File: Mathlib/Topology/ContinuousFunction/ZeroAtInfty.lean

full_name: ZeroAtInftyContinuousMap.zero_at_infty'
code: /-- `C₀(α, β)` is the type of continuous functions `α → β` which vanish at infinity from a
topological space to a metric space with a zero element.

When possible, instead of parametrizing results over `(f : C₀(α, β))`,
you should parametrize over `(F : Type*) [ZeroAtInftyContinuousMapClass F α β] (f : F)`.

When you extend this structure, make sure to extend `ZeroAtInftyContinuousMapClass`. -/
structure ZeroAtInftyContinuousMap (α : Type u) (β : Type v) [TopologicalSpace α] [Zero β]
    [TopologicalSpace β] extends ContinuousMap α β : Type max u v where
  /-- The function tends to zero along the `cocompact` filter. -/
  zero_at_infty' : Tendsto toFun (cocompact α) (𝓝 0)
goal: Filter.Tendsto self.toFun (Filter.cocompact α) (𝓝 0)

full_name: ZeroAtInftyContinuousMap
code: /-- `C₀(α, β)` is the type of continuous functions `α → β` which vanish at infinity from a
topological space to a metric space with a zero element.

When possible, instead of parametrizing results over `(f : C₀(α, β))`,
you should parametrize over `(F : Type*) [ZeroAtInftyContinuousMapClass F α β] (f : F)`.

When you extend this structure, make sure to extend `ZeroAtInftyContinuousMapClass`. -/
structure ZeroAtInftyContinuousMap (α : Type u) (β : Type v) [TopologicalSpace α] [Zero β]
    [TopologicalSpace β] extends ContinuousMap α β : Type max u v where
  /-- The function tends to zero along the `cocompact` filter. -/
  zero_at_infty' : Tendsto toFun (cocompact α) (𝓝 0)
goal: Type (max u v)

full_name: ZeroAtInfty._aux_Mathlib_Topology_ContinuousFunction_ZeroAtInfty___unexpand_ZeroAtInftyContinuousMap_1
code: @[inherit_doc]
scoped[ZeroAtInfty] notation (priority := 2000) "C₀(" α ", " β ")" => ZeroAtInftyContinuousMap α β
goal: Lean.PrettyPrinter.Unexpander

full_name: ZeroAtInfty.«_aux_Mathlib_Topology_ContinuousFunction_ZeroAtInfty___macroRules_ZeroAtInfty_termC₀(_,_)_1»
code: @[inherit_doc]
scoped[ZeroAtInfty] notation (priority := 2000) "C₀(" α ", " β ")" => ZeroAtInftyContinuousMap α β
goal: Lean.Macro

full_name: ZeroAtInfty._aux_Mathlib_Topology_ContinuousFunction_ZeroAtInfty___unexpand_ZeroAtInftyContinuousMap_2
code: @[inherit_doc]
scoped[ZeroAtInfty] notation α " →C₀ " β => ZeroAtInftyContinuousMap α β
goal: Lean.PrettyPrinter.Unexpander

full_name: ZeroAtInfty.«_aux_Mathlib_Topology_ContinuousFunction_ZeroAtInfty___macroRules_ZeroAtInfty_term_→C₀__1»
code: @[inherit_doc]
scoped[ZeroAtInfty] notation α " →C₀ " β => ZeroAtInftyContinuousMap α β
goal: Lean.Macro

full_name: ZeroAtInftyContinuousMapClass
code: /-- `ZeroAtInftyContinuousMapClass F α β` states that `F` is a type of continuous maps which
vanish at infinity.

You should also extend this typeclass when you extend `ZeroAtInftyContinuousMap`. -/
class ZeroAtInftyContinuousMapClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α]
    [Zero β] [TopologicalSpace β] [FunLike F α β] extends ContinuousMapClass F α β : Prop where
  /-- Each member of the class tends to zero along the `cocompact` filter. -/
  zero_at_infty (f : F) : Tendsto f (cocompact α) (𝓝 0)
goal: Prop

full_name: ZeroAtInftyContinuousMapClass.zero_at_infty
code: /-- `ZeroAtInftyContinuousMapClass F α β` states that `F` is a type of continuous maps which
vanish at infinity.

