Loogle!
Result
Found 14006 definitions mentioning HMul.hMul. Of these, 509 have a name containing "comm". Of these, 494 match your pattern(s). Of these, only the first 200 are shown.
- Nat.instCommutativeHMul Init.Data.Nat.Basic
Std.Commutative fun x x_1 => x * x_1 - Nat.mul_comm Init.Data.Nat.Basic
∀ (n m : ℕ), n * m = m * n - Nat.mul_left_comm Init.Data.Nat.Basic
∀ (n m k : ℕ), n * (m * k) = m * (n * k) - Int.instCommutativeHMul Init.Data.Int.Lemmas
Std.Commutative fun x x_1 => x * x_1 - Int.mul_comm Init.Data.Int.Lemmas
∀ (a b : ℤ), a * b = b * a - Int.neg_mul_comm Init.Data.Int.Lemmas
∀ (a b : ℤ), -a * b = a * -b - Int.mul_left_comm Init.Data.Int.Lemmas
∀ (a b c : ℤ), a * (b * c) = b * (a * c) - Int.mul_right_comm Init.Data.Int.Lemmas
∀ (a b c : ℤ), a * b * c = a * c * b - Lean.Omega.LinearCombo.smul_eval_comm Init.Omega.LinearCombo
∀ (lc : Lean.Omega.LinearCombo) (i : ℤ) (v : Lean.Omega.Coeffs), (i * lc).eval v = lc.eval v * i - Nat.mul_right_comm Init.Data.Nat.Lemmas
∀ (n m k : ℕ), n * m * k = n * k * m - Nat.mul_mul_mul_comm Init.Data.Nat.Lemmas
∀ (a b c d : ℕ), a * b * (c * d) = a * c * (b * d) - Fin.instCommutativeHMul Init.Data.Fin.Lemmas
∀ {n : ℕ}, Std.Commutative fun x x_1 => x * x_1 - Fin.mul_comm Init.Data.Fin.Lemmas
∀ {n : ℕ} (a b : Fin n), a * b = b * a - BitVec.instCommutativeHMul Init.Data.BitVec.Lemmas
∀ {w : ℕ}, Std.Commutative fun x y => x * y - BitVec.instLawfulCommIdentityHMulOfNatOfNatNat Init.Data.BitVec.Lemmas
∀ {w : ℕ}, Std.LawfulCommIdentity (fun x y => x * y) 1#w - BitVec.mul_comm Init.Data.BitVec.Lemmas
∀ {w : ℕ} (x y : BitVec w), x * y = y * x - CommMagma.mk Mathlib.Algebra.Group.Defs
{G : Type u} → [toMul : Mul G] → (∀ (a b : G), a * b = b * a) → CommMagma G - mul_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u_1} [inst : CommMagma G] (a b : G), a * b = b * a - CommMagma.mul_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u} [self : CommMagma G] (a b : G), a * b = b * a - CommSemigroup.mk Mathlib.Algebra.Group.Defs
{G : Type u} → [toSemigroup : Semigroup G] → (∀ (a b : G), a * b = b * a) → CommSemigroup G - CommSemigroup.mul_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u} [self : CommSemigroup G] (a b : G), a * b = b * a - CommMonoid.mk Mathlib.Algebra.Group.Defs
{M : Type u} → [toMonoid : Monoid M] → (∀ (a b : M), a * b = b * a) → CommMonoid M - CommMonoid.mul_comm Mathlib.Algebra.Group.Defs
∀ {M : Type u} [self : CommMonoid M] (a b : M), a * b = b * a - CancelCommMonoid.mk Mathlib.Algebra.Group.Defs
{M : Type u} → [toLeftCancelMonoid : LeftCancelMonoid M] → (∀ (a b : M), a * b = b * a) → CancelCommMonoid M - CancelCommMonoid.mul_comm Mathlib.Algebra.Group.Defs
∀ {M : Type u} [self : CancelCommMonoid M] (a b : M), a * b = b * a - CommGroup.mk Mathlib.Algebra.Group.Defs
{G : Type u} → [toGroup : Group G] → (∀ (a b : G), a * b = b * a) → CommGroup G - DivisionCommMonoid.mk Mathlib.Algebra.Group.Defs
{G : Type u} → [toDivisionMonoid : DivisionMonoid G] → (∀ (a b : G), a * b = b * a) → DivisionCommMonoid G - CommGroup.mul_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u} [self : CommGroup G] (a b : G), a * b = b * a - DivisionCommMonoid.mul_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u} [self : DivisionCommMonoid G] (a b : G), a * b = b * a - inv_mul_cancel_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u_1} [inst : CommGroup G] (a b : G), a⁻¹ * b * a = b - inv_mul_cancel_comm_assoc Mathlib.Algebra.Group.Defs
∀ {G : Type u_1} [inst : CommGroup G] (a b : G), a⁻¹ * (b * a) = b - mul_inv_cancel_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u_1} [inst : CommGroup G] (a b : G), a * b * a⁻¹ = b - mul_inv_cancel_comm_assoc Mathlib.Algebra.Group.Defs
∀ {G : Type u_1} [inst : CommGroup G] (a b : G), a * (b * a⁻¹) = b - pow_mul_comm' Mathlib.Algebra.Group.Defs
∀ {M : Type u_2} [inst : Monoid M] (a : M) (n : ℕ), a ^ n * a = a * a ^ n - pow_mul_comm Mathlib.Algebra.Group.Defs
∀ {M : Type u_2} [inst : Monoid M] (a : M) (m n : ℕ), a ^ m * a ^ n = a ^ n * a ^ m - Commute.eq Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : Mul S] {a b : S}, Commute a b → a * b = b * a - commute_iff_eq Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : Mul S] (a b : S), Commute a b ↔ a * b = b * a - Commute.mul_left Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : Semigroup S] {a b c : S}, Commute a c → Commute b c → Commute (a * b) c - Commute.mul_right Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : Semigroup S] {a b c : S}, Commute a b → Commute a c → Commute a (b * c) - Commute.left_comm Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : Semigroup S] {a b : S}, Commute a b → ∀ (c : S), a * (b * c) = b * (a * c) - Commute.right_comm Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : Semigroup S] {b c : S}, Commute b c → ∀ (a : S), a * b * c = a * c * b - Commute.mul_inv_cancel Mathlib.Algebra.Group.Commute.Defs
∀ {G : Type u_1} [inst : Group G] {a b : G}, Commute a b → a * b * a⁻¹ = b - Commute.mul_inv_cancel_assoc Mathlib.Algebra.Group.Commute.Defs
∀ {G : Type u_1} [inst : Group G] {a b : G}, Commute a b → a * (b * a⁻¹) = b - Commute.inv Mathlib.Algebra.Group.Commute.Defs
∀ {G : Type u_1} [inst : DivisionMonoid G] {a b : G}, Commute a b → (a * b)⁻¹ = a⁻¹ * b⁻¹ - Commute.mul_inv Mathlib.Algebra.Group.Commute.Defs
∀ {G : Type u_1} [inst : DivisionMonoid G] {a b : G}, Commute a b → (a * b)⁻¹ = a⁻¹ * b⁻¹ - Commute.mul_pow Mathlib.Algebra.Group.Commute.Defs
∀ {M : Type u_2} [inst : Monoid M] {a b : M}, Commute a b → ∀ (n : ℕ), (a * b) ^ n = a ^ n * b ^ n - Commute.mul_mul_mul_comm Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : Semigroup S] {b c : S}, Commute b c → ∀ (a d : S), a * b * (c * d) = a * c * (b * d) - Commute.mul_zpow Mathlib.Algebra.Group.Commute.Defs
