# 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_zero {α : Type u_3} {β : Type u_4} [Zero β] [] :
= {0}
@[simp]
theorem Set.range_one {α : Type u_3} {β : Type u_4} [One β] [] :
= {1}
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 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
∀ {x : Add M} {x_1 : } (f g : AddHom M N), f + g = fun (x_2 : M) => f x_2 + g x_2
theorem MulHom.coe_mul {M : Type u_3} {N : Type u_4} :
∀ {x : Mul M} {x_1 : } (f g : M →ₙ* N), f * g = fun (x_2 : M) => f x_2 * g x_2
theorem Pi.addHom.proof_1 {I : Type u_1} {f : IType u_2} {γ : Type u_3} [(i : I) → Add (f i)] [Add γ] (g : (i : I) → AddHom γ (f i)) (x : γ) (y : γ) :
(fun (x : γ) (i : I) => (g i) x) (x + y) = (fun (x : γ) (i : I) => (g i) x) x + (fun (x : γ) (i : I) => (g i) x) y
def Pi.addHom {I : Type u} {f : IType 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
• = { toFun := fun (x : γ) (i : I) => (g i) x, map_add' := }
Instances For
@[simp]
theorem Pi.mulHom_apply {I : Type u} {f : IType 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
@[simp]
theorem Pi.addHom_apply {I : Type u} {f : IType 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
def Pi.mulHom {I : Type u} {f : IType 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
• = { toFun := fun (x : γ) (i : I) => (g i) x, map_mul' := }
Instances For
theorem Pi.addHom_injective {I : Type u} {f : IType v} {γ : Type w} [] [(i : I) → Add (f i)] [Add γ] (g : (i : I) → AddHom γ (f i)) (hg : ∀ (i : I), Function.Injective (g i)) :
abbrev Pi.addHom_injective.match_1 {I : Type u_1} (motive : Prop) :
∀ (x : ), (∀ (i : I), motive )motive x
Equations
• =
Instances For
theorem Pi.mulHom_injective {I : Type u} {f : IType v} {γ : Type w} [] [(i : I) → Mul (f i)] [Mul γ] (g : (i : I) → γ →ₙ* f i) (hg : ∀ (i : I), Function.Injective (g i)) :
theorem Pi.addMonoidHom.proof_2 {I : Type u_1} {f : IType u_2} {γ : Type u_3} [(i : I) → AddZeroClass (f i)] [] (g : (i : I) → γ →+ f i) (x : γ) (y : γ) :
(Pi.addHom fun (i : I) => (g i)).toFun (x + y) = (Pi.addHom fun (i : I) => (g i)).toFun x + (Pi.addHom fun (i : I) => (g i)).toFun y
theorem Pi.addMonoidHom.proof_1 {I : Type u_1} {f : IType u_2} {γ : Type u_3} [(i : I) → AddZeroClass (f i)] [] (g : (i : I) → γ →+ f i) :
(fun (x : γ) (i : I) => (g i) x) 0 = 0
def Pi.addMonoidHom {I : Type u} {f : IType v} {γ : Type w} [(i : I) → AddZeroClass (f i)] [] (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
• = let __src := Pi.addHom fun (i : I) => (g i); { toFun := fun (x : γ) (i : I) => (g i) x, map_zero' := , map_add' := }
Instances For
@[simp]
theorem Pi.monoidHom_apply {I : Type u} {f : IType v} {γ : Type w} [(i : I) → MulOneClass (f i)] [] (g : (i : I) → γ →* f i) (x : γ) (i : I) :
(Pi.monoidHom g) x i = (g i) x
@[simp]
theorem Pi.addMonoidHom_apply {I : Type u} {f : IType v} {γ : Type w} [(i : I) → AddZeroClass (f i)] [] (g : (i : I) → γ →+ f i) (x : γ) (i : I) :
x i = (g i) x
def Pi.monoidHom {I : Type u} {f : IType v} {γ : Type w} [(i : I) → MulOneClass (f i)] [] (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
• = let __src := Pi.mulHom fun (i : I) => (g i); { toFun := fun (x : γ) (i : I) => (g i) x, map_one' := , map_mul' := }
Instances For
theorem Pi.addMonoidHom_injective {I : Type u} {f : IType v} {γ : Type w} [] [(i : I) → AddZeroClass (f i)] [] (g : (i : I) → γ →+ f i) (hg : ∀ (i : I), Function.Injective (g i)) :
theorem Pi.monoidHom_injective {I : Type u} {f : IType v} {γ : Type w} [] [(i : I) → MulOneClass (f i)] [] (g : (i : I) → γ →* f i) (hg : ∀ (i : I), Function.Injective (g i)) :
def Pi.evalAddHom {I : Type u} (f : IType 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
• = { toFun := fun (g : (i : I) → f i) => g i, map_add' := }
Instances For
@[simp]
theorem Pi.evalAddHom_apply {I : Type u} (f : IType 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 : IType v) [(i : I) → Mul (f i)] (i : I) (g : (i : I) → f i) :
(Pi.evalMulHom f i) g = g i
def Pi.evalMulHom {I : Type u} (f : IType 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
• = { toFun := fun (g : (i : I) → f i) => g i, map_mul' := }
Instances For
theorem Pi.constAddHom.proof_1 (α : Type u_1) (β : Type u_2) [Add β] :
∀ (x x_1 : β), Function.const α (x + x_1) = Function.const α (x + x_1)
def Pi.constAddHom (α : Type u_3) (β : Type u_4) [Add β] :

