddGroup G] {a b c : G} (h : b = a + c) : -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 add_neg_eq_of_eq_add {G : Type u_3} [AddGroup 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_add_of_add_neg_eq {G : Type u_3} [AddGroup G] {a b c : G} (h : a + -c = 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} [AddGroup 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} [AddGroup 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} [AddGroup 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} [AddGroup 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} [AddGroup G] {a b : G} : a + b = 0 ↔ -a = b

theorem mul_eq_one_iff_eq_inv' {G : Type u_3} [Group G] {a b : G} : a * b = 1 ↔ b = a⁻¹

Variant of mul_eq_one_iff_eq_inv with swapped equality.

theorem add_eq_zero_iff_eq_neg' {G : Type u_3} [AddGroup G] {a b : G} : a + b = 0 ↔ b = -a

theorem mul_eq_one_iff_inv_eq' {G : Type u_3} [Group G] {a b : G} : a * b = 1 ↔ b⁻¹ = a

Variant of mul_eq_one_iff_inv_eq with swapped equality.

theorem add_eq_zero_iff_neg_eq' {G : Type u_3} [AddGroup G] {a b : G} : a + b = 0 ↔ -b = a

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} [AddGroup 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 neg_eq_iff_add_eq_zero {G : Type u_3} [AddGroup G] {a b : G} : -a = b ↔ a + b = 0

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_add_neg_iff_add_eq {G : Type u_3} [AddGroup G] {a b c : G} : a = b + -c ↔ 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} [AddGroup 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 neg_add_eq_iff_eq_add {G : Type u_3} [AddGroup G] {a b c : G} : -a + b = c ↔ b = a + c

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 add_neg_eq_iff_eq_add {G : Type u_3} [AddGroup G] {a b c : G} : a + -b = 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} [AddGroup 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} [AddGroup G] {a b : G} : -a + b = 0 ↔ a = b

@[simp]

theorem conj_eq_one_iff {G : Type u_3} [Group G] {a b : G} : a * b * a⁻¹ = 1 ↔ b = 1

@[simp]

theorem conj_eq_zero_iff {G : Type u_3} [AddGroup G] {a b : G} : a + b + -a = 0 ↔ b = 0

theorem div_left_injective {G : Type u_3} [Group G] {b : G} : Function.Injective fun (a : G) => a / b

theorem sub_left_injective {G : Type u_3} [AddGroup G] {b : G} : Function.Injective fun (a : G) => a - b

theorem div_right_injective {G : Type u_3} [Group G] {b : G} : Function.Injective fun (a : G) => b / a

theorem sub_right_injective {G : Type u_3} [AddGroup G] {b : G} : Function.Injective fun (a : G) => b - a

@[simp]

theorem div_mul_cancel {G : Type u_3} [Group G] (a b : G) : a / b * b = a

@[simp]

theorem sub_add_cancel {G : Type u_3} [AddGroup 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} [AddGroup G] (a : G) : a - a = 0

@[simp]

theorem mul_div_cancel_right {G : Type u_3} [Group G] (a b : G) : a * b / b = a

@[simp]

theorem add_sub_cancel_right {G : Type u_3} [AddGroup G] (a b : G) : a + b - b = a

@[simp]

theorem div_mul_cancel_right {G : Type u_3} [Group G] (a b : G) : a / (b * a) = b⁻¹

@[simp]

theorem sub_add_cancel_right {G : Type u_3} [AddGroup G] (a b : G) : a - (b + a) = -b

@[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} [AddGroup G] (a b c : G) : a + c - (b + c) = a - b

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

theorem eq_sub_of_add_eq {G : Type u_3} [AddGroup 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} [AddGroup 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} [AddGroup 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} [AddGroup 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} [AddGroup 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} [AddGroup 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} [AddGroup 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} [AddGroup G] (a b c : G) : a - c - (b - c) = a - b

@[deprecated div_div_div_cancel_right (since := "2024-08-24")]

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

Alias of div_div_div_cancel_right.

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} [AddGroup 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} [AddGroup G] {a b : G} : a = b → a - b = 0

Alias of the reverse direction of sub_eq_zero.

theorem div_ne_one {G : Type u_3} [Group G] {a b : G} : a / b ≠ 1 ↔ a ≠ b

theorem sub_ne_zero {G : Type u_3} [AddGroup G] {a b : G} : a - b ≠ 0 ↔ a ≠ b

@[simp]

theorem div_eq_self {G : Type u_3} [Group G] {a b : G} : a / b = a ↔ b = 1

@[simp]

theorem sub_eq_self {G : Type u_3} [AddGroup G] {a b : G} : a - b = a ↔ b = 0

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} [AddGroup 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 sub_eq_iff_eq_add {G : Type u_3} [AddGroup G] {a b c : G} : a - b = c ↔ a = c + b