∀ {G : Type u_1} [inst : DivisionMonoid G] {a b : G}, Commute a b → ∀ (n : ℤ), (a * b) ^ n = a ^ n * b ^ n - Rat.mul_comm Batteries.Data.Rat.Lemmas
∀ (a b : ℚ), a * b = b * a - Nat.div_mul_div_comm Mathlib.Data.Nat.Defs
∀ {a b c d : ℕ}, b ∣ a → d ∣ c → a / b * (c / d) = a * c / (b * d) - Nat.mul_div_mul_comm_of_dvd_dvd Mathlib.Data.Nat.Defs
∀ {a b c d : ℕ}, b ∣ a → d ∣ c → a * c / (b * d) = a / b * (c / d) - CommMagma.to_isCommutative Mathlib.Algebra.Group.Basic
∀ {G : Type u_3} [inst : CommMagma G], Std.Commutative fun x x_1 => x * x_1 - mul_left_comm Mathlib.Algebra.Group.Basic
∀ {G : Type u_3} [inst : CommSemigroup G] (a b c : G), a * (b * c) = b * (a * c) - mul_right_comm Mathlib.Algebra.Group.Basic
∀ {G : Type u_3} [inst : CommSemigroup G] (a b c : G), a * b * c = a * c * b - div_mul_comm Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : DivisionCommMonoid α] (a b c : α), a / b * c = c / b * a - mul_comm_div Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : DivisionCommMonoid α] (a b c : α), a / b * c = a * (c / b) - mul_div_left_comm Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : DivisionCommMonoid α] (a b c : α), a * (b / c) = b * (a / c) - mul_div_right_comm Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : DivisionCommMonoid α] (a b c : α), a * b / c = a / c * b - mul_mul_mul_comm Mathlib.Algebra.Group.Basic
∀ {G : Type u_3} [inst : CommSemigroup G] (a b c d : G), a * b * (c * d) = a * c * (b * d) - zpow_mul_comm Mathlib.Algebra.Group.Basic
∀ {G : Type u_3} [inst : Group G] (a : G) (m n : ℤ), a ^ m * a ^ n = a ^ n * a ^ m - div_mul_div_comm Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : DivisionCommMonoid α] (a b c d : α), a / b * (c / d) = a * c / (b * d) - mul_div_mul_comm Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : DivisionCommMonoid α] (a b c d : α), a * b / (c * d) = a / c * (b / d) - Function.Injective.commMagma Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Mul M₁] → [inst_1 : CommMagma M₂] → (f : M₁ → M₂) → Function.Injective f → (∀ (x y : M₁), f (x * y) = f x * f y) → CommMagma M₁ - Function.Surjective.commMagma Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Mul M₂] → [inst_1 : CommMagma M₁] → (f : M₁ → M₂) → Function.Surjective f → (∀ (x y : M₁), f (x * y) = f x * f y) → CommMagma M₂ - Function.Injective.commSemigroup Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Mul M₁] → [inst_1 : CommSemigroup M₂] → (f : M₁ → M₂) → Function.Injective f → (∀ (x y : M₁), f (x * y) = f x * f y) → CommSemigroup M₁ - Function.Surjective.commSemigroup Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Mul M₂] → [inst_1 : CommSemigroup M₁] → (f : M₁ → M₂) → Function.Surjective f → (∀ (x y : M₁), f (x * y) = f x * f y) → CommSemigroup M₂ - Function.Injective.commMagma.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Mul M₁] [inst_1 : CommMagma M₂] (f : M₁ → M₂) (hf : Function.Injective f) (mul : ∀ (x y : M₁), f (x * y) = f x * f y), Function.Injective.commMagma f hf mul = CommMagma.mk ⋯ - Function.Surjective.commMagma.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Mul M₂] [inst_1 : CommMagma M₁] (f : M₁ → M₂) (hf : Function.Surjective f) (mul : ∀ (x y : M₁), f (x * y) = f x * f y), Function.Surjective.commMagma f hf mul = CommMagma.mk ⋯ - Function.Injective.commSemigroup.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Mul M₁] [inst_1 : CommSemigroup M₂] (f : M₁ → M₂) (hf : Function.Injective f) (mul : ∀ (x y : M₁), f (x * y) = f x * f y), Function.Injective.commSemigroup f hf mul = let __spread.0 := Function.Injective.commMagma f hf mul; CommSemigroup.mk ⋯ - Function.Surjective.commSemigroup.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Mul M₂] [inst_1 : CommSemigroup M₁] (f : M₁ → M₂) (hf : Function.Surjective f) (mul : ∀ (x y : M₁), f (x * y) = f x * f y), Function.Surjective.commSemigroup f hf mul = let __spread.0 := Function.Surjective.commMagma f hf mul; CommSemigroup.mk ⋯ - Function.Injective.commMonoid Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Mul M₁] → [inst_1 : One M₁] → [inst_2 : Pow M₁ ℕ] → [inst_3 : CommMonoid M₂] → (f : M₁ → M₂) → Function.Injective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = f x * f y) → (∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n) → CommMonoid M₁ - Function.Surjective.commMonoid Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Mul M₂] → [inst_1 : One M₂] → [inst_2 : Pow M₂ ℕ] → [inst_3 : CommMonoid M₁] → (f : M₁ → M₂) → Function.Surjective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = f x * f y) → (∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n) → CommMonoid M₂ - Function.Injective.cancelCommMonoid Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Mul M₁] → [inst_1 : One M₁] → [inst_2 : Pow M₁ ℕ] → [inst_3 : CancelCommMonoid M₂] → (f : M₁ → M₂) → Function.Injective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = f x * f y) → (∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n) → CancelCommMonoid M₁ - Function.Injective.commMonoid.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Mul M₁] [inst_1 : One M₁] [inst_2 : Pow M₁ ℕ] [inst_3 : CommMonoid M₂] (f : M₁ → M₂) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : M₁), f (x * y) = f x * f y) (npow : ∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n), Function.Injective.commMonoid f hf one mul npow = let __src := Function.Injective.monoid f hf one mul npow; let __src_1 := Function.Injective.commSemigroup f hf mul; CommMonoid.mk ⋯ - Function.Surjective.commMonoid.