Function.const as an AddHom.

Equations
• = { toFun := , map_add' := }
Instances For
@[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 Pi.constMulHom (α : Type u_3) (β : Type u_4) [Mul β] :
β →ₙ* αβ

Function.const as a MulHom.

Equations
• = { toFun := , map_mul' := }
Instances For
theorem AddHom.coeFn.proof_1 (α : Type u_1) (β : Type u_2) [Add α] [] :
∀ (x x_1 : AddHom α β), (fun (g : AddHom α β) => g) (x + x_1) = (fun (g : AddHom α β) => g) (x + x_1)
def AddHom.coeFn (α : Type u_3) (β : Type u_4) [Add α] [] :

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

See also AddHom.eval.

Equations
• = { toFun := fun (g : AddHom α β) => g, map_add' := }
Instances For
@[simp]
theorem AddHom.coeFn_apply (α : Type u_3) (β : Type u_4) [Add α] [] (g : AddHom α β) (a : α) :
(AddHom.coeFn α β) g a = g a
@[simp]
theorem MulHom.coeFn_apply (α : Type u_3) (β : Type u_4) [Mul α] [] (g : α →ₙ* β) (a : α) :
(MulHom.coeFn α β) g a = g a
def MulHom.coeFn (α : Type u_3) (β : Type u_4) [Mul α] [] :
(α →ₙ* β) →ₙ* αβ

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

See also MulHom.eval.

Equations
• = { toFun := fun (g : α →ₙ* β) => g, map_mul' := }
Instances For
theorem AddHom.compLeft.proof_1 {α : Type u_3} {β : Type u_2} [Add α] [Add β] (f : AddHom α β) (I : Type u_1) :
∀ (x x_1 : Iα), (fun (h : Iα) => f h) (x + x_1) = (fun (h : Iα) => f h) x + (fun (h : Iα) => f h) x_1
def AddHom.compLeft {α : Type u_3} {β : Type u_4} [Add α] [Add β] (f : AddHom α β) (I : Type u_5) :

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 : Iα) => f h, map_add' := }
Instances For
@[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 : Iα) :
∀ (a : I), (f.compLeft I) h a = (f h) a
@[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 : Iα) :
∀ (a : I), (f.compLeft I) h a = (f h) a
def MulHom.compLeft {α : Type u_3} {β : Type u_4} [Mul α] [Mul β] (f : α →ₙ* β) (I : Type u_5) :
(Iα) →ₙ* Iβ