You should also extend this typeclass when you extend `ZeroAtInftyContinuousMap`. -/
class ZeroAtInftyContinuousMapClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α]
    [Zero β] [TopologicalSpace β] [FunLike F α β] extends ContinuousMapClass F α β : Prop where
  /-- Each member of the class tends to zero along the `cocompact` filter. -/
  zero_at_infty (f : F) : Tendsto f (cocompact α) (𝓝 0)
goal: Filter.Tendsto (⇑f) (Filter.cocompact α) (𝓝 0)

full_name: ZeroAtInftyContinuousMap.instFunLike
code: instance instFunLike : FunLike C₀(α, β) α β 
goal: FunLike (α →C₀ β) α β

full_name: ZeroAtInftyContinuousMap.instZeroAtInftyContinuousMapClass
code: instance instZeroAtInftyContinuousMapClass : ZeroAtInftyContinuousMapClass C₀(α, β) α β 
goal: ZeroAtInftyContinuousMapClass (α →C₀ β) α β

full_name: ZeroAtInftyContinuousMap.instCoeTC
code: instance instCoeTC : CoeTC F C₀(α, β) 
goal: CoeTC F (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.coe_toContinuousMap
code: @[simp]
theorem coe_toContinuousMap (f : C₀(α, β)) : (f.toContinuousMap : α → β) = f 
goal: ⇑f.toContinuousMap = ⇑f

full_name: ZeroAtInftyContinuousMap.ext
code: @[ext]
theorem ext {f g : C₀(α, β)} (h : ∀ x, f x = g x) : f = g 
goal: f = g

full_name: ZeroAtInftyContinuousMap.copy
code: /-- Copy of a `ZeroAtInftyContinuousMap` with a new `toFun` equal to the old one. Useful
to fix definitional equalities. -/
protected def copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : C₀(α, β) 
goal: α →C₀ β

full_name: ZeroAtInftyContinuousMap.coe_copy
code: @[simp]
theorem coe_copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
goal: ⇑(f.copy f' h) = f'

full_name: ZeroAtInftyContinuousMap.copy_eq
code: theorem copy_eq (f : C₀(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f 
goal: f.copy f' h = f

full_name: ZeroAtInftyContinuousMap.eq_of_empty
code: theorem eq_of_empty [IsEmpty α] (f g : C₀(α, β)) : f = g 
goal: f = g

full_name: ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty_symm_apply
code: /-- A continuous function on a compact space is automatically a continuous function vanishing at
infinity. -/
@[simps]
def ContinuousMap.liftZeroAtInfty [CompactSpace α] : C(α, β) ≃ C₀(α, β) 
goal: ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty.symm f = ↑f

full_name: ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty
code: /-- A continuous function on a compact space is automatically a continuous function vanishing at
infinity. -/
@[simps]
def ContinuousMap.liftZeroAtInfty [CompactSpace α] : C(α, β) ≃ C₀(α, β) 
goal: C(α, β) ≃ α →C₀ β

full_name: ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty_apply_toFun
code: /-- A continuous function on a compact space is automatically a continuous function vanishing at
infinity. -/
@[simps]
def ContinuousMap.liftZeroAtInfty [CompactSpace α] : C(α, β) ≃ C₀(α, β) 
goal: (ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty f) a = f a

full_name: ZeroAtInftyContinuousMap.zeroAtInftyContinuousMapClass.ofCompact
code: /-- A continuous function on a compact space is automatically a continuous function vanishing at
infinity. This is not an instance to avoid type class loops. -/
lemma zeroAtInftyContinuousMapClass.ofCompact {G : Type*} [FunLike G α β]
    [ContinuousMapClass G α β] [CompactSpace α] : ZeroAtInftyContinuousMapClass G α β 
goal: ZeroAtInftyContinuousMapClass G α β

