Extra lemmas about products of monoids and groups

This file proves lemmas about the instances defined in Algebra.Group.Pi.Basic that require more imports.

@[simp]

theorem Set.range_one {α : Type u_5} {β : Type u_6} [One β] [Nonempty α] : range 1 = {1}

@[simp]

theorem Set.range_zero {α : Type u_5} {β : Type u_6} [Zero β] [Nonempty α] : range 0 = {0}

theorem Set.preimage_one {α : Type u_5} {β : Type u_6} [One β] (s : Set β) [Decidable (1 ∈ s)] : 1 ⁻¹' s = if 1 ∈ s then univ else ∅

theorem Set.preimage_zero {α : Type u_5} {β : Type u_6} [Zero β] (s : Set β) [Decidable (0 ∈ s)] : 0 ⁻¹' s = if 0 ∈ s then univ else ∅

theorem Pi.one_mono {α : Type u_5} {β : Type u_6} [Preorder α] [Preorder β] [One β] : Monotone 1

theorem Pi.zero_mono {α : Type u_5} {β : Type u_6} [Preorder α] [Preorder β] [Zero β] : Monotone 0

theorem Pi.one_anti {α : Type u_5} {β : Type u_6} [Preorder α] [Preorder β] [One β] : Antitone 1

theorem Pi.zero_anti {α : Type u_5} {β : Type u_6} [Preorder α] [Preorder β] [Zero β] : Antitone 0

theorem Pi.isSelfInv_iff {ι : Type u_5} {G : ι → Type u_6} [(i : ι) → Inv (G i)] {x : (i : ι) → G i} : IsSelfInv x ↔ ∀ (i : ι), IsSelfInv (x i)

theorem Pi.isSelfNeg_iff {ι : Type u_5} {G : ι → Type u_6} [(i : ι) → Neg (G i)] {x : (i : ι) → G i} : IsSelfNeg x ↔ ∀ (i : ι), IsSelfNeg (x i)

theorem MulHom.coe_mul {M : Type u_5} {N : Type u_6} {x✝ : Mul M} {x✝¹ : CommSemigroup N} (f g : M →ₙ* N) : ⇑f * ⇑g = fun (x : M) => f x * g x

theorem AddHom.coe_add {M : Type u_5} {N : Type u_6} {x✝ : Add M} {x✝¹ : AddCommSemigroup N} (f g : M →ₙ+ N) : ⇑f + ⇑g = fun (x : M) => f x + g x

def MulHom.coeFn (α : Type u_5) (β : Type u_6) [Mul α] [CommSemigroup β] : (α →ₙ* β) →ₙ* α → β

Coercion of a MulHom into a function is itself a MulHom.

See also MulHom.eval.

Equations

• MulHom.coeFn α β = { toFun := fun (g : α →ₙ* β) => ⇑g, map_mul' := ⋯ }

def AddHom.coeFn (α : Type u_5) (β : Type u_6) [Add α] [AddCommSemigroup β] : (α →ₙ+ β) →ₙ+ α → β

Coercion of an AddHom into a function is itself an AddHom.

See also AddHom.eval.

Equations

• AddHom.coeFn α β = { toFun := fun (g : α →ₙ+ β) => ⇑g, map_add' := ⋯ }

@[simp]

theorem MulHom.coeFn_apply (α : Type u_5) (β : Type u_6) [Mul α] [CommSemigroup β] (g : α →ₙ* β) (a : α) : (coeFn α β) g a = g a

@[simp]

theorem AddHom.coeFn_apply (α : Type u_5) (β : Type u_6) [Add α] [AddCommSemigroup β] (g : α →ₙ+ β) (a : α) : (coeFn α β) g a = g a

def MonoidHom.coeFn (α : Type u_5) (β : Type u_6) [MulOneClass α] [CommMonoid β] : (α →* β) →* α → β

Coercion of a MonoidHom into a function is itself a MonoidHom.

See also MonoidHom.eval.

Equations

• MonoidHom.coeFn α β = { toFun := fun (g : α →* β) => ⇑g, map_one' := ⋯, map_mul' := ⋯ }

def AddMonoidHom.coeFn (α : Type u_5) (β : Type u_6) [AddZeroClass α] [AddCommMonoid β] : (α →+ β) →+ α → β

Coercion of an AddMonoidHom into a function is itself an AddMonoidHom.

