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_3} {β : Type u_4} [One β] [Nonempty α] : Set.range 1 = {1}

@[simp]

theorem Set.range_zero {α : Type u_3} {β : Type u_4} [Zero β] [Nonempty α] : Set.range 0 = {0}

theorem Set.preimage_one {α : Type u_3} {β : Type u_4} [One β] (s : Set β) [Decidable (1 ∈ s)] : 1 ⁻¹' s = if 1 ∈ s then Set.univ else ∅

theorem Set.preimage_zero {α : Type u_3} {β : Type u_4} [Zero β] (s : Set β) [Decidable (0 ∈ s)] : 0 ⁻¹' s = if 0 ∈ s then Set.univ else ∅

theorem Pi.one_mono {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [One β] : Monotone 1

theorem Pi.zero_mono {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [Zero β] : Monotone 0

theorem Pi.one_anti {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [One β] : Antitone 1

theorem Pi.zero_anti {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [Zero β] : Antitone 0

theorem MulHom.coe_mul {M : Type u_3} {N : Type u_4} : ∀ {x : Mul M} {x_1 : CommSemigroup N} (f g : M →ₙ* N), ⇑f * ⇑g = fun (x_2 : M) => f x_2 * g x_2

theorem AddHom.coe_add {M : Type u_3} {N : Type u_4} : ∀ {x : Add M} {x_1 : AddCommSemigroup N} (f g : AddHom M N), ⇑f + ⇑g = fun (x_2 : M) => f x_2 + g x_2

def Pi.mulHom {I : Type u} {f : I → Type v} {γ : Type w} [(i : I) → Mul (f i)] [Mul γ] (g : (i : I) → γ →ₙ* f i) : γ →ₙ* (i : I) → f i

A family of MulHom's f a : γ →ₙ* β a defines a MulHom Pi.mulHom f : γ →ₙ* Π a, β a given by Pi.mulHom f x b = f b x.

Equations

• Pi.mulHom g = { toFun := fun (x : γ) (i : I) => (g i) x, map_mul' := ⋯ }

def Pi.addHom {I : Type u} {f : I → Type v} {γ : Type w} [(i : I) → Add (f i)] [Add γ] (g : (i : I) → AddHom γ (f i)) : AddHom γ ((i : I) → f i)

A family of AddHom's f a : γ → β a defines an AddHom Pi.addHom f : γ → Π a, β a given by Pi.addHom f x b = f b x.

Equations

• Pi.addHom g = { toFun := fun (x : γ) (i : I) => (g i) x, map_add' := ⋯ }

@[simp]

theorem Pi.addHom_apply {I : Type u} {f : I → Type v} {γ : Type w} [(i : I) → Add (f i)] [Add γ] (g : (i : I) → AddHom γ (f i)) (x : γ) (i : I) : (Pi.addHom g) x i = (g i) x

@[simp]

theorem Pi.mulHom_apply {I : Type u} {f : I → Type v} {γ : Type w} [(i : I) → Mul (f i)] [Mul γ] (g : (i : I) → γ →ₙ* f i) (x : γ) (i : I) : (Pi.mulHom g) x i = (g i) x

theorem Pi.mulHom_injective {I : Type u} {f : I → Type v} {γ : Type w} [Nonempty I] [(i : I) → Mul (f i)] [Mul γ] (g : (i : I) → γ →ₙ* f i) (hg : ∀ (i : I), Function.Injective ⇑(g i)) : Function.Injective ⇑(Pi.mulHom g)

theorem Pi.addHom_injective {I : Type u} {f : I → Type v} {γ : Type w} [Nonempty I] [(i : I) → Add (f i)] [Add γ] (g : (i : I) → AddHom γ (f i)) (hg : ∀ (i : I), Function.Injective ⇑(g i)) : Function.Injective ⇑(Pi.addHom g)

def Pi.monoidHom {I : Type u} {f : I → Type v} {γ : Type w} [(i : I) → MulOneClass (f i)] [MulOneClass γ] (g : (i : I) → γ →* f i) : γ →* (i : I) → f i

A family of monoid homomorphisms f a : γ →* β a defines a monoid homomorphism Pi.monoidHom f : γ →* Π a, β a given by Pi.monoidHom f x b = f b x.

Equations

• Pi.monoidHom g = { toFun := fun (x : γ) (i : I) => (g i) x, map_one' := ⋯, map_mul' := ⋯ }

def Pi.addMonoidHom {I : Type u} {f : I → Type v} {γ : Type w} [(i : I) → AddZeroClass (f i)] [AddZeroClass γ] (g : (i : I) → γ →+ f i) : γ →+ (i : I) → f i

A family of additive monoid homomorphisms f a : γ →+ β a defines a monoid homomorphism Pi.addMonoidHom f : γ →+ Π a, β a given by Pi.addMonoidHom f x b = f b x.