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 : Iα) => f h, map_mul' := }
Instances For
theorem Pi.evalAddMonoidHom.proof_1 {I : Type u_2} (f : IType u_1) [(i : I) → AddZeroClass (f i)] (i : I) :
0 i = 0
def Pi.evalAddMonoidHom {I : Type u} (f : IType 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
• = { toFun := fun (g : (i : I) → f i) => g i, map_zero' := , map_add' := }
Instances For
theorem Pi.evalAddMonoidHom.proof_2 {I : Type u_2} (f : IType u_1) [(i : I) → AddZeroClass (f i)] (i : I) :
∀ (x x_1 : (i : I) → f i), (x + x_1) i = x i + x_1 i
@[simp]
theorem Pi.evalAddMonoidHom_apply {I : Type u} (f : IType v) [(i : I) → AddZeroClass (f i)] (i : I) (g : (i : I) → f i) :
g = g i
@[simp]
theorem Pi.evalMonoidHom_apply {I : Type u} (f : IType v) [(i : I) → MulOneClass (f i)] (i : I) (g : (i : I) → f i) :
(Pi.evalMonoidHom f i) g = g i
def Pi.evalMonoidHom {I : Type u} (f : IType 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
• = { toFun := fun (g : (i : I) → f i) => g i, map_one' := , map_mul' := }
Instances For
def Pi.constAddMonoidHom (α : Type u_3) (β : Type u_4) [] :
β →+ αβ

Function.const as an AddMonoidHom.

Equations
• = { toFun := , map_zero' := , map_add' := }
Instances For
theorem Pi.constAddMonoidHom.proof_2 (α : Type u_1) (β : Type u_2) [] :
∀ (x x_1 : β), { toFun := , map_zero' := }.toFun (x + x_1) = { toFun := , map_zero' := }.toFun (x + x_1)
theorem Pi.constAddMonoidHom.proof_1 (α : Type u_1) (β : Type u_2) [] :
=
@[simp]
theorem Pi.constMonoidHom_apply (α : Type u_3) (β : Type u_4) [] (a : β) :
∀ (a_1 : α), a a_1 = Function.const α a a_1
@[simp]
theorem Pi.constAddMonoidHom_apply (α : Type u_3) (β : Type u_4) [] (a : β) :
∀ (a_1 : α), a a_1 = Function.const α a a_1
def Pi.constMonoidHom (α : Type u_3) (β : Type u_4) [] :
β →* αβ

Function.const as a MonoidHom.

Equations
• = { toFun := , map_one' := , map_mul' := }
Instances For
theorem AddMonoidHom.coeFn.proof_1 (α : Type u_1) (β : Type u_2) [] [] :
(fun (g : α →+ β) => g) 0 = (fun (g : α →+ β) => g) 0
def AddMonoidHom.coeFn (α : Type u_3) (β : Type u_4) [] [] :
(α →+ β) →+ αβ

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

See also AddMonoidHom.eval.

Equations
• = { toFun := fun (g : α →+ β) => g, map_zero' := , map_add' := }
Instances For
theorem AddMonoidHom.coeFn.proof_2 (α : Type u_1) (β : Type u_2) [] [] :
∀ (x x_1 : α →+ β), { toFun := fun (g : α →+ β) => g, map_zero' := }.toFun (x + x_1) = { toFun := fun (g : α →+ β) => g, map_zero' := }.toFun (x + x_1)
@[simp]
theorem MonoidHom.coeFn_apply (α : Type u_3) (β : Type u_4) [] [] (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) [] [] (g : α →+ β) (a : α) :
g a = g a
def MonoidHom.coeFn (α : Type u_3) (β : Type u_4) [] [] :
(α →* β) →* αβ

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

See also MonoidHom.eval.

Equations
• = { toFun := fun (g : α →* β) => g, map_one' := , map_mul' := }
Instances For
def AddMonoidHom.compLeft {α : Type u_3} {β : Type u_4} [] [] (f : α →+ β) (I : Type u_5) :
(Iα) →+ Iβ

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 : Iα) => f h, map_zero' := , map_add' := }
Instances For
theorem AddMonoidHom.compLeft.proof_1 {α : Type u_3} {β : Type u_2} [] [] (f : α →+ β) (I : Type u_1) :
(fun (h : Iα) => f h) 0 = 0
theorem AddMonoidHom.compLeft.proof_2 {α : Type u_3} {β : Type u_2} [] [] (f : α →+ β) (I : Type u_1) :
∀ (x x_1 : Iα), { toFun := fun (h : Iα) => f h, map_zero' := }.toFun (x + x_1) = { toFun := fun (h : Iα) => f h, map_zero' := }.toFun x + { toFun := fun (h : Iα) => f h, map_zero' := }.toFun x_1
@[simp]
theorem AddMonoidHom.compLeft_apply {α : Type u_3} {β : Type u_4} [] [] (f : α →+ β) (I : Type u_5) (h : Iα) :
∀ (a : I), (f.compLeft I) h a = (f h) a
@[simp]
theorem MonoidHom.compLeft_apply {α : Type u_3} {β : Type u_4} [] [] (f : α →* β) (I : Type u_5) (h : Iα) :
∀ (a : I), (f.compLeft I) h a = (f h) a
def MonoidHom.compLeft {α : Type u_3} {β : Type u_4} [] [] (f : α →* β) (I : Type u_5) :
(Iα) →* Iβ