full_name: ZeroAtInftyContinuousMap.instZero
code: instance instZero [Zero β] : Zero C₀(α, β) 
goal: Zero (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instInhabited
code: instance instInhabited [Zero β] : Inhabited C₀(α, β) 
goal: Inhabited (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.coe_zero
code: @[simp]
theorem coe_zero [Zero β] : ⇑(0 : C₀(α, β)) = 0 
goal: ⇑0 = 0

full_name: ZeroAtInftyContinuousMap.zero_apply
code: theorem zero_apply [Zero β] : (0 : C₀(α, β)) x = 0 
goal: 0 x = 0

full_name: ZeroAtInftyContinuousMap.instMul
code: instance instMul [MulZeroClass β] [ContinuousMul β] : Mul C₀(α, β) 
goal: Mul (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.coe_mul
code: @[simp]
theorem coe_mul [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : ⇑(f * g) = f * g 
goal: ⇑(f * g) = ⇑f * ⇑g

full_name: ZeroAtInftyContinuousMap.mul_apply
code: theorem mul_apply [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : (f * g) x = f x * g x 
goal: (f * g) x = f x * g x

full_name: ZeroAtInftyContinuousMap.instMulZeroClass
code: instance instMulZeroClass [MulZeroClass β] [ContinuousMul β] : MulZeroClass C₀(α, β) 
goal: MulZeroClass (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instSemigroupWithZero
code: instance instSemigroupWithZero [SemigroupWithZero β] [ContinuousMul β] :
    SemigroupWithZero C₀(α, β) 
goal: SemigroupWithZero (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instAdd
code: instance instAdd [AddZeroClass β] [ContinuousAdd β] : Add C₀(α, β) 
goal: Add (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.coe_add
code: @[simp]
theorem coe_add [AddZeroClass β] [ContinuousAdd β] (f g : C₀(α, β)) : ⇑(f + g) = f + g 
goal: ⇑(f + g) = ⇑f + ⇑g

full_name: ZeroAtInftyContinuousMap.add_apply
code: theorem add_apply [AddZeroClass β] [ContinuousAdd β] (f g : C₀(α, β)) : (f + g) x = f x + g x 
goal: (f + g) x = f x + g x

full_name: ZeroAtInftyContinuousMap.instAddZeroClass
code: instance instAddZeroClass [AddZeroClass β] [ContinuousAdd β] : AddZeroClass C₀(α, β) 
goal: AddZeroClass (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instSMul
code: instance instSMul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] :
    SMul R C₀(α, β) 
goal: SMul R (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.coe_smul
code: @[simp, norm_cast]
theorem coe_smul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R)
    (f : C₀(α, β)) : ⇑(r • f) = r • ⇑f 
goal: ⇑(r • f) = r • ⇑f

full_name: ZeroAtInftyContinuousMap.smul_apply
code: theorem smul_apply [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β]
    (r : R) (f : C₀(α, β)) (x : α) : (r • f) x = r • f x 
goal: (r • f) x = r • f x

full_name: ZeroAtInftyContinuousMap.instAddMonoid
code: instance instAddMonoid : AddMonoid C₀(α, β) 
goal: AddMonoid (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instAddCommMonoid
code: instance instAddCommMonoid [AddCommMonoid β] [ContinuousAdd β] : AddCommMonoid C₀(α, β) 
goal: AddCommMonoid (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNeg
code: instance instNeg : Neg C₀(α, β) 
goal: Neg (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.coe_neg
code: @[simp]
theorem coe_neg : ⇑(-f) = -f 
goal: ⇑(-f) = -⇑f

full_name: ZeroAtInftyContinuousMap.neg_apply
code: theorem neg_apply : (-f) x = -f x 
goal: (-f) x = -f x