See also AddMonoidHom.eval.

Equations

• AddMonoidHom.coeFn α β = { toFun := fun (g : α →+ β) => ⇑g, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem AddMonoidHom.coeFn_apply (α : Type u_5) (β : Type u_6) [AddZeroClass α] [AddCommMonoid β] (g : α →+ β) (a : α) : (coeFn α β) g a = g a

@[simp]

theorem MonoidHom.coeFn_apply (α : Type u_5) (β : Type u_6) [MulOneClass α] [CommMonoid β] (g : α →* β) (a : α) : (coeFn α β) g a = g a

def OneHom.mulSingle {I : Type u} (f : I → Type v) [DecidableEq I] [(i : I) → One (f i)] (i : I) : OneHom (f i) ((i : I) → f i)

The one-preserving homomorphism including a single value into a dependent family of values, as functions supported at a point.

This is the OneHom version of Pi.mulSingle.

Equations

• OneHom.mulSingle f i = { toFun := Pi.mulSingle i, map_one' := ⋯ }

def ZeroHom.single {I : Type u} (f : I → Type v) [DecidableEq I] [(i : I) → Zero (f i)] (i : I) : ZeroHom (f i) ((i : I) → f i)

The zero-preserving homomorphism including a single value into a dependent family of values, as functions supported at a point.

This is the ZeroHom version of Pi.single.

Equations

• ZeroHom.single f i = { toFun := Pi.single i, map_zero' := ⋯ }

@[simp]

theorem OneHom.mulSingle_apply {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → One (f i)] (i : I) (x : f i) : (mulSingle f i) x = Pi.mulSingle i x

@[simp]

theorem ZeroHom.single_apply {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → Zero (f i)] (i : I) (x : f i) : (single f i) x = Pi.single i x

@[simp]

theorem OneHom.coe_mulSingle {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → One (f i)] (i : I) : ⇑(mulSingle f i) = Pi.mulSingle i

@[simp]

theorem ZeroHom.coe_single {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → Zero (f i)] (i : I) : ⇑(single f i) = Pi.single i

def MonoidHom.mulSingle {I : Type u} (f : I → Type v) [DecidableEq I] [(i : I) → MulOneClass (f i)] (i : I) : f i →* (i : I) → f i

The monoid homomorphism including a single monoid into a dependent family of additive monoids, as functions supported at a point.

This is the MonoidHom version of Pi.mulSingle.

Equations

• MonoidHom.mulSingle f i = { toOneHom := OneHom.mulSingle f i, map_mul' := ⋯ }

def AddMonoidHom.single {I : Type u} (f : I → Type v) [DecidableEq I] [(i : I) → AddZeroClass (f i)] (i : I) : f i →+ (i : I) → f i

The additive monoid homomorphism including a single additive monoid into a dependent family of additive monoids, as functions supported at a point.

This is the AddMonoidHom version of Pi.single.

Equations

• AddMonoidHom.single f i = { toZeroHom := ZeroHom.single f i, map_add' := ⋯ }

@[simp]

theorem MonoidHom.mulSingle_apply {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → MulOneClass (f i)] (i : I) (x : f i) : (mulSingle f i) x = Pi.mulSingle i x

@[simp]

theorem AddMonoidHom.single_apply {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddZeroClass (f i)] (i : I) (x : f i) : (single f i) x = Pi.single i x

@[simp]

theorem MonoidHom.coe_mulSingle {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → MulOneClass (f i)] (i : I) : ⇑(mulSingle f i) = Pi.mulSingle i

@[simp]

theorem AddMonoidHom.coe_single {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddZeroClass (f i)] (i : I) : ⇑(single f i) = Pi.single i

theorem Pi.mulSingle_sup {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → SemilatticeSup (f i)] [(i : I) → One (f i)] (i : I) (x y : f i) : mulSingle i (x ⊔ y) = mulSingle i x ⊔ mulSingle i y

theorem Pi.single_sup {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → SemilatticeSup (f i)] [(i : I) → Zero (f i)] (i : I) (x y : f i) : single i (x ⊔ y) = single i x ⊔ single i y

theorem Pi.mulSingle_inf {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → SemilatticeInf (f i)] [(i : I) → One (f i)] (i : I) (x y : f i) : mulSingle i (x ⊓ y) = mulSingle i x ⊓ mulSingle i y