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 : Iα) => f h, map_one' := , map_mul' := }
Instances For
def ZeroHom.single {I : Type u} (f : IType v) [] [(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
• = { toFun := , map_zero' := }
Instances For
def OneHom.mulSingle {I : Type u} (f : IType v) [] [(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
• = { toFun := , map_one' := }
Instances For
@[simp]
theorem ZeroHom.single_apply {I : Type u} (f : IType v) [] [(i : I) → Zero (f i)] (i : I) (x : f i) :
(ZeroHom.single f i) x =
@[simp]
theorem OneHom.mulSingle_apply {I : Type u} (f : IType v) [] [(i : I) → One (f i)] (i : I) (x : f i) :
theorem AddMonoidHom.single.proof_1 {I : Type u_1} (f : IType u_2) [] [(i : I) → AddZeroClass (f i)] (i : I) (x₁ : f i) (x₂ : f i) :
Pi.single i (x₁ + x₂) = fun (j : I) => Pi.single i x₁ j + Pi.single i x₂ j
def AddMonoidHom.single {I : Type u} (f : IType v) [] [(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
• = let __src := ; { toZeroHom := __src, map_add' := }
Instances For
def MonoidHom.mulSingle {I : Type u} (f : IType v) [] [(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
• = let __src := ; { toOneHom := __src, map_mul' := }
Instances For
@[simp]
theorem AddMonoidHom.single_apply {I : Type u} (f : IType v) [] [(i : I) → AddZeroClass (f i)] (i : I) (x : f i) :
x =
@[simp]
theorem MonoidHom.mulSingle_apply {I : Type u} (f : IType v) [] [(i : I) → MulOneClass (f i)] (i : I) (x : f i) :
x =
theorem Pi.single_sup {I : Type u} {f : IType v} [] [(i : I) → SemilatticeSup (f i)] [(i : I) → Zero (f i)] (i : I) (x : f i) (y : f i) :
Pi.single i (x y) =
theorem Pi.mulSingle_sup {I : Type u} {f : IType v} [] [(i : I) → SemilatticeSup (f i)] [(i : I) → One (f i)] (i : I) (x : f i) (y : f i) :
theorem Pi.single_inf {I : Type u} {f : IType v} [] [(i : I) → SemilatticeInf (f i)] [(i : I) → Zero (f i)] (i : I) (x : f i) (y : f i) :
Pi.single i (x y) =
theorem Pi.mulSingle_inf {I : Type u} {f : IType v} [] [(i : I) → SemilatticeInf (f i)] [(i : I) → One (f i)] (i : I) (x : f i) (y : f i) :
theorem Pi.single_add {I : Type u} {f : IType v} [] [(i : I) → AddZeroClass (f i)] (i : I) (x : f i) (y : f i) :
Pi.single i (x + y) = +
theorem Pi.mulSingle_mul {I : Type u} {f : IType v} [] [(i : I) → MulOneClass (f i)] (i : I) (x : f i) (y : f i) :
Pi.mulSingle i (x * y) = *
theorem Pi.single_neg {I : Type u} {f : IType v} [] [(i : I) → AddGroup (f i)] (i : I) (x : f i) :
Pi.single i (-x) = -
theorem Pi.mulSingle_inv {I : Type u} {f : IType v} [] [(i : I) → Group (f i)] (i : I) (x : f i) :
theorem Pi.single_sub {I : Type u} {f : IType v} [] [(i : I) → AddGroup (f i)] (i : I) (x : f i) (y : f i) :
Pi.single i (x - y) = -
theorem Pi.mulSingle_div {I : Type u} {f : IType v} [] [(i : I) → Group (f i)] (i : I) (x : f i) (y : f i) :
Pi.mulSingle i (x / y) = /
theorem Pi.single_addCommute {I : Type u} {f : IType v} [] [(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_commute {I : Type u} {f : IType v} [] [(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_apply_addCommute {I : Type u} {f : IType v} [] [(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.mulSingle_apply_commute {I : Type u} {f : IType v} [] [(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.update_eq_sub_add_single {I : Type u} {f : IType v} (i : I) [] [(i : I) → AddGroup (f i)] (g : (i : I) → f i) (x : f i) :
= g - Pi.single i (g i) +
theorem Pi.update_eq_div_mul_mulSingle {I : Type u} {f : IType v} (i : I) [] [(i : I) → Group (f i)] (g : (i : I) → f i) (x : f i) :