full_name: ZeroAtInftyContinuousMap.instSub
code: instance instSub : Sub C₀(α, β) 
goal: Sub (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.coe_sub
code: @[simp]
theorem coe_sub : ⇑(f - g) = f - g 
goal: ⇑(f - g) = ⇑f - ⇑g

full_name: ZeroAtInftyContinuousMap.sub_apply
code: theorem sub_apply : (f - g) x = f x - g x 
goal: (f - g) x = f x - g x

full_name: ZeroAtInftyContinuousMap.instAddGroup
code: instance instAddGroup : AddGroup C₀(α, β) 
goal: AddGroup (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instAddCommGroup
code: instance instAddCommGroup [AddCommGroup β] [TopologicalAddGroup β] : AddCommGroup C₀(α, β) 
goal: AddCommGroup (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instIsCentralScalar
code: instance instIsCentralScalar [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [SMulWithZero Rᵐᵒᵖ β]
    [ContinuousConstSMul R β] [IsCentralScalar R β] : IsCentralScalar R C₀(α, β) 
goal: IsCentralScalar R (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instSMulWithZero
code: instance instSMulWithZero [Zero β] {R : Type*} [Zero R] [SMulWithZero R β]
    [ContinuousConstSMul R β] : SMulWithZero R C₀(α, β) 
goal: SMulWithZero R (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instMulActionWithZero
code: instance instMulActionWithZero [Zero β] {R : Type*} [MonoidWithZero R] [MulActionWithZero R β]
    [ContinuousConstSMul R β] : MulActionWithZero R C₀(α, β) 
goal: MulActionWithZero R (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instModule
code: instance instModule [AddCommMonoid β] [ContinuousAdd β] {R : Type*} [Semiring R] [Module R β]
    [ContinuousConstSMul R β] : Module R C₀(α, β) 
goal: Module R (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNonUnitalNonAssocSemiring
code: instance instNonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] [TopologicalSemiring β] :
    NonUnitalNonAssocSemiring C₀(α, β) 
goal: NonUnitalNonAssocSemiring (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNonUnitalSemiring
code: instance instNonUnitalSemiring [NonUnitalSemiring β] [TopologicalSemiring β] :
    NonUnitalSemiring C₀(α, β) 
goal: NonUnitalSemiring (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNonUnitalCommSemiring
code: instance instNonUnitalCommSemiring [NonUnitalCommSemiring β] [TopologicalSemiring β] :
    NonUnitalCommSemiring C₀(α, β) 
goal: NonUnitalCommSemiring (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNonUnitalNonAssocRing
code: instance instNonUnitalNonAssocRing [NonUnitalNonAssocRing β] [TopologicalRing β] :
    NonUnitalNonAssocRing C₀(α, β) 
goal: NonUnitalNonAssocRing (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNonUnitalRing
code: instance instNonUnitalRing [NonUnitalRing β] [TopologicalRing β] : NonUnitalRing C₀(α, β) 
goal: NonUnitalRing (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNonUnitalCommRing
code: instance instNonUnitalCommRing [NonUnitalCommRing β] [TopologicalRing β] :
    NonUnitalCommRing C₀(α, β) 
goal: NonUnitalCommRing (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instIsScalarTower
code: instance instIsScalarTower {R : Type*} [Semiring R] [NonUnitalNonAssocSemiring β]
    [TopologicalSemiring β] [Module R β] [ContinuousConstSMul R β] [IsScalarTower R β β] :
    IsScalarTower R C₀(α, β) C₀(α, β) 
goal: IsScalarTower R (α →C₀ β) (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instSMulCommClass
code: instance instSMulCommClass {R : Type*} [Semiring R] [NonUnitalNonAssocSemiring β]
    [TopologicalSemiring β] [Module R β] [ContinuousConstSMul R β] [SMulCommClass R β β] :
    SMulCommClass R C₀(α, β) C₀(α, β) 
goal: SMulCommClass R (α →C₀ β) (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.uniformContinuous
code: theorem uniformContinuous (f : F) : UniformContinuous (f : β → γ) 
goal: UniformContinuous ⇑f