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

theorem leftInverse_div_mul_left {G : Type u_3} [Group G] (c : G) : Function.LeftInverse (fun (x : G) => x / c) fun (x : G) => x * c

theorem leftInverse_sub_add_left {G : Type u_3} [AddGroup G] (c : G) : Function.LeftInverse (fun (x : G) => x - c) fun (x : G) => x + c

theorem leftInverse_mul_left_div {G : Type u_3} [Group G] (c : G) : Function.LeftInverse (fun (x : G) => x * c) fun (x : G) => x / c

theorem leftInverse_add_left_sub {G : Type u_3} [AddGroup G] (c : G) : Function.LeftInverse (fun (x : G) => x + c) fun (x : G) => x - c

theorem leftInverse_mul_right_inv_mul {G : Type u_3} [Group G] (c : G) : Function.LeftInverse (fun (x : G) => c * x) fun (x : G) => c⁻¹ * x

theorem leftInverse_add_right_neg_add {G : Type u_3} [AddGroup G] (c : G) : Function.LeftInverse (fun (x : G) => c + x) fun (x : G) => -c + x

theorem leftInverse_inv_mul_mul_right {G : Type u_3} [Group G] (c : G) : Function.LeftInverse (fun (x : G) => c⁻¹ * x) fun (x : G) => c * x

theorem leftInverse_neg_add_add_right {G : Type u_3} [AddGroup G] (c : G) : Function.LeftInverse (fun (x : G) => -c + x) fun (x : G) => c + x

@[simp]

theorem pow_natAbs_eq_one {G : Type u_3} [Group G] {a : G} {n : ℤ} : a ^ n.natAbs = 1 ↔ a ^ n = 1

@[simp]

theorem natAbs_nsmul_eq_zero {G : Type u_3} [AddGroup G] {a : G} {n : ℤ} : n.natAbs • a = 0 ↔ n • a = 0

theorem pow_sub {G : Type u_3} [Group G] (a : G) {m n : ℕ} (h : n ≤ m) : a ^ (m - n) = a ^ m * (a ^ n)⁻¹

theorem sub_nsmul {G : Type u_3} [AddGroup G] (a : G) {m n : ℕ} (h : n ≤ m) : (m - n) • a = m • a + -(n • a)

theorem inv_pow_sub {G : Type u_3} [Group G] (a : G) {m n : ℕ} (h : n ≤ m) : a⁻¹ ^ (m - n) = (a ^ m)⁻¹ * a ^ n

theorem sub_nsmul_neg {G : Type u_3} [AddGroup G] (a : G) {m n : ℕ} (h : n ≤ m) : (m - n) • -a = -(m • a) + n • a

theorem zpow_add_one {G : Type u_3} [Group G] (a : G) (n : ℤ) : a ^ (n + 1) = a ^ n * a

theorem add_one_zsmul {G : Type u_3} [AddGroup G] (a : G) (n : ℤ) : (n + 1) • a = n • a + a

theorem zpow_sub_one {G : Type u_3} [Group G] (a : G) (n : ℤ) : a ^ (n - 1) = a ^ n * a⁻¹

theorem sub_one_zsmul {G : Type u_3} [AddGroup G] (a : G) (n : ℤ) : (n - 1) • a = n • a + -a

theorem zpow_add {G : Type u_3} [Group G] (a : G) (m n : ℤ) : a ^ (m + n) = a ^ m * a ^ n

theorem add_zsmul {G : Type u_3} [AddGroup G] (a : G) (m n : ℤ) : (m + n) • a = m • a + n • a

theorem zpow_one_add {G : Type u_3} [Group G] (a : G) (n : ℤ) : a ^ (1 + n) = a * a ^ n

theorem one_add_zsmul {G : Type u_3} [AddGroup G] (a : G) (n : ℤ) : (1 + n) • a = a + n • a

theorem mul_self_zpow {G : Type u_3} [Group G] (a : G) (n : ℤ) : a * a ^ n = a ^ (n + 1)

theorem add_zsmul_self {G : Type u_3} [AddGroup G] (a : G) (n : ℤ) : a + n • a = (n + 1) • a

theorem mul_zpow_self {G : Type u_3} [Group G] (a : G) (n : ℤ) : a ^ n * a = a ^ (n + 1)

theorem add_self_zsmul {G : Type u_3} [AddGroup G] (a : G) (n : ℤ) : n • a + a = (n + 1) • a

theorem zpow_sub {G : Type u_3} [Group G] (a : G) (m n : ℤ) : a ^ (m - n) = a ^ m * (a ^ n)⁻¹

theorem sub_zsmul {G : Type u_3} [AddGroup G] (a : G) (m n : ℤ) : (m - n) • a = m • a + -(n • a)