Equations

• Pi.addMonoidHom g = { toFun := fun (x : γ) (i : I) => (g i) x, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem Pi.addMonoidHom_apply {I : Type u} {f : I → Type v} {γ : Type w} [(i : I) → AddZeroClass (f i)] [AddZeroClass γ] (g : (i : I) → γ →+ f i) (x : γ) (i : I) : (Pi.addMonoidHom g) x i = (g i) x

@[simp]

theorem Pi.monoidHom_apply {I : Type u} {f : I → Type v} {γ : Type w} [(i : I) → MulOneClass (f i)] [MulOneClass γ] (g : (i : I) → γ →* f i) (x : γ) (i : I) : (Pi.monoidHom g) x i = (g i) x

theorem Pi.monoidHom_injective {I : Type u} {f : I → Type v} {γ : Type w} [Nonempty I] [(i : I) → MulOneClass (f i)] [MulOneClass γ] (g : (i : I) → γ →* f i) (hg : ∀ (i : I), Function.Injective ⇑(g i)) : Function.Injective ⇑(Pi.monoidHom g)

theorem Pi.addMonoidHom_injective {I : Type u} {f : I → Type v} {γ : Type w} [Nonempty I] [(i : I) → AddZeroClass (f i)] [AddZeroClass γ] (g : (i : I) → γ →+ f i) (hg : ∀ (i : I), Function.Injective ⇑(g i)) : Function.Injective ⇑(Pi.addMonoidHom g)

def Pi.evalMulHom {I : Type u} (f : I → Type v) [(i : I) → Mul (f i)] (i : I) : ((i : I) → f i) →ₙ* f i

Evaluation of functions into an indexed collection of semigroups at a point is a semigroup homomorphism. This is Function.eval i as a MulHom.

Equations

• Pi.evalMulHom f i = { toFun := fun (g : (i : I) → f i) => g i, map_mul' := ⋯ }

def Pi.evalAddHom {I : Type u} (f : I → Type v) [(i : I) → Add (f i)] (i : I) : AddHom ((i : I) → f i) (f i)

Evaluation of functions into an indexed collection of additive semigroups at a point is an additive semigroup homomorphism. This is Function.eval i as an AddHom.

Equations

• Pi.evalAddHom f i = { toFun := fun (g : (i : I) → f i) => g i, map_add' := ⋯ }

@[simp]

theorem Pi.evalAddHom_apply {I : Type u} (f : I → Type v) [(i : I) → Add (f i)] (i : I) (g : (i : I) → f i) : (Pi.evalAddHom f i) g = g i

@[simp]

theorem Pi.evalMulHom_apply {I : Type u} (f : I → Type v) [(i : I) → Mul (f i)] (i : I) (g : (i : I) → f i) : (Pi.evalMulHom f i) g = g i

def Pi.constMulHom (α : Type u_3) (β : Type u_4) [Mul β] : β →ₙ* α → β

Function.const as a MulHom.

Equations

• Pi.constMulHom α β = { toFun := Function.const α, map_mul' := ⋯ }

def Pi.constAddHom (α : Type u_3) (β : Type u_4) [Add β] : AddHom β (α → β)

Function.const as an AddHom.

Equations

• Pi.constAddHom α β = { toFun := Function.const α, map_add' := ⋯ }

@[simp]

theorem Pi.constAddHom_apply (α : Type u_3) (β : Type u_4) [Add β] (a : β) : ∀ (a_1 : α), (Pi.constAddHom α β) a a_1 = Function.const α a a_1

@[simp]

theorem Pi.constMulHom_apply (α : Type u_3) (β : Type u_4) [Mul β] (a : β) : ∀ (a_1 : α), (Pi.constMulHom α β) a a_1 = Function.const α a a_1

def MulHom.coeFn (α : Type u_3) (β : Type u_4) [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_3) (β : Type u_4) [Add α] [AddCommSemigroup β] : AddHom (AddHom α β) (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 : AddHom α β) => ⇑g, map_add' := ⋯ }

@[simp]

theorem MulHom.coeFn_apply (α : Type u_3) (β : Type u_4) [Mul α] [CommSemigroup β] (g : α →ₙ* β) (a : α) : (MulHom.coeFn α β) g a = g a

@[simp]

theorem AddHom.coeFn_apply (α : Type u_3) (β : Type u_4) [Add α] [AddCommSemigroup β] (g : AddHom α β) (a : α) : (AddHom.coeFn α β) g a = g a

def MulHom.compLeft {α : Type u_3} {β : Type u_4} [Mul α] [Mul β] (f : α →ₙ* β) (I : Type u_5) : (I → α) →ₙ* I → β

Semigroup homomorphism between the function spaces I → α and I → β, induced by a semigroup homomorphism f between α and β.