full_name: ZeroAtInftyContinuousMap.bounded
code: protected theorem bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x) (f y) ≤ C 
goal: ∃ C, ∀ (x y : α), dist (f x) (f y) ≤ C

full_name: ZeroAtInftyContinuousMap.isBounded_range
code: theorem isBounded_range (f : C₀(α, β)) : IsBounded (range f) 
goal: Bornology.IsBounded (Set.range ⇑f)

full_name: ZeroAtInftyContinuousMap.isBounded_image
code: theorem isBounded_image (f : C₀(α, β)) (s : Set α) : IsBounded (f '' s) 
goal: Bornology.IsBounded (⇑f '' s)

full_name: ZeroAtInftyContinuousMap.instBoundedContinuousMapClass
code: instance (priority := 100) instBoundedContinuousMapClass : BoundedContinuousMapClass F α β 
goal: BoundedContinuousMapClass F α β

full_name: ZeroAtInftyContinuousMap.toBCF
code: /-- Construct a bounded continuous function from a continuous function vanishing at infinity. -/
@[simps!]
def toBCF (f : C₀(α, β)) : α →ᵇ β 
goal: α →ᵇ β

full_name: ZeroAtInftyContinuousMap.toBCF_apply
code: /-- Construct a bounded continuous function from a continuous function vanishing at infinity. -/
@[simps!]
def toBCF (f : C₀(α, β)) : α →ᵇ β 
goal: f.toBCF a = f a

full_name: ZeroAtInftyContinuousMap.toBCF_toFun
code: /-- Construct a bounded continuous function from a continuous function vanishing at infinity. -/
@[simps!]
def toBCF (f : C₀(α, β)) : α →ᵇ β 
goal: f.toBCF a = f a

full_name: ZeroAtInftyContinuousMap.toBCF_injective
code: theorem toBCF_injective : Function.Injective (toBCF : C₀(α, β) → α →ᵇ β) 
goal: Function.Injective ZeroAtInftyContinuousMap.toBCF

full_name: ZeroAtInftyContinuousMap.instPseudoMetricSpace
code: /-- The type of continuous functions vanishing at infinity, with the uniform distance induced by the
inclusion `ZeroAtInftyContinuousMap.toBCF`, is a pseudo-metric space. -/
noncomputable instance instPseudoMetricSpace : PseudoMetricSpace C₀(α, β) 
goal: PseudoMetricSpace (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instMetricSpace
code: /-- The type of continuous functions vanishing at infinity, with the uniform distance induced by the
inclusion `ZeroAtInftyContinuousMap.toBCF`, is a metric space. -/
noncomputable instance instMetricSpace {β : Type*} [MetricSpace β] [Zero β] :
    MetricSpace C₀(α, β) 
goal: MetricSpace (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.dist_toBCF_eq_dist
code: @[simp]
theorem dist_toBCF_eq_dist {f g : C₀(α, β)} : dist f.toBCF g.toBCF = dist f g 
goal: dist f.toBCF g.toBCF = dist f g

full_name: ZeroAtInftyContinuousMap.tendsto_iff_tendstoUniformly
code: /-- Convergence in the metric on `C₀(α, β)` is uniform convergence. -/
theorem tendsto_iff_tendstoUniformly {ι : Type*} {F : ι → C₀(α, β)} {f : C₀(α, β)} {l : Filter ι} :
    Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => F i) f l 
goal: Filter.Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => ⇑(F i)) (⇑f) l

full_name: ZeroAtInftyContinuousMap.isometry_toBCF
code: theorem isometry_toBCF : Isometry (toBCF : C₀(α, β) → α →ᵇ β) 
goal: Isometry ZeroAtInftyContinuousMap.toBCF

full_name: ZeroAtInftyContinuousMap.isClosed_range_toBCF
code: theorem isClosed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ β)) 
goal: IsClosed (Set.range ZeroAtInftyContinuousMap.toBCF)