theorem zpow_natCast_sub_natCast {G : Type u_3} [Group G] (a : G) (m n : ℕ) : a ^ (↑m - ↑n) = a ^ m / a ^ n

theorem natCast_sub_natCast_zsmul {G : Type u_3} [AddGroup G] (a : G) (m n : ℕ) : (↑m - ↑n) • a = m • a - n • a

theorem zpow_natCast_sub_one {G : Type u_3} [Group G] (a : G) (n : ℕ) : a ^ (↑n - 1) = a ^ n / a

theorem natCast_sub_one_zsmul {G : Type u_3} [AddGroup G] (a : G) (n : ℕ) : (↑n - 1) • a = n • a - a

theorem zpow_one_sub_natCast {G : Type u_3} [Group G] (a : G) (n : ℕ) : a ^ (1 - ↑n) = a / a ^ n

theorem one_sub_natCast_zsmul {G : Type u_3} [AddGroup G] (a : G) (n : ℕ) : (1 - ↑n) • a = a - n • a

theorem zpow_mul_comm {G : Type u_3} [Group G] (a : G) (m n : ℤ) : a ^ m * a ^ n = a ^ n * a ^ m

theorem zsmul_add_comm {G : Type u_3} [AddGroup G] (a : G) (m n : ℤ) : m • a + n • a = n • a + m • a

theorem zpow_eq_zpow_emod {G : Type u_3} [Group G] {x : G} (m : ℤ) {n : ℤ} (h : x ^ n = 1) : x ^ m = x ^ (m % n)

theorem zpow_eq_zpow_emod' {G : Type u_3} [Group G] {x : G} (m : ℤ) {n : ℕ} (h : x ^ n = 1) : x ^ m = x ^ (m % ↑n)

theorem zpow_induction_left {G : Type u_3} [Group G] {g : G} {P : G → Prop} (h_one : P 1) (h_mul : ∀ (a : G), P a → P (g * a)) (h_inv : ∀ (a : G), P a → P (g⁻¹ * a)) (n : ℤ) : P (g ^ n)

To show a property of all powers of g it suffices to show it is closed under multiplication by g and g⁻¹ on the left. For subgroups generated by more than one element, see Subgroup.closure_induction_left.

theorem zsmul_induction_left {G : Type u_3} [AddGroup G] {g : G} {P : G → Prop} (h_one : P 0) (h_mul : ∀ (a : G), P a → P (g + a)) (h_inv : ∀ (a : G), P a → P (-g + a)) (n : ℤ) : P (n • g)

To show a property of all multiples of g it suffices to show it is closed under addition by g and -g on the left. For additive subgroups generated by more than one element, see AddSubgroup.closure_induction_left.

theorem zpow_induction_right {G : Type u_3} [Group G] {g : G} {P : G → Prop} (h_one : P 1) (h_mul : ∀ (a : G), P a → P (a * g)) (h_inv : ∀ (a : G), P a → P (a * g⁻¹)) (n : ℤ) : P (g ^ n)

To show a property of all powers of g it suffices to show it is closed under multiplication by g and g⁻¹ on the right. For subgroups generated by more than one element, see Subgroup.closure_induction_right.

theorem zsmul_induction_right {G : Type u_3} [AddGroup G] {g : G} {P : G → Prop} (h_one : P 0) (h_mul : ∀ (a : G), P a → P (a + g)) (h_inv : ∀ (a : G), P a → P (a + -g)) (n : ℤ) : P (n • g)

To show a property of all multiples of g it suffices to show it is closed under addition by g and -g on the right. For additive subgroups generated by more than one element, see AddSubgroup.closure_induction_right.

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

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

@[simp]

theorem mul_div_mul_left_eq_div {G : Type u_3} [CommGroup G] (a b c : G) : c * a / (c * b) = a / b

@[simp]

theorem add_sub_add_left_eq_sub {G : Type u_3} [AddCommGroup G] (a b c : G) : c + a - (c + b) = a - b

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

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

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

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

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

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

theorem div_div_self' {G : Type u_3} [CommGroup G] (a b : G) : a / (a / b) = b

theorem sub_sub_self {G : Type u_3} [AddCommGroup G] (a b : G) : a - (a - b) = b

theorem div_eq_div_mul_div {G : Type u_3} [CommGroup G] (a b c : G) : a / b = c / b * (a / c)

theorem sub_eq_sub_add_sub {G : Type u_3} [AddCommGroup G] (a b c : G) : a - b = c - b + (a - c)

@[simp]

theorem div_div_cancel {G : Type u_3} [CommGroup G] (a b : G) : a / (a / b) = b

@[simp]

theorem sub_sub_cancel {G : Type u_3} [AddCommGroup G] (a b : G) : a - (a - b) = b

@[simp]

theorem div_div_cancel_left {G : Type u_3} [CommGroup G] (a b : G) : a / b / a = b⁻¹