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Mul M₂] [inst_1 : One M₂] [inst_2 : Pow M₂ ℕ] [inst_3 : CommMonoid M₁] (f : M₁ → M₂) (hf : Function.Surjective f) (one : f 1 = 1) (mul : ∀ (x y : M₁), f (x * y) = f x * f y) (npow : ∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n), Function.Surjective.commMonoid f hf one mul npow = let __src := Function.Surjective.commSemigroup f hf mul; let __src_1 := Function.Surjective.monoid f hf one mul npow; CommMonoid.mk ⋯ - Function.Injective.divisionCommMonoid Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Mul M₁] → [inst_1 : One M₁] → [inst_2 : Pow M₁ ℕ] → [inst_3 : Inv M₁] → [inst_4 : Div M₁] → [inst_5 : Pow M₁ ℤ] → [inst_6 : DivisionCommMonoid M₂] → (f : M₁ → M₂) → Function.Injective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = f x * f y) → (∀ (x : M₁), f x⁻¹ = (f x)⁻¹) → (∀ (x y : M₁), f (x / y) = f x / f y) → (∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (x : M₁) (n : ℤ), f (x ^ n) = f x ^ n) → DivisionCommMonoid M₁ - Function.Injective.commGroup Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Mul M₁] → [inst_1 : One M₁] → [inst_2 : Pow M₁ ℕ] → [inst_3 : Inv M₁] → [inst_4 : Div M₁] → [inst_5 : Pow M₁ ℤ] → [inst_6 : CommGroup M₂] → (f : M₁ → M₂) → Function.Injective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = f x * f y) → (∀ (x : M₁), f x⁻¹ = (f x)⁻¹) → (∀ (x y : M₁), f (x / y) = f x / f y) → (∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (x : M₁) (n : ℤ), f (x ^ n) = f x ^ n) → CommGroup M₁ - Function.Surjective.commGroup Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Mul M₂] → [inst_1 : One M₂] → [inst_2 : Pow M₂ ℕ] → [inst_3 : Inv M₂] → [inst_4 : Div M₂] → [inst_5 : Pow M₂ ℤ] → [inst_6 : CommGroup M₁] → (f : M₁ → M₂) → Function.Surjective f → f 1 = 1 → (∀ (x y : M₁), f (x * y) = f x * f y) → (∀ (x : M₁), f x⁻¹ = (f x)⁻¹) → (∀ (x y : M₁), f (x / y) = f x / f y) → (∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (x : M₁) (n : ℤ), f (x ^ n) = f x ^ n) → CommGroup M₂ - Function.Injective.cancelCommMonoid.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Mul M₁] [inst_1 : One M₁] [inst_2 : Pow M₁ ℕ] [inst_3 : CancelCommMonoid M₂] (f : M₁ → M₂) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : M₁), f (x * y) = f x * f y) (npow : ∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n), Function.Injective.cancelCommMonoid f hf one mul npow = let __src := Function.Injective.leftCancelSemigroup f hf mul; let __src_1 := Function.Injective.commMonoid f hf one mul npow; CancelCommMonoid.mk ⋯ - Function.Injective.divisionCommMonoid.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Mul M₁] [inst_1 : One M₁] [inst_2 : Pow M₁ ℕ] [inst_3 : Inv M₁] [inst_4 : Div M₁] [inst_5 : Pow M₁ ℤ] [inst_6 : DivisionCommMonoid M₂] (f : M₁ → M₂) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : M₁), f (x * y) = f x * f y) (inv : ∀ (x : M₁), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : M₁), f (x / y) = f x / f y) (npow : ∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : M₁) (n : ℤ), f (x ^ n) = f x ^ n), Function.Injective.divisionCommMonoid f hf one mul inv div npow zpow = let __src := Function.Injective.divisionMonoid f hf one mul inv div npow zpow; let __src_1 := Function.Injective.commSemigroup f hf mul; DivisionCommMonoid.mk ⋯ - Function.Injective.commGroup.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Mul M₁] [inst_1 : One M₁] [inst_2 : Pow M₁ ℕ] [inst_3 : Inv M₁] [inst_4 : Div M₁] [inst_5 : Pow M₁ ℤ] [inst_6 : CommGroup M₂] (f : M₁ → M₂) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : M₁), f (x * y) = f x * f y) (inv : ∀ (x : M₁), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : M₁), f (x / y) = f x / f y) (npow : ∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : M₁) (n : ℤ), f (x ^ n) = f x ^ n), Function.Injective.commGroup f hf one mul inv div npow zpow = let __src := Function.Injective.commMonoid f hf one mul npow; let __src_1 := Function.Injective.group f hf one mul inv div npow zpow; CommGroup.mk ⋯ - Function.Surjective.commGroup.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Mul M₂] [inst_1 : One M₂] [inst_2 : Pow M₂ ℕ] [inst_3 : Inv M₂] [inst_4 : Div M₂] [inst_5 : Pow M₂ ℤ] [inst_6 : CommGroup M₁] (f : M₁ → M₂) (hf : Function.Surjective f) (one : f 1 = 1) (mul : ∀ (x y : M₁), f (x * y) = f x * f y) (inv : ∀ (x : M₁), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : M₁), f (x / y) = f x / f y) (npow : ∀ (x : M₁) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : M₁) (n : ℤ), f (x ^ n) = f x ^ n), Function.Surjective.commGroup f hf one mul inv div npow zpow = let __src := Function.Surjective.commMonoid f hf one mul npow; let __src_1 := Function.Surjective.group f hf one mul inv div npow zpow; CommGroup.mk ⋯ - CommGroupWithZero.zpow_succ' Mathlib.Algebra.GroupWithZero.Defs
∀ {G₀ : Type u_4} [self : CommGroupWithZero G₀] (n : ℕ) (a : G₀), CommGroupWithZero.zpow (Int.ofNat n.succ) a = CommGroupWithZero.zpow (Int.ofNat n) a * a - CommMonoidWithZero.mul_zero Mathlib.Algebra.GroupWithZero.Defs
∀ {M₀ : Type u_4} [self : CommMonoidWithZero M₀] (a : M₀), a * 0 = 0 - CommMonoidWithZero.zero_mul Mathlib.Algebra.GroupWithZero.Defs
∀ {M₀ : Type u_4} [self : CommMonoidWithZero M₀] (a : M₀), 0 * a = 0 - CommGroupWithZero.div_eq_mul_inv Mathlib.Algebra.GroupWithZero.Defs
∀ {G₀ : Type u_4} [self : CommGroupWithZero G₀] (a b : G₀), a / b = a * b⁻¹ - mul_eq_zero_comm Mathlib.Algebra.GroupWithZero.Defs
∀ {M₀ : Type u_1} [inst : MulZeroClass M₀] [inst_1 : NoZeroDivisors M₀] {a b : M₀}, a * b = 0 ↔ b * a = 0 - mul_ne_zero_comm Mathlib.Algebra.GroupWithZero.Defs
∀ {M₀ : Type u_1} [inst : MulZeroClass M₀] [inst_1 : NoZeroDivisors M₀] {a b : M₀}, a * b ≠ 0 ↔ b * a ≠ 0 - CommGroupWithZero.mul_inv_cancel Mathlib.Algebra.GroupWithZero.Defs
