Documentation

Mathlib.Algebra.Group.Pi.Lemmas

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.

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@[simp]
theorem Set.range_one {α : Type u_3} {β : Type u_4} [One β] [Nonempty α] :
range 1 = {1}
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@[simp]
theorem Set.range_zero {α : Type u_3} {β : Type u_4} [Zero β] [Nonempty α] :
range 0 = {0}
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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 univ else ∅
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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 univ else ∅
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theorem Pi.one_mono {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [One β] :
Monotone 1
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theorem Pi.zero_mono {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [Zero β] :
Monotone 0
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theorem Pi.one_anti {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [One β] :
Antitone 1
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theorem Pi.zero_anti {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [Zero β] :
Antitone 0
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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 : M) => f x * g x
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theorem AddHom.coe_add {M : Type u_3} {N : Type u_4} {x✝ : Add M} {x✝¹ : AddCommSemigroup N} (f g : M →ₙ+ N) :
⇑f + ⇑g = fun (x : M) => f x + g x
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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' := ⋯ }
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def Pi.addHom {I : Type u} {f : I → Type v} {γ : Type w} [(i : I) → Add (f i)] [Add γ] (g : (i : I) → γ →ₙ+ f i) :
γ →ₙ+ (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' := ⋯ }
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@[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) → γ →ₙ+ f i) (x : γ) (i : I) :
(addHom g) x i = (g i) x
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@[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) :
(mulHom g) x i = (g i) x
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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 ⇑(mulHom g)
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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) → γ →ₙ+ f i) (hg : ∀ (i : I), Function.Injective ⇑(g i)) :
Function.Injective ⇑(addHom g)
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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' := ⋯ }
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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' := ⋯ }
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@[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) :
(addMonoidHom g) x i = (g i) x
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@[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) :
(monoidHom g) x i = (g i) x
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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 ⇑(monoidHom g)
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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 ⇑(addMonoidHom g)
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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' := ⋯ }
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def Pi.evalAddHom {I : Type u} (f : I → Type v) [(i : I) → Add (f i)] (i : I) :
((i : I) → f i) →ₙ+ 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' := ⋯ }
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@[simp]
theorem Pi.evalAddHom_apply {I : Type u} (f : I → Type v) [(i : I) → Add (f i)] (i : I) (g : (i : I) → f i) :
(evalAddHom f i) g = g i
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@[simp]
theorem Pi.evalMulHom_apply {I : Type u} (f : I → Type v) [(i : I) → Mul (f i)] (i : I) (g : (i : I) → f i) :
(evalMulHom f i) g = g i
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def Pi.constMulHom (α : Type u_3) (β : Type u_4) [Mul β] :
β →ₙ* α → β

Function.const as a MulHom.

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

Function.const as an AddHom.

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@[simp]
theorem Pi.constAddHom_apply (α : Type u_3) (β : Type u_4) [Add β] (a : β) (a✝ : α) :
(constAddHom α β) a a✝ = Function.const α a a✝
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@[simp]
theorem Pi.constMulHom_apply (α : Type u_3) (β : Type u_4) [Mul β] (a : β) (a✝ : α) :
(constMulHom α β) a a✝ = Function.const α a a✝
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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.

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def AddHom.coeFn (α : Type u_3) (β : Type u_4) [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.