= g / Pi.mulSingle i (g i) *
theorem Pi.single_add_single_eq_single_add_single {I : Type u} [] {M : Type u_3} [] {k : I} {l : I} {m : I} {n : I} {u : M} {v : M} (hu : u 0) (hv : v 0) :
+ = + k = m l = n u = v k = n l = m u + v = 0 k = l m = n
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {I : Type u} [] {M : Type u_3} [] {k : I} {l : I} {m : I} {n : I} {u : M} {v : M} (hu : u 1) (hv : v 1) :
* = * k = m l = n u = v k = n l = m u * v = 1 k = l m = n
theorem AddSemiconjBy.pi {I : Type u} {f : IType 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)) :
theorem SemiconjBy.pi {I : Type u} {f : IType 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)) :
theorem Pi.addSemiconjBy_iff {I : Type u} {f : IType v} [(i : I) → Add (f i)] {x : (i : I) → f i} {y : (i : I) → f i} {z : (i : I) → f i} :
∀ (i : I), AddSemiconjBy (x i) (y i) (z i)
theorem Pi.semiconjBy_iff {I : Type u} {f : IType 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 AddCommute.pi {I : Type u} {f : IType v} [(i : I) → Add (f i)] {x : (i : I) → f i} {y : (i : I) → f i} (h : ∀ (i : I), AddCommute (x i) (y i)) :
theorem Commute.pi {I : Type u} {f : IType v} [(i : I) → Mul (f i)] {x : (i : I) → f i} {y : (i : I) → f i} (h : ∀ (i : I), Commute (x i) (y i)) :
theorem Pi.addCommute_iff {I : Type u} {f : IType v} [(i : I) → Add (f i)] {x : (i : I) → f i} {y : (i : I) → f i} :
∀ (i : I), AddCommute (x i) (y i)
theorem Pi.commute_iff {I : Type u} {f : IType 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)
@[simp]
theorem Function.update_zero {I : Type u} {f : IType v} [(i : I) → Zero (f i)] [] (i : I) :
= 0
@[simp]
theorem Function.update_one {I : Type u} {f : IType v} [(i : I) → One (f i)] [] (i : I) :
= 1
theorem Function.update_add {I : Type u} {f : IType v} [(i : I) → Add (f 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_mul {I : Type u} {f : IType v} [(i : I) → Mul (f 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_neg {I : Type u} {f : IType v} [(i : I) → Neg (f i)] [] (f₁ : (i : I) → f i) (i : I) (x₁ : f i) :
Function.update (-f₁) i (-x₁) = -Function.update f₁ i x₁
theorem Function.update_inv {I : Type u} {f : IType v} [(i : I) → Inv (f i)] [] (f₁ : (i : I) → f i) (i : I) (x₁ : f i) :
theorem Function.update_sub {I : Type u} {f : IType v} [(i : I) → Sub (f 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_div {I : Type u} {f : IType v} [(i : I) → Div (f 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_zero {ι : Type u_1} {α : Type u_2} [Zero α] [] {a : α} :
= 0 a = 0
@[simp]
theorem Function.const_eq_one {ι : Type u_1} {α : Type u_2} [One α] [] {a : α} :
= 1 a = 1
theorem Function.const_ne_zero {ι : Type u_1} {α : Type u_2} [Zero α] [] {a : α} :
0 a 0
theorem Function.const_ne_one {ι : Type u_1} {α : Type u_2} [One α] [] {a : α} :
1 a 1
theorem Set.piecewise_add {I : Type u} {f : IType 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_mul {I : Type u} {f : IType 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_neg {I : Type u} {f : IType 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_inv {I : Type u} {f : IType 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_sub {I : Type u} {f : IType 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
theorem Set.piecewise_div {I : Type u} {f : IType 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
noncomputable def Function.ExtendByZero.hom {ι : Type u_1} {η : Type v} (R : Type w) (s : ιη) [] :
(ιR) →+ ηR