∀ {G₀ : Type u_4} [self : CommGroupWithZero G₀] (a : G₀), a ≠ 0 → a * a⁻¹ = 1 - CommMonoidWithZero.mk Mathlib.Algebra.GroupWithZero.Defs
{M₀ : Type u_4} → [toCommMonoid : CommMonoid M₀] → [toZero : Zero M₀] → (∀ (a : M₀), 0 * a = 0) → (∀ (a : M₀), a * 0 = 0) → CommMonoidWithZero M₀ - CommGroupWithZero.mk Mathlib.Algebra.GroupWithZero.Defs
{G₀ : Type u_4} → [toCommMonoidWithZero : CommMonoidWithZero G₀] → [toInv : Inv G₀] → [toDiv : Div G₀] → autoParam (∀ (a b : G₀), a / b = a * b⁻¹) _auto✝ → (zpow : ℤ → G₀ → G₀) → autoParam (∀ (a : G₀), zpow 0 a = 1) _auto✝¹ → autoParam (∀ (n : ℕ) (a : G₀), zpow (Int.ofNat n.succ) a = zpow (Int.ofNat n) a * a) _auto✝² → autoParam (∀ (n : ℕ) (a : G₀), zpow (Int.negSucc n) a = (zpow (↑n.succ) a)⁻¹) _auto✝³ → [toNontrivial : Nontrivial G₀] → 0⁻¹ = 0 → (∀ (a : G₀), a ≠ 0 → a * a⁻¹ = 1) → CommGroupWithZero G₀ - mul_smul_comm Mathlib.GroupTheory.GroupAction.Defs
∀ {α : Type u_6} {β : Type u_7} [inst : Mul β] [inst_1 : SMul α β] [inst_2 : SMulCommClass α β β] (s : α) (x y : β), x * s • y = s • (x * y) - SMulCommClass.of_mul_smul_one Mathlib.GroupTheory.GroupAction.Defs
∀ {M : Type u_10} {N : Type u_11} [inst : Monoid N] [inst_1 : SMul M N], (∀ (x : M) (y : N), y * x • 1 = x • y) → SMulCommClass M N N - Commute.function_commute_mul_left Mathlib.Algebra.GroupPower.IterateHom
∀ {G : Type u_3} [inst : Semigroup G] {a b : G}, Commute a b → Function.Commute (fun x => a * x) fun x => b * x - Commute.function_commute_mul_right Mathlib.Algebra.GroupPower.IterateHom
∀ {G : Type u_3} [inst : Semigroup G] {a b : G}, Commute a b → Function.Commute (fun x => x * a) fun x => x * b - NonUnitalNonAssocCommRing.mk Mathlib.Algebra.Ring.Defs
{α : Type u} → [toNonUnitalNonAssocRing : NonUnitalNonAssocRing α] → (∀ (a b : α), a * b = b * a) → NonUnitalNonAssocCommRing α - NonUnitalNonAssocCommSemiring.mk Mathlib.Algebra.Ring.Defs
{α : Type u} → [toNonUnitalNonAssocSemiring : NonUnitalNonAssocSemiring α] → (∀ (a b : α), a * b = b * a) → NonUnitalNonAssocCommSemiring α - NonUnitalNonAssocCommRing.mul_comm Mathlib.Algebra.Ring.Defs
∀ {α : Type u} [self : NonUnitalNonAssocCommRing α] (a b : α), a * b = b * a - NonUnitalNonAssocCommSemiring.mul_comm Mathlib.Algebra.Ring.Defs
∀ {α : Type u} [self : NonUnitalNonAssocCommSemiring α] (a b : α), a * b = b * a - NonUnitalCommRing.mk Mathlib.Algebra.Ring.Defs
{α : Type u} → [toNonUnitalRing : NonUnitalRing α] → (∀ (a b : α), a * b = b * a) → NonUnitalCommRing α - NonUnitalCommSemiring.mk Mathlib.Algebra.Ring.Defs
{α : Type u} → [toNonUnitalSemiring : NonUnitalSemiring α] → (∀ (a b : α), a * b = b * a) → NonUnitalCommSemiring α - NonUnitalCommRing.mul_comm Mathlib.Algebra.Ring.Defs
∀ {α : Type u} [self : NonUnitalCommRing α] (a b : α), a * b = b * a - NonUnitalCommSemiring.mul_comm Mathlib.Algebra.Ring.Defs
∀ {α : Type u} [self : NonUnitalCommSemiring α] (a b : α), a * b = b * a - CommSemiring.mk Mathlib.Algebra.Ring.Defs
{R : Type u} → [toSemiring : Semiring R] → (∀ (a b : R), a * b = b * a) → CommSemiring R - CommSemiring.mul_comm Mathlib.Algebra.Ring.Defs
∀ {R : Type u} [self : CommSemiring R] (a b : R), a * b = b * a - CommRing.mk Mathlib.Algebra.Ring.Defs
{α : Type u} → [toRing : Ring α] → (∀ (a b : α), a * b = b * a) → CommRing α - neg_mul_comm Mathlib.Algebra.Ring.Defs
∀ {α : Type u} [inst : Mul α] [inst_1 : HasDistribNeg α] (a b : α), -a * b = a * -b - CommRing.mul_comm Mathlib.Algebra.Ring.Defs
∀ {α : Type u} [self : CommRing α] (a b : α), a * b = b * a - MonoidHom.commMonoid.eq_1 Mathlib.Algebra.Group.Hom.Instances
∀ {M : Type uM} {N : Type uN} [inst : MulOneClass M] [inst_1 : CommMonoid N], MonoidHom.commMonoid = CommMonoid.mk ⋯ - Commute.isUnit_mul_iff Mathlib.Algebra.Group.Commute.Units
∀ {M : Type u_1} [inst : Monoid M] {a b : M}, Commute a b → (IsUnit (a * b) ↔ IsUnit a ∧ IsUnit b) - Commute.div_eq_div_iff_of_isUnit Mathlib.Algebra.Group.Commute.Units
∀ {M : Type u_1} [inst : DivisionMonoid M] {a b c d : M}, Commute b d → IsUnit b → IsUnit d → (a / b = c / d ↔ a * d = c * b) - pow_inv_comm₀ Mathlib.Algebra.GroupWithZero.Commute
∀ {G₀ : Type u_8} [inst : GroupWithZero G₀] (a : G₀) (m n : ℕ), a⁻¹ ^ m * a ^ n = a ^ n * a⁻¹ ^ m - Function.Injective.commMonoidWithZero Mathlib.Algebra.GroupWithZero.InjSurj
{M₀ : Type u_1} → {M₀' : Type u_3} → [inst : Zero M₀'] → [inst_1 : Mul M₀'] → [inst_2 : One M₀'] → [inst_3 : Pow M₀' ℕ] → [inst_4 : CommMonoidWithZero M₀] → (f : M₀' → M₀) → Function.Injective f → f 0 = 0 → f 1 = 1 → (∀ (x y : M₀'), f (x * y) = f x * f y) → (∀ (x : M₀') (n : ℕ), f (x ^ n) = f x ^ n) → CommMonoidWithZero M₀' - Function.Surjective.commMonoidWithZero Mathlib.Algebra.GroupWithZero.InjSurj
{M₀ : Type u_1} → {M₀' : Type u_3} → [inst : Zero M₀'] → [inst_1 : Mul M₀'] → [inst_2 : One M₀'] → [inst_3 : Pow M₀' ℕ] → [inst_4 : CommMonoidWithZero M₀] → (f : M₀ → M₀') → Function.Surjective f → f 0 = 0 → f 1 = 1 → (∀ (x y : M₀), f (x * y) = f x * f y) → (∀ (x : M₀) (n : ℕ), f (x ^ n) = f x ^ n) → CommMonoidWithZero M₀' - Function.Injective.cancelCommMonoidWithZero Mathlib.Algebra.GroupWithZero.InjSurj
{M₀ : Type u_1} → {M₀' : Type u_3} → [inst : CancelCommMonoidWithZero M₀] → [inst_1 : Zero M₀'] → [inst_2 : Mul M₀'] → [inst_3 : One M₀'] → [inst_4 : Pow M₀' ℕ] → (f : M₀' → M₀) → Function.Injective f → f 0 = 0 → f 1 = 1 → (∀ (x y : M₀'), f (x * y) = f x * f y) → (∀ (x : M₀') (n : ℕ), f (x ^ n) = f x ^ n) → CancelCommMonoidWithZero M₀' - Function.Injective.commGroupWithZero Mathlib.Algebra.GroupWithZero.InjSurj