@[simp]

theorem sub_sub_cancel_left {G : Type u_3} [AddCommGroup G] (a b : G) : a - b - a = -b

theorem eq_div_iff_mul_eq'' {G : Type u_3} [CommGroup G] {a b c : G} : a = b / c ↔ c * a = b

theorem eq_sub_iff_add_eq' {G : Type u_3} [AddCommGroup G] {a b c : G} : a = b - c ↔ c + a = b

theorem div_eq_iff_eq_mul' {G : Type u_3} [CommGroup G] {a b c : G} : a / b = c ↔ a = b * c

theorem sub_eq_iff_eq_add' {G : Type u_3} [AddCommGroup G] {a b c : G} : a - b = c ↔ a = b + c

@[simp]

theorem mul_div_cancel_left {G : Type u_3} [CommGroup G] (a b : G) : a * b / a = b

@[simp]

theorem add_sub_cancel_left {G : Type u_3} [AddCommGroup G] (a b : G) : a + b - a = b

@[simp]

theorem mul_div_cancel {G : Type u_3} [CommGroup G] (a b : G) : a * (b / a) = b

@[simp]

theorem add_sub_cancel {G : Type u_3} [AddCommGroup G] (a b : G) : a + (b - a) = b

@[simp]

theorem div_mul_cancel_left {G : Type u_3} [CommGroup G] (a b : G) : a / (a * b) = b⁻¹

@[simp]

theorem sub_add_cancel_left {G : Type u_3} [AddCommGroup G] (a b : G) : a - (a + b) = -b

theorem mul_mul_inv_cancel'_right {G : Type u_3} [CommGroup G] (a b : G) : a * (b * a⁻¹) = b

theorem add_add_neg_cancel'_right {G : Type u_3} [AddCommGroup G] (a b : G) : a + (b + -a) = b

@[simp]

theorem mul_mul_div_cancel {G : Type u_3} [CommGroup G] (a b c : G) : a * c * (b / c) = a * b

@[simp]

theorem add_add_sub_cancel {G : Type u_3} [AddCommGroup G] (a b c : G) : a + c + (b - c) = a + b

@[simp]

theorem div_mul_mul_cancel {G : Type u_3} [CommGroup G] (a b c : G) : a / c * (b * c) = a * b

@[simp]

theorem sub_add_add_cancel {G : Type u_3} [AddCommGroup G] (a b c : G) : a - c + (b + c) = a + b

@[simp]

theorem div_mul_div_cancel' {G : Type u_3} [CommGroup G] (a b c : G) : a / b * (c / a) = c / b

@[simp]

theorem sub_add_sub_cancel' {G : Type u_3} [AddCommGroup G] (a b c : G) : a - b + (c - a) = c - b

@[deprecated div_mul_div_cancel' (since := "2024-08-24")]

theorem div_mul_div_cancel'' {G : Type u_3} [CommGroup G] (a b c : G) : a / b * (c / a) = c / b

Alias of div_mul_div_cancel'.

@[simp]

theorem mul_div_div_cancel {G : Type u_3} [CommGroup G] (a b c : G) : a * b / (a / c) = b * c

@[simp]

theorem add_sub_sub_cancel {G : Type u_3} [AddCommGroup G] (a b c : G) : a + b - (a - c) = b + c

@[simp]

theorem div_div_div_cancel_left {G : Type u_3} [CommGroup G] (a b c : G) : c / a / (c / b) = b / a

@[simp]

theorem sub_sub_sub_cancel_left {G : Type u_3} [AddCommGroup G] (a b c : G) : c - a - (c - b) = b - a

theorem div_eq_div_iff_mul_eq_mul {G : Type u_3} [CommGroup G] {a b c d : G} : a / b = c / d ↔ a * d = c * b

theorem sub_eq_sub_iff_add_eq_add {G : Type u_3} [AddCommGroup G] {a b c d : G} : a - b = c - d ↔ a + d = c + b

theorem div_eq_div_iff_div_eq_div {G : Type u_3} [CommGroup G] {a b c d : G} : a / b = c / d ↔ a / c = b / d

theorem sub_eq_sub_iff_sub_eq_sub {G : Type u_3} [AddCommGroup G] {a b c d : G} : a - b = c - d ↔ a - c = b - d

theorem multiplicative_of_symmetric_of_isTotal {α : Type u_1} {β : Type u_2} [Monoid β] (p r : α → α → Prop) [IsTotal 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_isTotal {α : Type u_1} {β : Type u_2} [AddMonoid β] (p r : α → α → Prop) [IsTotal 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_isTotal {α : Type u_1} {β : Type u_2} [Monoid β] (r : α → α → Prop) [IsTotal 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_isTotal {α : Type u_1} {β : Type u_2} [AddMonoid β] (r : α → α → Prop) [IsTotal 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.