theorem Pi.single_inf {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → SemilatticeInf (f i)] [(i : I) → Zero (f i)] (i : I) (x y : f i) : single i (x ⊓ y) = single i x ⊓ single i y

theorem Pi.mulSingle_mul {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → MulOneClass (f i)] (i : I) (x y : f i) : mulSingle i (x * y) = mulSingle i x * mulSingle i y

theorem Pi.single_add {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddZeroClass (f i)] (i : I) (x y : f i) : single i (x + y) = single i x + single i y

theorem Pi.mulSingle_inv {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → Group (f i)] (i : I) (x : f i) : mulSingle i x⁻¹ = (mulSingle i x)⁻¹

theorem Pi.single_neg {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddGroup (f i)] (i : I) (x : f i) : single i (-x) = -single i x

theorem Pi.mulSingle_div {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → Group (f i)] (i : I) (x y : f i) : mulSingle i (x / y) = mulSingle i x / mulSingle i y

theorem Pi.single_sub {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddGroup (f i)] (i : I) (x y : f i) : single i (x - y) = single i x - single i y

theorem Pi.mulSingle_pow {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → Monoid (f i)] (i : I) (x : f i) (n : ℕ) : mulSingle i (x ^ n) = mulSingle i x ^ n

theorem Pi.single_nsmul {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddMonoid (f i)] (i : I) (x : f i) (n : ℕ) : single i (n • x) = n • single i x

theorem Pi.mulSingle_zpow {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → Group (f i)] (i : I) (x : f i) (n : ℤ) : mulSingle i (x ^ n) = mulSingle i x ^ n

theorem Pi.single_zsmul {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddGroup (f i)] (i : I) (x : f i) (n : ℤ) : single i (n • x) = n • single i x

theorem Pi.mulSingle_commute {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → MulOneClass (f i)] : Pairwise fun (i j : I) => ∀ (x : f i) (y : f j), Commute (mulSingle i x) (mulSingle j y)

The injection into a pi group at different indices commutes.

For injections of commuting elements at the same index, see Commute.map

theorem Pi.single_addCommute {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddZeroClass (f i)] : Pairwise fun (i j : I) => ∀ (x : f i) (y : f j), AddCommute (single i x) (single j y)

The injection into an additive pi group at different indices commutes.

For injections of commuting elements at the same index, see AddCommute.map

theorem Pi.mulSingle_apply_commute {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → MulOneClass (f i)] (x : (i : I) → f i) (i j : I) : Commute (mulSingle i (x i)) (mulSingle j (x j))

The injection into a pi group with the same values commutes.

theorem Pi.single_apply_addCommute {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddZeroClass (f i)] (x : (i : I) → f i) (i j : I) : AddCommute (single i (x i)) (single j (x j))

The injection into an additive pi group with the same values commutes.

theorem Pi.update_eq_div_mul_mulSingle {I : Type u} {f : I → Type v} (i : I) [DecidableEq I] [(i : I) → Group (f i)] (g : (i : I) → f i) (x : f i) : Function.update g i x = g / mulSingle i (g i) * mulSingle i x

theorem Pi.update_eq_sub_add_single {I : Type u} {f : I → Type v} (i : I) [DecidableEq I] [(i : I) → AddGroup (f i)] (g : (i : I) → f i) (x : f i) : Function.update g i x = g - single i (g i) + single i x

theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {I : Type u} [DecidableEq I] {M : Type u_5} [CommMonoid M] {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v ↔ k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n

theorem Pi.single_add_single_eq_single_add_single {I : Type u} [DecidableEq I] {M : Type u_5} [AddCommMonoid M] {k l m n : I} {u v : M} (hu : u ≠ 0) (hv : v ≠ 0) : single k u + single l v = single m u + single n v ↔ k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u + v = 0 ∧ k = l ∧ m = n

theorem SemiconjBy.pi {I : Type u} {f : I → Type v} [(i : I) → Mul (f i)] {x y z : (i : I) → f i} (h : ∀ (i : I), SemiconjBy (x i) (y i) (z i)) : SemiconjBy x y z

theorem AddSemiconjBy.pi {I : Type u} {f : I → Type v} [(i : I) → Add (f i)] {x y z : (i : I) → f i} (h : ∀ (i : I), AddSemiconjBy (x i) (y i) (z i)) : AddSemiconjBy x y z