{G₀ : Type u_2} → {G₀' : Type u_4} → [inst : CommGroupWithZero G₀] → [inst_1 : Zero G₀'] → [inst_2 : Mul G₀'] → [inst_3 : One G₀'] → [inst_4 : Inv G₀'] → [inst_5 : Div G₀'] → [inst_6 : Pow G₀' ℕ] → [inst_7 : Pow G₀' ℤ] → (f : G₀' → G₀) → Function.Injective f → f 0 = 0 → f 1 = 1 → (∀ (x y : G₀'), f (x * y) = f x * f y) → (∀ (x : G₀'), f x⁻¹ = (f x)⁻¹) → (∀ (x y : G₀'), f (x / y) = f x / f y) → (∀ (x : G₀') (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (x : G₀') (n : ℤ), f (x ^ n) = f x ^ n) → CommGroupWithZero G₀' - Function.Surjective.commGroupWithZero Mathlib.Algebra.GroupWithZero.InjSurj
{G₀ : Type u_2} → {G₀' : Type u_4} → [inst : CommGroupWithZero G₀] → [inst_1 : Zero G₀'] → [inst_2 : Mul G₀'] → [inst_3 : One G₀'] → [inst_4 : Inv G₀'] → [inst_5 : Div G₀'] → [inst_6 : Pow G₀' ℕ] → [inst_7 : Pow G₀' ℤ] → 0 ≠ 1 → (f : G₀ → G₀') → Function.Surjective f → f 0 = 0 → f 1 = 1 → (∀ (x y : G₀), f (x * y) = f x * f y) → (∀ (x : G₀), f x⁻¹ = (f x)⁻¹) → (∀ (x y : G₀), f (x / y) = f x / f y) → (∀ (x : G₀) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (x : G₀) (n : ℤ), f (x ^ n) = f x ^ n) → CommGroupWithZero G₀' - Function.Injective.nonUnitalNonAssocCommSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : SMul ℕ β] → (f : β → α) → Function.Injective f → [inst_4 : NonUnitalNonAssocCommSemiring α] → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → NonUnitalNonAssocCommSemiring β - Function.Surjective.nonUnitalNonAssocCommSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : SMul ℕ β] → (f : α → β) → Function.Surjective f → [inst_4 : NonUnitalNonAssocCommSemiring α] → f 0 = 0 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → NonUnitalNonAssocCommSemiring β - Function.Injective.nonUnitalCommSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : SMul ℕ β] → [inst_4 : NonUnitalCommSemiring α] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → NonUnitalCommSemiring β - Function.Surjective.nonUnitalCommSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : SMul ℕ β] → (f : α → β) → Function.Surjective f → [inst_4 : NonUnitalCommSemiring α] → f 0 = 0 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → NonUnitalCommSemiring β - Function.Injective.commSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : One β] → [inst_2 : Add β] → [inst_3 : Mul β] → [inst_4 : SMul ℕ β] → [inst_5 : Pow β ℕ] → [inst_6 : NatCast β] → (f : β → α) → Function.Injective f → [inst_7 : CommSemiring α] → f 0 = 0 → f 1 = 1 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → (∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (n : ℕ), f ↑n = ↑n) → CommSemiring β - Function.Surjective.commSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : One β] → [inst_2 : Add β] → [inst_3 : Mul β] → [inst_4 : SMul ℕ β] → [inst_5 : Pow β ℕ] → [inst_6 : NatCast β] → (f : α → β) → Function.Surjective f → [inst_7 : CommSemiring α] → f 0 = 0 → f 1 = 1 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → (∀ (x : α) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (n : ℕ), f ↑n = ↑n) → CommSemiring β - Function.Injective.nonUnitalNonAssocCommRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : Neg β] → [inst_4 : Sub β] → [inst_5 : SMul ℕ β] → [inst_6 : SMul ℤ β] → [inst_7 : NonUnitalNonAssocCommRing α] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (x : β), f (-x) = -f x) → (∀ (x y : β), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → (∀ (n : ℤ) (x : β), f (n • x) = n • f x) → NonUnitalNonAssocCommRing β - Function.Surjective.nonUnitalNonAssocCommRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : Neg β] → [inst_4 : Sub β] → [inst_5 : SMul ℕ β] → [inst_6 : SMul ℤ β] → (f : α → β) → Function.Surjective f → [inst_7 : NonUnitalNonAssocCommRing α] → f 0 = 0 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (x : α), f (-x) = -f x) → (∀ (x y : α), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → (∀ (n : ℤ) (x : α), f (n • x) = n • f x) → NonUnitalNonAssocCommRing β - Function.Injective.nonUnitalCommRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : Neg β] → [inst_4 : Sub β] → [inst_5 : SMul ℕ β] → [inst_6 : SMul ℤ β] → [inst_7 : NonUnitalCommRing α] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (x : β), f (-x) = -f x) → (∀ (x y : β), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → (∀ (n : ℤ) (x : β), f (n • x) = n • f x) → NonUnitalCommRing β - Function.Surjective.nonUnitalCommRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : Neg β] → [inst_4 : Sub β] → [inst_5 : SMul ℕ β] → [inst_6 : SMul ℤ β] → (f : α → β) → Function.Surjective f → [inst_7 : NonUnitalCommRing α] → f 0 = 0 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (x : α), f (-x) = -f x) → (∀ (x y : α), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → (∀ (n : ℤ) (x : α), f (n • x) = n • f x) → NonUnitalCommRing β - Function.Injective.commRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : One β] → [inst_2 : Add β] → [inst_3 : Mul β] → [inst_4 : Neg β] → [inst_5 : Sub β] → [inst_6 : SMul ℕ β] → [inst_7 : SMul ℤ β] → [inst_8 : Pow β ℕ] → [inst_9 : NatCast β] → [inst_10 : IntCast β] → (f : β → α) → Function.Injective f → [inst_11 : CommRing α] → f 0 = 0 → f 1 = 1 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (x : β), f (-x) = -f x) → (∀ (x y : β), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → (∀ (n : ℤ) (x : β), f (n • x) = n • f x) → (∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (n : ℕ), f ↑n = ↑n) → (∀ (n : ℤ), f ↑n = ↑n) → CommRing β - Function.Surjective.commRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : One β] → [inst_2 : Add β] → [inst_3 : Mul β] → [inst_4 : Neg β] → [inst_5 : Sub β] → [inst_6 : SMul ℕ β] → [inst_7 : SMul ℤ β] → [inst_8 : Pow β ℕ] → [inst_9 : NatCast β] → [inst_10 : IntCast β] → (f : α → β) → Function.Surjective f → [inst_11 : CommRing α] → f 0 = 0 → f 1 = 1 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (x : α), f (-x) = -f x) → (∀ (x y : α), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → (∀ (n : ℤ) (x : α), f (n • x) = n • f x) → (∀ (x : α) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (n : ℕ), f ↑n = ↑n) → (∀ (n : ℤ), f ↑n = ↑n) → CommRing β - Commute.mul_self_eq_mul_self_iff Mathlib.Algebra.Ring.Commute