Equations

• f.compLeft I = { toFun := fun (h : I → α) => ⇑f ∘ h, map_mul' := ⋯ }

def AddHom.compLeft {α : Type u_3} {β : Type u_4} [Add α] [Add β] (f : AddHom α β) (I : Type u_5) : AddHom (I → α) (I → β)

Additive semigroup homomorphism between the function spaces I → α and I → β, induced by an additive semigroup homomorphism f between α and β

Equations

• f.compLeft I = { toFun := fun (h : I → α) => ⇑f ∘ h, map_add' := ⋯ }

@[simp]

theorem MulHom.compLeft_apply {α : Type u_3} {β : Type u_4} [Mul α] [Mul β] (f : α →ₙ* β) (I : Type u_5) (h : I → α) : ∀ (a : I), (f.compLeft I) h a = (⇑f ∘ h) a

@[simp]

theorem AddHom.compLeft_apply {α : Type u_3} {β : Type u_4} [Add α] [Add β] (f : AddHom α β) (I : Type u_5) (h : I → α) : ∀ (a : I), (f.compLeft I) h a = (⇑f ∘ h) a

def Pi.evalMonoidHom {I : Type u} (f : I → Type v) [(i : I) → MulOneClass (f i)] (i : I) : ((i : I) → f i) →* f i

Evaluation of functions into an indexed collection of monoids at a point is a monoid homomorphism. This is Function.eval i as a MonoidHom.

Equations

• Pi.evalMonoidHom f i = { toFun := fun (g : (i : I) → f i) => g i, map_one' := ⋯, map_mul' := ⋯ }

def Pi.evalAddMonoidHom {I : Type u} (f : I → Type v) [(i : I) → AddZeroClass (f i)] (i : I) : ((i : I) → f i) →+ f i

Evaluation of functions into an indexed collection of additive monoids at a point is an additive monoid homomorphism. This is Function.eval i as an AddMonoidHom.

Equations

• Pi.evalAddMonoidHom f i = { toFun := fun (g : (i : I) → f i) => g i, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem Pi.evalAddMonoidHom_apply {I : Type u} (f : I → Type v) [(i : I) → AddZeroClass (f i)] (i : I) (g : (i : I) → f i) : (Pi.evalAddMonoidHom f i) g = g i

@[simp]

theorem Pi.evalMonoidHom_apply {I : Type u} (f : I → Type v) [(i : I) → MulOneClass (f i)] (i : I) (g : (i : I) → f i) : (Pi.evalMonoidHom f i) g = g i

def Pi.constMonoidHom (α : Type u_3) (β : Type u_4) [MulOneClass β] : β →* α → β

Function.const as a MonoidHom.

Equations

• Pi.constMonoidHom α β = { toFun := Function.const α, map_one' := ⋯, map_mul' := ⋯ }

def Pi.constAddMonoidHom (α : Type u_3) (β : Type u_4) [AddZeroClass β] : β →+ α → β

Function.const as an AddMonoidHom.

Equations

• Pi.constAddMonoidHom α β = { toFun := Function.const α, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem Pi.constMonoidHom_apply (α : Type u_3) (β : Type u_4) [MulOneClass β] (a : β) : ∀ (a_1 : α), (Pi.constMonoidHom α β) a a_1 = Function.const α a a_1

@[simp]

theorem Pi.constAddMonoidHom_apply (α : Type u_3) (β : Type u_4) [AddZeroClass β] (a : β) : ∀ (a_1 : α), (Pi.constAddMonoidHom α β) a a_1 = Function.const α a a_1

def MonoidHom.coeFn (α : Type u_3) (β : Type u_4) [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_3) (β : Type u_4) [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 MonoidHom.coeFn_apply (α : Type u_3) (β : Type u_4) [MulOneClass α] [CommMonoid β] (g : α →* β) (a : α) : (MonoidHom.coeFn α β) g a = g a

@[simp]

theorem AddMonoidHom.coeFn_apply (α : Type u_3) (β : Type u_4) [AddZeroClass α] [AddCommMonoid β] (g : α →+ β) (a : α) : (AddMonoidHom.coeFn α β) g a = g a

def MonoidHom.compLeft {α : Type u_3} {β : Type u_4} [MulOneClass α] [MulOneClass β] (f : α →* β) (I : Type u_5) : (I → α) →* I → β

Monoid homomorphism between the function spaces I → α and I → β, induced by a monoid homomorphism f between α and β.