theorem Pi.semiconjBy_iff {I : Type u} {f : I → Type v} [(i : I) → Mul (f i)] {x y z : (i : I) → f i} : SemiconjBy x y z ↔ ∀ (i : I), SemiconjBy (x i) (y i) (z i)

theorem Pi.addSemiconjBy_iff {I : Type u} {f : I → Type v} [(i : I) → Add (f i)] {x y z : (i : I) → f i} : AddSemiconjBy x y z ↔ ∀ (i : I), AddSemiconjBy (x i) (y i) (z i)

theorem Commute.pi {I : Type u} {f : I → Type v} [(i : I) → Mul (f i)] {x y : (i : I) → f i} (h : ∀ (i : I), Commute (x i) (y i)) : Commute x y

theorem AddCommute.pi {I : Type u} {f : I → Type v} [(i : I) → Add (f i)] {x y : (i : I) → f i} (h : ∀ (i : I), AddCommute (x i) (y i)) : AddCommute x y

theorem Pi.commute_iff {I : Type u} {f : I → Type v} [(i : I) → Mul (f i)] {x y : (i : I) → f i} : Commute x y ↔ ∀ (i : I), Commute (x i) (y i)

theorem Pi.addCommute_iff {I : Type u} {f : I → Type v} [(i : I) → Add (f i)] {x y : (i : I) → f i} : AddCommute x y ↔ ∀ (i : I), AddCommute (x i) (y i)

@[simp]

theorem Function.update_one {I : Type u} {f : I → Type v} [(i : I) → One (f i)] [DecidableEq I] (i : I) : update 1 i 1 = 1

@[simp]

theorem Function.update_zero {I : Type u} {f : I → Type v} [(i : I) → Zero (f i)] [DecidableEq I] (i : I) : update 0 i 0 = 0

theorem Function.update_mul {I : Type u} {f : I → Type v} [(i : I) → Mul (f i)] [DecidableEq I] (f₁ f₂ : (i : I) → f i) (i : I) (x₁ x₂ : f i) : update (f₁ * f₂) i (x₁ * x₂) = update f₁ i x₁ * update f₂ i x₂

theorem Function.update_add {I : Type u} {f : I → Type v} [(i : I) → Add (f i)] [DecidableEq I] (f₁ f₂ : (i : I) → f i) (i : I) (x₁ x₂ : f i) : update (f₁ + f₂) i (x₁ + x₂) = update f₁ i x₁ + update f₂ i x₂

theorem Function.update_inv {I : Type u} {f : I → Type v} [(i : I) → Inv (f i)] [DecidableEq I] (f₁ : (i : I) → f i) (i : I) (x₁ : f i) : update f₁⁻¹ i x₁⁻¹ = (update f₁ i x₁)⁻¹

theorem Function.update_neg {I : Type u} {f : I → Type v} [(i : I) → Neg (f i)] [DecidableEq I] (f₁ : (i : I) → f i) (i : I) (x₁ : f i) : update (-f₁) i (-x₁) = -update f₁ i x₁

theorem Function.update_div {I : Type u} {f : I → Type v} [(i : I) → Div (f i)] [DecidableEq I] (f₁ f₂ : (i : I) → f i) (i : I) (x₁ x₂ : f i) : update (f₁ / f₂) i (x₁ / x₂) = update f₁ i x₁ / update f₂ i x₂

theorem Function.update_sub {I : Type u} {f : I → Type v} [(i : I) → Sub (f i)] [DecidableEq I] (f₁ f₂ : (i : I) → f i) (i : I) (x₁ x₂ : f i) : update (f₁ - f₂) i (x₁ - x₂) = update f₁ i x₁ - update f₂ i x₂

@[simp]

theorem Function.const_eq_one {ι : Type u_1} {α : Type u_2} [One α] [Nonempty ι] {a : α} : const ι a = 1 ↔ a = 1