Function.extend s f 0 as a bundled hom.

Equations
• = { toFun := fun (f : ιR) => , map_zero' := , map_add' := }
Instances For
theorem Function.ExtendByZero.hom.proof_1 {ι : Type u_3} {η : Type u_1} (R : Type u_2) (s : ιη) [] :
= 0
theorem Function.ExtendByZero.hom.proof_2 {ι : Type u_3} {η : Type u_1} (R : Type u_2) (s : ιη) [] (f : ιR) (g : ιR) :
{ toFun := fun (f : ιR) => , map_zero' := }.toFun (f + g) = { toFun := fun (f : ιR) => , map_zero' := }.toFun f + { toFun := fun (f : ιR) => , map_zero' := }.toFun g
@[simp]
theorem Function.ExtendByOne.hom_apply {ι : Type u_1} {η : Type v} (R : Type w) (s : ιη) [] (f : ιR) :
∀ (a : η), 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 : ιη) [] (f : ιR) :
∀ (a : η), f a = Function.extend s f 0 a
noncomputable def Function.ExtendByOne.hom {ι : Type u_1} {η : Type v} (R : Type w) (s : ιη) [] :
(ιR) →* ηR

Function.extend s f 1 as a bundled hom.

Equations
• = { toFun := fun (f : ιR) => , map_one' := , map_mul' := }
Instances For
theorem Pi.single_mono {I : Type u} {f : IType v} (i : I) [] [(i : I) → Preorder (f i)] [(i : I) → Zero (f i)] :
theorem Pi.mulSingle_mono {I : Type u} {f : IType v} (i : I) [] [(i : I) → Preorder (f i)] [(i : I) → One (f i)] :
theorem Pi.single_strictMono {I : Type u} {f : IType v} (i : I) [] [(i : I) → Preorder (f i)] [(i : I) → Zero (f i)] :
theorem Pi.mulSingle_strictMono {I : Type u} {f : IType v} (i : I) [] [(i : I) → Preorder (f i)] [(i : I) → One (f i)] :
@[simp]
theorem Sigma.curry_zero {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [(a : α) → (b : β a) → Zero (γ a b)] :
= 0
@[simp]
theorem Sigma.curry_one {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [(a : α) → (b : β a) → One (γ a b)] :
= 1
@[simp]
theorem Sigma.uncurry_zero {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [(a : α) → (b : β a) → Zero (γ a b)] :
@[simp]
theorem Sigma.uncurry_one {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [(a : α) → (b : β a) → One (γ a b)] :
@[simp]
theorem Sigma.curry_add {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType 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) =
@[simp]
theorem Sigma.curry_mul {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType 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) =
@[simp]
theorem Sigma.uncurry_add {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType 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) :
@[simp]
theorem Sigma.uncurry_mul {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType 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) :
@[simp]
theorem Sigma.curry_neg {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [(a : α) → (b : β a) → Neg (γ a b)] (x : (i : (a : α) × β a) → γ i.fst i.snd) :
@[simp]
theorem Sigma.curry_inv {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [(a : α) → (b : β a) → Inv (γ a b)] (x : (i : (a : α) × β a) → γ i.fst i.snd) :
@[simp]
theorem Sigma.uncurry_neg {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [(a : α) → (b : β a) → Neg (γ a b)] (x : (a : α) → (b : β a) → γ a b) :
@[simp]
theorem Sigma.uncurry_inv {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [(a : α) → (b : β a) → Inv (γ a b)] (x : (a : α) → (b : β a) → γ a b) :
@[simp]
theorem Sigma.curry_single {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [] [(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.curry_mulSingle {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [] [(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) :
@[simp]
theorem Sigma.uncurry_single_single {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [] [(a : α) → DecidableEq (β a)] [(a : α) → (b : β a) → Zero (γ a b)] (a : α) (b : β a) (x : γ a b) :
@[simp]
theorem Sigma.uncurry_mulSingle_mulSingle {α : Type u_3} {β : αType u_4} {γ : (a : α) → β aType u_5} [] [(a : α) → DecidableEq (β a)] [(a : α) → (b : β a) → One (γ a b)] (a : α) (b : β a) (x : γ a b) :