Equations

• f.compLeft I = { toFun := fun (h : I → α) => ⇑f ∘ h, map_one' := ⋯, map_mul' := ⋯ }

def AddMonoidHom.compLeft {α : Type u_3} {β : Type u_4} [AddZeroClass α] [AddZeroClass β] (f : α →+ β) (I : Type u_5) : (I → α) →+ I → β

Additive monoid homomorphism between the function spaces I → α and I → β, induced by an additive monoid homomorphism f between α and β

Equations

• f.compLeft I = { toFun := fun (h : I → α) => ⇑f ∘ h, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem AddMonoidHom.compLeft_apply {α : Type u_3} {β : Type u_4} [AddZeroClass α] [AddZeroClass β] (f : α →+ β) (I : Type u_5) (h : I → α) : ∀ (a : I), (f.compLeft I) h a = (⇑f ∘ h) a

@[simp]

theorem MonoidHom.compLeft_apply {α : Type u_3} {β : Type u_4} [MulOneClass α] [MulOneClass β] (f : α →* β) (I : Type u_5) (h : I → α) : ∀ (a : I), (f.compLeft I) h a = (⇑f ∘ h) 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) : (OneHom.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) : (ZeroHom.single f i) x = Pi.single i x

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) : (MonoidHom.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) : (AddMonoidHom.single f i) x = Pi.single i x

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 : f i) (y : f i) : Pi.mulSingle i (x ⊔ y) = Pi.mulSingle i x ⊔ Pi.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 : f i) (y : f i) : Pi.single i (x ⊔ y) = Pi.single i x ⊔ Pi.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 : f i) (y : f i) : Pi.mulSingle i (x ⊓ y) = Pi.mulSingle i x ⊓ Pi.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 : f i) (y : f i) : Pi.single i (x ⊓ y) = Pi.single i x ⊓ Pi.single i y

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

theorem Pi.single_add {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddZeroClass (f i)] (i : I) (x : f i) (y : f i) : Pi.single i (x + y) = Pi.single i x + Pi.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) : Pi.mulSingle i x⁻¹ = (Pi.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) : Pi.single i (-x) = -Pi.single i x

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

theorem Pi.single_sub {I : Type u} {f : I → Type v} [DecidableEq I] [(i : I) → AddGroup (f i)] (i : I) (x : f i) (y : f i) : Pi.single i (x - y) = Pi.single i x - Pi.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) : Pi.mulSingle i (x ^ n) = Pi.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) : Pi.single i (n • x) = n • Pi.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) : Pi.mulSingle i (x ^ n) = Pi.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) : Pi.single i (n • x) = n • Pi.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 (Pi.mulSingle i x) (Pi.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 (Pi.single i x) (Pi.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 : I) (j : I) : Commute (Pi.mulSingle i (x i)) (Pi.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 : I) (j : I) : AddCommute (Pi.single i (x i)) (Pi.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 / Pi.mulSingle i (g i) * Pi.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 - Pi.single i (g i) + Pi.single i x

theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {I : Type u} [DecidableEq I] {M : Type u_3} [CommMonoid M] {k : I} {l : I} {m : I} {n : I} {u : M} {v : M} (hu : u ≠ 1) (hv : v ≠ 1) : Pi.mulSingle k u * Pi.mulSingle l v = Pi.mulSingle m u * Pi.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_3} [AddCommMonoid M] {k : I} {l : I} {m : I} {n : I} {u : M} {v : M} (hu : u ≠ 0) (hv : v ≠ 0) : Pi.single k u + Pi.single l v = Pi.single m u + Pi.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 : (i : I) → f i} {y : (i : I) → f i} {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 : (i : I) → f i} {y : (i : I) → f i} {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 : (i : I) → f i} {y : (i : I) → f i} {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 : (i : I) → f i} {y : (i : I) → f i} {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 : (i : I) → f i} {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 : (i : I) → f i} {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 : (i : I) → f i} {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 : (i : I) → f i} {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) : Function.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) : Function.update 0 i 0 = 0

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

theorem Function.update_add {I : Type u} {f : I → Type v} [(i : I) → Add (f i)] [DecidableEq I] (f₁ : (i : I) → f i) (f₂ : (i : I) → f i) (i : I) (x₁ : f i) (x₂ : f i) : Function.update (f₁ + f₂) i (x₁ + x₂) = Function.update f₁ i x₁ + Function.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) : Function.update f₁⁻¹ i x₁⁻¹ = (Function.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) : Function.update (-f₁) i (-x₁) = -Function.update f₁ i x₁