@[simp]

theorem Function.const_eq_zero {ι : Type u_1} {α : Type u_2} [Zero α] [Nonempty ι] {a : α} : const ι a = 0 ↔ a = 0

theorem Function.const_ne_one {ι : Type u_1} {α : Type u_2} [One α] [Nonempty ι] {a : α} : const ι a ≠ 1 ↔ a ≠ 1

theorem Function.const_ne_zero {ι : Type u_1} {α : Type u_2} [Zero α] [Nonempty ι] {a : α} : const ι a ≠ 0 ↔ a ≠ 0

theorem Set.piecewise_mul {I : Type u} {f : I → Type v} [(i : I) → Mul (f i)] (s : Set I) [(i : I) → Decidable (i ∈ s)] (f₁ f₂ g₁ g₂ : (i : I) → f i) : s.piecewise (f₁ * f₂) (g₁ * g₂) = s.piecewise f₁ g₁ * s.piecewise f₂ g₂

theorem Set.piecewise_add {I : Type u} {f : I → Type v} [(i : I) → Add (f i)] (s : Set I) [(i : I) → Decidable (i ∈ s)] (f₁ f₂ g₁ g₂ : (i : I) → f i) : s.piecewise (f₁ + f₂) (g₁ + g₂) = s.piecewise f₁ g₁ + s.piecewise f₂ g₂

theorem Set.piecewise_inv {I : Type u} {f : I → Type v} [(i : I) → Inv (f i)] (s : Set I) [(i : I) → Decidable (i ∈ s)] (f₁ g₁ : (i : I) → f i) : s.piecewise f₁⁻¹ g₁⁻¹ = (s.piecewise f₁ g₁)⁻¹

theorem Set.piecewise_neg {I : Type u} {f : I → Type v} [(i : I) → Neg (f i)] (s : Set I) [(i : I) → Decidable (i ∈ s)] (f₁ g₁ : (i : I) → f i) : s.piecewise (-f₁) (-g₁) = -s.piecewise f₁ g₁

theorem Set.piecewise_div {I : Type u} {f : I → Type v} [(i : I) → Div (f i)] (s : Set I) [(i : I) → Decidable (i ∈ s)] (f₁ f₂ g₁ g₂ : (i : I) → f i) : s.piecewise (f₁ / f₂) (g₁ / g₂) = s.piecewise f₁ g₁ / s.piecewise f₂ g₂

theorem Set.piecewise_sub {I : Type u} {f : I → Type v} [(i : I) → Sub (f i)] (s : Set I) [(i : I) → Decidable (i ∈ s)] (f₁ f₂ g₁ g₂ : (i : I) → f i) : s.piecewise (f₁ - f₂) (g₁ - g₂) = s.piecewise f₁ g₁ - s.piecewise f₂ g₂

noncomputable def Function.ExtendByOne.hom {ι : Type u_1} {η : Type v} (R : Type w) (s : ι → η) [MulOneClass R] : (ι → R) →* η → R

Function.extend s f 1 as a bundled hom.

Equations

• Function.ExtendByOne.hom R s = { toFun := fun (f : ι → R) => Function.extend s f 1, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem Function.ExtendByZero.hom_apply {ι : Type u_1} {η : Type v} (R : Type w) (s : ι → η) [AddZeroClass R] (f : ι → R) (a✝ : η) : (hom R s) f a✝ = extend s f 0 a✝

@[simp]

theorem Function.ExtendByOne.hom_apply {ι : Type u_1} {η : Type v} (R : Type w) (s : ι → η) [MulOneClass R] (f : ι → R) (a✝ : η) : (hom R s) f a✝ = extend s f 1 a✝

noncomputable def Function.ExtendByZero.hom {ι : Type u_1} {η : Type v} (R : Type w) (s : ι → η) [AddZeroClass R] : (ι → R) →+ η → R

Function.extend s f 0 as a bundled hom.

Equations

• Function.ExtendByZero.hom R s = { toFun := fun (f : ι → R) => Function.extend s f 0, map_zero' := ⋯, map_add' := ⋯ }

theorem Pi.mulSingle_mono {I : Type u} {f : I → Type v} (i : I) [DecidableEq I] [(i : I) → Preorder (f i)] [(i : I) → One (f i)] : Monotone (mulSingle i)

theorem Pi.single_mono {I : Type u} {f : I → Type v} (i : I) [DecidableEq I] [(i : I) → Preorder (f i)] [(i : I) → Zero (f i)] : Monotone (single i)

theorem Pi.mulSingle_strictMono {I : Type u} {f : I → Type v} (i : I) [DecidableEq I] [(i : I) → Preorder (f i)] [(i : I) → One (f i)] : StrictMono (mulSingle i)