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@[simp]
theorem MulHom.coeFn_apply (α : Type u_3) (β : Type u_4) [Mul α] [CommSemigroup β] (g : α →ₙ* β) (a : α) :
(coeFn α β) g a = g a
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@[simp]
theorem AddHom.coeFn_apply (α : Type u_3) (β : Type u_4) [Add α] [AddCommSemigroup β] (g : α →ₙ+ β) (a : α) :
(coeFn α β) g a = g a
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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' := ⋯ }
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def AddHom.compLeft {α : Type u_3} {β : Type u_4} [Add α] [Add β] (f : α →ₙ+ β) (I : Type u_5) :
(I → α) →ₙ+ 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' := ⋯ }
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@[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✝
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@[simp]
theorem AddHom.compLeft_apply {α : Type u_3} {β : Type u_4} [Add α] [Add β] (f : α →ₙ+ β) (I : Type u_5) (h : I → α) (a✝ : I) :
(f.compLeft I) h a✝ = (⇑f ∘ h) a✝
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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' := ⋯ }
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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' := ⋯ }
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@[simp]
theorem Pi.evalAddMonoidHom_apply {I : Type u} (f : I → Type v) [(i : I) → AddZeroClass (f i)] (i : I) (g : (i : I) → f i) :
(evalAddMonoidHom f i) g = g i
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@[simp]
theorem Pi.evalMonoidHom_apply {I : Type u} (f : I → Type v) [(i : I) → MulOneClass (f i)] (i : I) (g : (i : I) → f i) :
(evalMonoidHom f i) g = g i
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def Pi.constMonoidHom (α : Type u_3) (β : Type u_4) [MulOneClass β] :
β →* α → β

Function.const as a MonoidHom.

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

Function.const as an AddMonoidHom.

Equations
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@[simp]
theorem Pi.constMonoidHom_apply (α : Type u_3) (β : Type u_4) [MulOneClass β] (a : β) (a✝ : α) :
(constMonoidHom α β) a a✝ = Function.const α a a✝
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@[simp]
theorem Pi.constAddMonoidHom_apply (α : Type u_3) (β : Type u_4) [AddZeroClass β] (a : β) (a✝ : α) :
(constAddMonoidHom α β) a a✝ = Function.const α a a✝
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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
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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
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@[simp]
theorem MonoidHom.coeFn_apply (α : Type u_3) (β : Type u_4) [MulOneClass α] [CommMonoid β] (g : α →* β) (a : α) :
(coeFn α β) g a = g a
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@[simp]
theorem AddMonoidHom.coeFn_apply (α : Type u_3) (β : Type u_4) [AddZeroClass α] [AddCommMonoid β] (g : α →+ β) (a : α) :
(coeFn α β) g a = g a
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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' := ⋯ }
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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' := ⋯ }
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@[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✝
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@[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✝
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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.

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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
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@[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
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@[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
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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
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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
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@[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
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@[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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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

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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

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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.

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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.

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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
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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
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theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {I : Type u} [DecidableEq I] {M : Type u_3} [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
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theorem Pi.single_add_single_eq_single_add_single {I : Type u} [DecidableEq I] {M : Type u_3} [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
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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
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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
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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)
source
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)
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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
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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
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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)
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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)
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@[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
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@[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
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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
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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
source
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
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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
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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
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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
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@[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
source
@[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
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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
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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
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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
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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
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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
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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
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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
source
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
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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
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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
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@[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✝
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@[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✝
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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)
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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)
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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)
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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)
source
@[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)] :
curry 1 = 1
source
@[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)] :
curry 0 = 0
source
@[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)] :
uncurry 1 = 1
source
@[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)] :
uncurry 0 = 0
source
@[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 y : (i : (a : α) × β a) → γ i.fst i.snd) :
curry (x * y) = curry x * curry y
source
@[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 y : (i : (a : α) × β a) → γ i.fst i.snd) :
curry (x + y) = curry x + curry y
source
@[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 y : (a : α) → (b : β a) → γ a b) :
uncurry (x * y) = uncurry x * uncurry y
source
@[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 y : (a : α) → (b : β a) → γ a b) :
uncurry (x + y) = uncurry x + uncurry y
source
@[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) :
curry x⁻¹ = (curry x)⁻¹
source
@[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) :
curry (-x) = -curry x
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@[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) :
uncurry x⁻¹ = (uncurry x)⁻¹
source
@[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) :
uncurry (-x) = -uncurry x
source
@[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) :
curry (Pi.mulSingle i x) = Pi.mulSingle i.fst (Pi.mulSingle i.snd x)
source
@[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) :
curry (Pi.single i x) = Pi.single i.fst (Pi.single i.snd x)
source
@[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) :
uncurry (Pi.mulSingle a (Pi.mulSingle b x)) = Pi.mulSingle ⟨a, b⟩ x
source
@[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) :
uncurry (Pi.single a (Pi.single b x)) = Pi.single ⟨a, b⟩ x