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

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

@[simp]

theorem Function.const_eq_one {ι : Type u_1} {α : Type u_2} [One α] [Nonempty ι] {a : α} : Function.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} : Function.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} : Function.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} : Function.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 : (i : I) → f i) (f₂ : (i : I) → f i) (g₁ : (i : I) → f i) (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 : (i : I) → f i) (f₂ : (i : I) → f i) (g₁ : (i : I) → f i) (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 : (i : I) → f i) (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 : (i : I) → f i) (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 : (i : I) → f i) (f₂ : (i : I) → f i) (g₁ : (i : I) → f i) (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 : (i : I) → f i) (f₂ : (i : I) → f i) (g₁ : (i : I) → f i) (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' := ⋯ }

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' := ⋯ }

@[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), (Function.ExtendByOne.hom R s) f a = Function.extend s f 1 a

@[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), (Function.ExtendByZero.hom R s) f a = Function.extend s f 0 a

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 (Pi.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 (Pi.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 (Pi.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 (Pi.single i)

@[simp]

theorem Sigma.curry_one {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → One (γ a b)] : Sigma.curry 1 = 1

@[simp]

theorem Sigma.curry_zero {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → Zero (γ a b)] : Sigma.curry 0 = 0

@[simp]

theorem Sigma.uncurry_one {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → One (γ a b)] : Sigma.uncurry 1 = 1

@[simp]

theorem Sigma.uncurry_zero {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → Zero (γ a b)] : Sigma.uncurry 0 = 0

@[simp]

theorem Sigma.curry_mul {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → Mul (γ a b)] (x : (i : (a : α) × β a) → γ i.fst i.snd) (y : (i : (a : α) × β a) → γ i.fst i.snd) : Sigma.curry (x * y) = Sigma.curry x * Sigma.curry y

@[simp]

theorem Sigma.curry_add {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → Add (γ a b)] (x : (i : (a : α) × β a) → γ i.fst i.snd) (y : (i : (a : α) × β a) → γ i.fst i.snd) : Sigma.curry (x + y) = Sigma.curry x + Sigma.curry y

@[simp]

theorem Sigma.uncurry_mul {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → Mul (γ a b)] (x : (a : α) → (b : β a) → γ a b) (y : (a : α) → (b : β a) → γ a b) : Sigma.uncurry (x * y) = Sigma.uncurry x * Sigma.uncurry y

@[simp]

theorem Sigma.uncurry_add {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → Add (γ a b)] (x : (a : α) → (b : β a) → γ a b) (y : (a : α) → (b : β a) → γ a b) : Sigma.uncurry (x + y) = Sigma.uncurry x + Sigma.uncurry y

@[simp]

theorem Sigma.curry_inv {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → Inv (γ a b)] (x : (i : (a : α) × β a) → γ i.fst i.snd) : Sigma.curry x⁻¹ = (Sigma.curry x)⁻¹

@[simp]

theorem Sigma.curry_neg {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → Neg (γ a b)] (x : (i : (a : α) × β a) → γ i.fst i.snd) : Sigma.curry (-x) = -Sigma.curry x

@[simp]

theorem Sigma.uncurry_inv {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → Inv (γ a b)] (x : (a : α) → (b : β a) → γ a b) : Sigma.uncurry x⁻¹ = (Sigma.uncurry x)⁻¹

@[simp]

theorem Sigma.uncurry_neg {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [(a : α) → (b : β a) → Neg (γ a b)] (x : (a : α) → (b : β a) → γ a b) : Sigma.uncurry (-x) = -Sigma.uncurry x

@[simp]

theorem Sigma.curry_mulSingle {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [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) : Sigma.curry (Pi.mulSingle i x) = Pi.mulSingle i.fst (Pi.mulSingle i.snd x)

@[simp]

theorem Sigma.curry_single {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [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) : Sigma.curry (Pi.single i x) = Pi.single i.fst (Pi.single i.snd x)

@[simp]

theorem Sigma.uncurry_mulSingle_mulSingle {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [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) : Sigma.uncurry (Pi.mulSingle a (Pi.mulSingle b x)) = Pi.mulSingle ⟨a, b⟩ x

@[simp]

theorem Sigma.uncurry_single_single {α : Type u_3} {β : α → Type u_4} {γ : (a : α) → β a → Type u_5} [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) : Sigma.uncurry (Pi.single a (Pi.single b x)) = Pi.single ⟨a, b⟩ x