theorem Pi.single_strictMono {I : Type u} {f : I → Type v} (i : I) [DecidableEq I] [(i : I) → Preorder (f i)] [(i : I) → Zero (f i)] : StrictMono (single i)

theorem Pi.mulSingle_comp_equiv {α : Type u_2} {m : Type u_5} {n : Type u_6} [DecidableEq n] [DecidableEq m] [One α] (σ : n ≃ m) (i : m) (x : α) : mulSingle i x ∘ ⇑σ = mulSingle (σ.symm i) x

theorem Pi.single_comp_equiv {α : Type u_2} {m : Type u_5} {n : Type u_6} [DecidableEq n] [DecidableEq m] [Zero α] (σ : n ≃ m) (i : m) (x : α) : single i x ∘ ⇑σ = single (σ.symm i) x

@[simp]

theorem Sigma.curry_one {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → One (γ a b)] : curry 1 = 1

@[simp]

theorem Sigma.curry_zero {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → Zero (γ a b)] : curry 0 = 0

@[simp]

theorem Sigma.uncurry_one {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → One (γ a b)] : uncurry 1 = 1

@[simp]

theorem Sigma.uncurry_zero {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → Zero (γ a b)] : uncurry 0 = 0

@[simp]

theorem Sigma.curry_mul {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → Mul (γ a b)] (x y : (i : (a : α) × β a) → γ i.fst i.snd) : curry (x * y) = curry x * curry y

@[simp]

theorem Sigma.curry_add {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → Add (γ a b)] (x y : (i : (a : α) × β a) → γ i.fst i.snd) : curry (x + y) = curry x + curry y

@[simp]

theorem Sigma.uncurry_mul {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → Mul (γ a b)] (x y : (a : α) → (b : β a) → γ a b) : uncurry (x * y) = uncurry x * uncurry y

@[simp]

theorem Sigma.uncurry_add {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → Add (γ a b)] (x y : (a : α) → (b : β a) → γ a b) : uncurry (x + y) = uncurry x + uncurry y

@[simp]

theorem Sigma.curry_inv {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → Inv (γ a b)] (x : (i : (a : α) × β a) → γ i.fst i.snd) : curry x⁻¹ = (curry x)⁻¹

@[simp]

theorem Sigma.curry_neg {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → Neg (γ a b)] (x : (i : (a : α) × β a) → γ i.fst i.snd) : curry (-x) = -curry x

@[simp]

theorem Sigma.uncurry_inv {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → Inv (γ a b)] (x : (a : α) → (b : β a) → γ a b) : uncurry x⁻¹ = (uncurry x)⁻¹

@[simp]

theorem Sigma.uncurry_neg {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [(a : α) → (b : β a) → Neg (γ a b)] (x : (a : α) → (b : β a) → γ a b) : uncurry (-x) = -uncurry x

@[simp]

theorem Sigma.curry_mulSingle {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [DecidableEq α] [(a : α) → DecidableEq (β a)] [(a : α) → (b : β a) → One (γ a b)] (i : (a : α) × β a) (x : γ i.fst i.snd) : curry (Pi.mulSingle i x) = Pi.mulSingle i.fst (Pi.mulSingle i.snd x)

@[simp]

theorem Sigma.curry_single {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [DecidableEq α] [(a : α) → DecidableEq (β a)] [(a : α) → (b : β a) → Zero (γ a b)] (i : (a : α) × β a) (x : γ i.fst i.snd) : curry (Pi.single i x) = Pi.single i.fst (Pi.single i.snd x)

@[simp]

theorem Sigma.uncurry_mulSingle_mulSingle {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [DecidableEq α] [(a : α) → DecidableEq (β a)] [(a : α) → (b : β a) → One (γ a b)] (a : α) (b : β a) (x : γ a b) : uncurry (Pi.mulSingle a (Pi.mulSingle b x)) = Pi.mulSingle ⟨a, b⟩ x

@[simp]

theorem Sigma.uncurry_single_single {α : Type u_5} {β : α → Type u_6} {γ : (a : α) → β a → Type u_7} [DecidableEq α] [(a : α) → DecidableEq (β a)] [(a : α) → (b : β a) → Zero (γ a b)] (a : α) (b : β a) (x : γ a b) : uncurry (Pi.single a (Pi.single b x)) = Pi.single ⟨a, b⟩ x