∀ {R : Type x} [inst : NonUnitalNonAssocRing R] [inst_1 : NoZeroDivisors R] {a b : R}, Commute a b → (a * a = b * b ↔ a = b ∨ a = -b) - Commute.mul_self_sub_mul_self_eq Mathlib.Algebra.Ring.Commute
∀ {R : Type x} [inst : NonUnitalNonAssocRing R] {a b : R}, Commute a b → a * a - b * b = (a + b) * (a - b) - Commute.mul_self_sub_mul_self_eq' Mathlib.Algebra.Ring.Commute
∀ {R : Type x} [inst : NonUnitalNonAssocRing R] {a b : R}, Commute a b → a * a - b * b = (a - b) * (a + b) - Commute.sq_sub_sq Mathlib.Algebra.Ring.Commute
∀ {R : Type x} [inst : Ring R] {a b : R}, Commute a b → a ^ 2 - b ^ 2 = (a + b) * (a - b) - Commute.cast_nat_mul_self Mathlib.Data.Nat.Cast.Commute
∀ {α : Type u_1} [inst : Semiring α] (a : α) (n : ℕ), Commute (↑n * a) a - Commute.self_cast_nat_mul Mathlib.Data.Nat.Cast.Commute
∀ {α : Type u_1} [inst : Semiring α] (a : α) (n : ℕ), Commute a (↑n * a) - Nat.cast_comm Mathlib.Data.Nat.Cast.Commute
∀ {α : Type u_1} [inst : NonAssocSemiring α] (n : ℕ) (x : α), ↑n * x = x * ↑n - Commute.cast_nat_mul_left Mathlib.Data.Nat.Cast.Commute
∀ {α : Type u_1} [inst : Semiring α] {a b : α}, Commute a b → ∀ (n : ℕ), Commute (↑n * a) b - Commute.cast_nat_mul_right Mathlib.Data.Nat.Cast.Commute
∀ {α : Type u_1} [inst : Semiring α] {a b : α}, Commute a b → ∀ (n : ℕ), Commute a (↑n * b) - Commute.self_cast_nat_mul_cast_nat_mul Mathlib.Data.Nat.Cast.Commute
∀ {α : Type u_1} [inst : Semiring α] (a : α) (m n : ℕ), Commute (↑m * a) (↑n * a) - Commute.cast_nat_mul_cast_nat_mul Mathlib.Data.Nat.Cast.Commute
∀ {α : Type u_1} [inst : Semiring α] {a b : α}, Commute a b → ∀ (m n : ℕ), Commute (↑m * a) (↑n * b) - Commute.inv_mul_cancel Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : Group G] {a b : G}, Commute a b → a⁻¹ * b * a = b - Commute.inv_mul_cancel_assoc Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : Group G] {a b : G}, Commute a b → a⁻¹ * (b * a) = b - Commute.conj Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : Group G] {a b : G}, Commute a b → ∀ (h : G), Commute (h * a * h⁻¹) (h * b * h⁻¹) - Commute.conj_iff Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : Group G] {a b : G} (h : G), Commute (h * a * h⁻¹) (h * b * h⁻¹) ↔ Commute a b - pow_inv_comm Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : Group G] (a : G) (m n : ℕ), a⁻¹ ^ m * a ^ n = a ^ n * a⁻¹ ^ m - Commute.div_mul_div_comm Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : DivisionMonoid G] {a b c d : G}, Commute b d → Commute b⁻¹ c → a / b * (c / d) = a * c / (b * d) - Commute.mul_div_mul_comm Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : DivisionMonoid G] {a b c d : G}, Commute c d → Commute b c⁻¹ → a * b / (c * d) = a / c * (b / d) - Commute.div_eq_div_iff Mathlib.Algebra.GroupWithZero.Units.Lemmas
∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a b c d : G₀}, Commute b d → b ≠ 0 → d ≠ 0 → (a / b = c / d ↔ a * d = c * b) - OrderedCommMonoid.toCovariantClassLeft Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_1} [inst : OrderedCommMonoid α], CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 - OrderedCancelCommMonoid.toContravariantClassLeLeft Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_1} [inst : OrderedCancelCommMonoid α], ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 - OrderedCancelCommMonoid.toContravariantClassLeft Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_1} [inst : OrderedCancelCommMonoid α], ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1 - OrderedCommMonoid.toCovariantClassRight Mathlib.Algebra.Order.Monoid.Defs
∀ (M : Type u_3) [inst : OrderedCommMonoid M], CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 - OrderedCancelCommMonoid.toContravariantClassRight Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_1} [inst : OrderedCancelCommMonoid α], ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1 - OrderedCommMonoid.mk Mathlib.Algebra.Order.Monoid.Defs
{α : Type u_3} → [toCommMonoid : CommMonoid α] → [toPartialOrder : PartialOrder α] → (∀ (a b : α), a ≤ b → ∀ (c : α), c * a ≤ c * b) → OrderedCommMonoid α - OrderedCommMonoid.mul_le_mul_left Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_3} [self : OrderedCommMonoid α] (a b : α), a ≤ b → ∀ (c : α), c * a ≤ c * b - OrderedCancelCommMonoid.mk Mathlib.Algebra.Order.Monoid.Defs
{α : Type u_3} → [toOrderedCommMonoid : OrderedCommMonoid α] → (∀ (a b c : α), a * b ≤ a * c → b ≤ c) → OrderedCancelCommMonoid α - OrderedCancelCommMonoid.le_of_mul_le_mul_left Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_3} [self : OrderedCancelCommMonoid α] (a b c : α), a * b ≤ a * c → b ≤ c - OrderedCommGroup.to_contravariantClass_left_le Mathlib.Algebra.Order.Group.Defs
∀ (α : Type u) [inst : OrderedCommGroup α], ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 - OrderedCommGroup.to_covariantClass_left_le Mathlib.Algebra.Order.Group.Defs
∀ (α : Type u) [inst : OrderedCommGroup α], CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 - OrderedCommGroup.to_contravariantClass_right_le Mathlib.Algebra.Order.Group.Defs
∀ (α : Type u) [inst : OrderedCommGroup α], ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 - OrderedCommGroup.le_of_mul_le_mul_left Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u_1} [inst : Mul α] [inst_1 : LE α] [inst_2 : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}, a * b ≤ a * c → b ≤ c - OrderedCommGroup.lt_of_mul_lt_mul_left Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u_1} [inst : Mul α] [inst_1 : LT α] [inst_2 : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}, a * b < a * c → b < c - OrderedCommGroup.mul_lt_mul_left' Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u_1} [inst : Mul α] [inst_1 : LT α] [inst_2 : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {b c : α}, b < c → ∀ (a : α), a * b < a * c - OrderedCommGroup.mk Mathlib.Algebra.Order.Group.Defs
{α : Type u} → [toCommGroup : CommGroup α] → [toPartialOrder : PartialOrder α] → (∀ (a b : α), a ≤ b → ∀ (c : α), c * a ≤ c * b) → OrderedCommGroup α - OrderedCommGroup.mul_le_mul_left Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [self : OrderedCommGroup α] (a b : α), a ≤ b → ∀ (c : α), c * a ≤ c * b - LinearOrderedCommGroup.mul_lt_mul_left' Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [inst : LinearOrderedCommGroup α] (a b : α), a < b → ∀ (c : α), c * a < c * b - div_le_comm Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [inst : CommGroup α] [inst_1 : LE α] [inst_2 : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}, a / b ≤ c ↔ a / c ≤ b - div_lt_comm Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [inst : CommGroup α] [inst_1 : LT α] [inst_2 : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}, a / b < c ↔ a / c < b - le_div_comm Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [inst : CommGroup α] [inst_1 : LE α] [inst_2 : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}, a ≤ b / c ↔ c ≤ b / a - lt_div_comm Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [inst : CommGroup α] [inst_1 : LT α] [inst_2 : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}, a < b / c ↔ c < b / a - CanonicallyOrderedCommMonoid.le_self_mul Mathlib.Algebra.Order.Monoid.Canonical.Defs
∀ {α : Type u_1} [self : CanonicallyOrderedCommMonoid α] (a b : α), a ≤ a * b - CanonicallyOrderedCommMonoid.exists_mul_of_le Mathlib.Algebra.Order.Monoid.Canonical.Defs
∀ {α : Type u_1} [self : CanonicallyOrderedCommMonoid α] {a b : α}, a ≤ b → ∃ c, b = a * c - CanonicallyOrderedCommMonoid.mk Mathlib.Algebra.Order.Monoid.Canonical.Defs
{α : Type u_1} → [toOrderedCommMonoid : OrderedCommMonoid α] → [toOrderBot : OrderBot α] → (∀ {a b : α}, a ≤ b → ∃ c, b = a * c) → (∀ (a b : α), a ≤ a * b) → CanonicallyOrderedCommMonoid α - OrderedCommSemiring.mk Mathlib.Algebra.Order.Ring.Defs
{α : Type u} → [toOrderedSemiring : OrderedSemiring α] → (∀ (a b : α), a * b = b * a) → OrderedCommSemiring α - StrictOrderedCommSemiring.mk Mathlib.Algebra.Order.Ring.Defs
{α : Type u} → [toStrictOrderedSemiring : StrictOrderedSemiring α] → (∀ (a b : α), a * b = b * a) → StrictOrderedCommSemiring α - OrderedCommSemiring.mul_comm Mathlib.Algebra.Order.Ring.Defs
∀ {α : Type u} [self : OrderedCommSemiring α] (a b : α), a * b = b * a - StrictOrderedCommSemiring.mul_comm Mathlib.Algebra.Order.Ring.Defs
∀ {α : Type u} [self : StrictOrderedCommSemiring α] (a b : α), a * b = b * a - OrderedCommRing.mk Mathlib.Algebra.Order.Ring.Defs
{α : Type u} → [toOrderedRing : OrderedRing α] → (∀ (a b : α), a * b = b * a) → OrderedCommRing α - StrictOrderedCommRing.mk Mathlib.Algebra.Order.Ring.Defs
{α : Type u_2} → [toStrictOrderedRing : StrictOrderedRing α] → (∀ (a b : α), a * b = b * a) → StrictOrderedCommRing α - OrderedCommRing.mul_comm Mathlib.Algebra.Order.Ring.Defs
∀ {α : Type u} [self : OrderedCommRing α] (a b : α), a * b = b * a - StrictOrderedCommRing.mul_comm Mathlib.Algebra.Order.Ring.Defs
∀ {α : Type u_2} [self : StrictOrderedCommRing α] (a b : α), a * b = b * a - LinearOrderedCommRing.mk Mathlib.Algebra.Order.Ring.Defs
{α : Type u} → [toLinearOrderedRing : LinearOrderedRing α] → (∀ (a b : α), a * b = b * a) → LinearOrderedCommRing α - LinearOrderedCommRing.mul_comm Mathlib.Algebra.Order.Ring.Defs
∀ {α : Type u} [self : LinearOrderedCommRing α] (a b : α), a * b = b * a - CommGroup.toDistribLattice Mathlib.Algebra.Order.Group.Lattice
(α : Type u_3) → [inst : Lattice α] → [inst_1 : CommGroup α] → [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] → DistribLattice α - CommGroup.toDistribLattice.eq_1 Mathlib.Algebra.Order.Group.Lattice
∀ (α : Type u_3) [inst : Lattice α] [inst_1 : CommGroup α] [inst_2 : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1], CommGroup.toDistribLattice α = DistribLattice.mk ⋯ - Function.Injective.orderedCommMonoid Mathlib.Algebra.Order.Monoid.Basic
{α : Type u} → [inst : OrderedCommMonoid α] → {β : Type u_2} → [inst_1 : One β] → [inst_2 : Mul β] → [inst_3 : Pow β ℕ] → (f : β → α) → Function.Injective f → f 1 = 1 → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) → OrderedCommMonoid β - Function.Injective.orderedCancelCommMonoid Mathlib.Algebra.Order.Monoid.Basic
{α : Type u} → {β : Type u_1} → [inst : OrderedCancelCommMonoid α] → [inst_1 : One β] → [inst_2 : Mul β] → [inst_3 : Pow β ℕ] → (f : β → α) → Function.Injective f → f 1 = 1 → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) → OrderedCancelCommMonoid β - Function.Injective.orderedCancelCommMonoid.eq_1 Mathlib.Algebra.Order.Monoid.Basic
∀ {α : Type u} {β : Type u_1} [inst : OrderedCancelCommMonoid α] [inst_1 : One β] [inst_2 : Mul β] [inst_3 : Pow β ℕ] (f : β → α) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n), Function.Injective.orderedCancelCommMonoid f hf one mul npow = OrderedCancelCommMonoid.mk ⋯ - Function.Injective.orderedCommMonoid.eq_1 Mathlib.Algebra.Order.Monoid.Basic
∀ {α : Type u} [inst : OrderedCommMonoid α] {β : Type u_2} [inst_1 : One β] [inst_2 : Mul β] [inst_3 : Pow β ℕ] (f : β → α) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n), Function.Injective.orderedCommMonoid f hf one mul npow = OrderedCommMonoid.mk ⋯ - Function.Injective.linearOrderedCommMonoid Mathlib.Algebra.Order.Monoid.Basic
{α : Type u} → [inst : LinearOrderedCommMonoid α] → {β : Type u_2} → [inst_1 : One β] → [inst_2 : Mul β] → [inst_3 : Pow β ℕ] → [inst_4 : Sup β] → [inst_5 : Inf β] → (f : β → α) → Function.Injective f → f 1 = 1 → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) → (∀ (x y : β), f (x ⊓ y) = min (f x) (f y)) → LinearOrderedCommMonoid β - Function.Injective.linearOrderedCancelCommMonoid Mathlib.Algebra.Order.Monoid.Basic
{α : Type u} → {β : Type u_1} → [inst : LinearOrderedCancelCommMonoid α] → [inst_1 : One β] → [inst_2 : Mul β] → [inst_3 : Pow β ℕ] → [inst_4 : Sup β] → [inst_5 : Inf β] → (f : β → α) → Function.Injective f → f 1 = 1 → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) → (∀ (x y : β), f (x ⊓ y) = min (f x) (f y)) → LinearOrderedCancelCommMonoid β - Function.Injective.linearOrderedCommMonoid.eq_1 Mathlib.Algebra.Order.Monoid.Basic
∀ {α : Type u} [inst : LinearOrderedCommMonoid α] {β : Type u_2} [inst_1 : One β] [inst_2 : Mul β] [inst_3 : Pow β ℕ] [inst_4 : Sup β] [inst_5 : Inf β] (f : β → α) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (sup : ∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) (inf : ∀ (x y : β), f (x ⊓ y) = min (f x) (f y)), Function.Injective.linearOrderedCommMonoid f hf one mul npow sup inf = let __spread.0 := LinearOrder.lift f hf sup inf; LinearOrderedCommMonoid.mk ⋯ LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT ⋯ ⋯ ⋯ - Function.Injective.linearOrderedCancelCommMonoid.eq_1 Mathlib.Algebra.Order.Monoid.Basic
∀ {α : Type u} {β : Type u_1} [inst : LinearOrderedCancelCommMonoid α] [inst_1 : One β] [inst_2 : Mul β] [inst_3 : Pow β ℕ] [inst_4 : Sup β] [inst_5 : Inf β] (f : β → α) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (hsup : ∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x ⊓ y) = min (f x) (f y)), Function.Injective.linearOrderedCancelCommMonoid f hf one mul npow hsup hinf = let __spread.0 := Function.Injective.linearOrderedCommMonoid f hf one mul npow hsup hinf; LinearOrderedCancelCommMonoid.mk ⋯ LinearOrderedCommMonoid.decidableLE LinearOrderedCommMonoid.decidableEq LinearOrderedCommMonoid.decidableLT ⋯ ⋯ ⋯ - LinearOrderedCommGroupWithZero.zpow_succ' Mathlib.Algebra.Order.GroupWithZero.Canonical
∀ {α : Type u_2} [self : LinearOrderedCommGroupWithZero α] (n : ℕ) (a : α), LinearOrderedCommGroupWithZero.zpow (Int.ofNat n.succ) a = LinearOrderedCommGroupWithZero.zpow (Int.ofNat n) a * a - LinearOrderedCommMonoidWithZero.mul_zero Mathlib.Algebra.Order.GroupWithZero.Canonical
∀ {α : Type u_2} [self : LinearOrderedCommMonoidWithZero α] (a : α), a * 0 = 0 - LinearOrderedCommMonoidWithZero.zero_mul Mathlib.Algebra.Order.GroupWithZero.Canonical
∀ {α : Type u_2} [self : LinearOrderedCommMonoidWithZero α] (a : α), 0 * a = 0 - LinearOrderedCommGroupWithZero.div_eq_mul_inv Mathlib.Algebra.Order.GroupWithZero.Canonical
∀ {α : Type u_2} [self : LinearOrderedCommGroupWithZero α] (a b : α), a / b = a * b⁻¹ - LinearOrderedCommGroupWithZero.mul_inv_cancel Mathlib.Algebra.Order.GroupWithZero.Canonical
∀ {α : Type u_2} [self : LinearOrderedCommGroupWithZero α] (a : α), a ≠ 0 → a * a⁻¹ = 1 - LinearOrderedCommMonoidWithZero.mk Mathlib.Algebra.Order.GroupWithZero.Canonical
{α : Type u_2} → [toLinearOrderedCommMonoid : LinearOrderedCommMonoid α] → [toZero : Zero α] → (∀ (a : α), 0 * a = 0) → (∀ (a : α), a * 0 = 0) → 0 ≤ 1 → LinearOrderedCommMonoidWithZero α - Function.Injective.linearOrderedCommMonoidWithZero Mathlib.Algebra.Order.GroupWithZero.Canonical
{α : Type u_1} → [inst : LinearOrderedCommMonoidWithZero α] → {β : Type u_2} → [inst_1 : Zero β] → [inst_2 : One β] → [inst_3 : Mul β] → [inst_4 : Pow β ℕ] → [inst_5 : Sup β] → [inst_6 : Inf β] → (f : β → α) → Function.Injective f → f 0 = 0 → f 1 = 1 → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) → (∀ (x y : β), f (x ⊓ y) = min (f x) (f y)) → LinearOrderedCommMonoidWithZero β
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About
Loogle searches of Lean and Mathlib definitions and theorems.
You may also want to try the CLI version, the VS
Code extension, the lean.nvim
integration or the Zulip bot.
Usage
Loogle finds definitions and lemmas in various ways:
By constant:
🔍Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
🔍"differ"
finds all lemmas that have"differ"
somewhere in their lemma name.By subexpression:
🔍_ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.The pattern can also be non-linear, as in
🔍Real.sqrt ?a * Real.sqrt ?a
If the pattern has parameters, they are matched in any order. Both of these will find
List.map
:
🔍(?a -> ?b) -> List ?a -> List ?b
🔍List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍|- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all→
and∀
) has the given shape.As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍|- _ < _ → tsum _ < tsum _
will findtsum_lt_tsum
even though the hypothesisf i < g i
is not the last.
If you pass more than one such search filter, separated by commas
Loogle will return lemmas which match all of them. The
search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
woould find all lemmas which mention the constants Real.sin
and tsum
, have "two"
as a substring of the
lemma name, include a product and a power somewhere in the type,
and have a hypothesis of the form _ < _
(if
there were any such lemmas). Metavariables (?a
) are
assigned independently in each filter.
The #lucky
button will directly send you to the
documentation of the first hit.
Source code
You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is currently provided by Joachim Breitner <mail@joachim-breitner.de>.
This is Loogle revision fa2ddf5
serving mathlib revision c44d42e