full_name: ZeroAtInftyContinuousMap.instCompleteSpace
code: /-- Continuous functions vanishing at infinity taking values in a complete space form a
complete space. -/
instance instCompleteSpace [CompleteSpace β] : CompleteSpace C₀(α, β) 
goal: CompleteSpace (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instSeminormedAddCommGroup
code: noncomputable instance instSeminormedAddCommGroup [SeminormedAddCommGroup β] :
    SeminormedAddCommGroup C₀(α, β) 
goal: SeminormedAddCommGroup (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNormedAddCommGroup
code: noncomputable instance instNormedAddCommGroup [NormedAddCommGroup β] :
    NormedAddCommGroup C₀(α, β) 
goal: NormedAddCommGroup (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.norm_toBCF_eq_norm
code: @[simp]
theorem norm_toBCF_eq_norm {f : C₀(α, β)} : ‖f.toBCF‖ = ‖f‖ 
goal: ‖f.toBCF‖ = ‖f‖

full_name: ZeroAtInftyContinuousMap.instNormedSpace
code: instance : NormedSpace 𝕜 C₀(α, β) 
goal: NormedSpace 𝕜 (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNonUnitalSeminormedRing
code: noncomputable instance instNonUnitalSeminormedRing [NonUnitalSeminormedRing β] :
    NonUnitalSeminormedRing C₀(α, β) 
goal: NonUnitalSeminormedRing (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNonUnitalNormedRing
code: noncomputable instance instNonUnitalNormedRing [NonUnitalNormedRing β] :
    NonUnitalNormedRing C₀(α, β) 
goal: NonUnitalNormedRing (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNonUnitalSeminormedCommRing
code: noncomputable instance instNonUnitalSeminormedCommRing [NonUnitalSeminormedCommRing β] :
    NonUnitalSeminormedCommRing C₀(α, β) 
goal: NonUnitalSeminormedCommRing (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNonUnitalNormedCommRing
code: noncomputable instance instNonUnitalNormedCommRing [NonUnitalNormedCommRing β] :
    NonUnitalNormedCommRing C₀(α, β) 
goal: NonUnitalNormedCommRing (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instStar
code: instance instStar : Star C₀(α, β) 
goal: Star (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.coe_star
code: @[simp]
theorem coe_star (f : C₀(α, β)) : ⇑(star f) = star (⇑f) 
goal: ⇑(star f) = star ⇑f

full_name: ZeroAtInftyContinuousMap.star_apply
code: theorem star_apply (f : C₀(α, β)) (x : α) : (star f) x = star (f x) 
goal: (star f) x = star (f x)

full_name: ZeroAtInftyContinuousMap.instStarAddMonoid
code: instance instStarAddMonoid [ContinuousAdd β] : StarAddMonoid C₀(α, β) 
goal: StarAddMonoid (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instNormedStarGroup
code: instance instNormedStarGroup : NormedStarGroup C₀(α, β) 
goal: NormedStarGroup (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instStarModule
code: instance instStarModule : StarModule 𝕜 C₀(α, β) 
goal: StarModule 𝕜 (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instStarRing
code: instance instStarRing : StarRing C₀(α, β) 
goal: StarRing (α →C₀ β)

full_name: ZeroAtInftyContinuousMap.instCstarRing
code: instance instCstarRing [NonUnitalNormedRing β] [StarRing β] [CstarRing β] : CstarRing C₀(α, β) 
goal: CstarRing (α →C₀ β)

full_name: ZeroAtInftyContinuousMap._aux_Mathlib_Topology_ContinuousFunction_ZeroAtInfty___unexpand_CocompactMap_1
code: local notation α " →co " β => CocompactMap α β
goal: Lean.PrettyPrinter.Unexpander

full_name: ZeroAtInftyContinuousMap.«_aux_Mathlib_Topology_ContinuousFunction_ZeroAtInfty___macroRules_ZeroAtInftyContinuousMap_term_→co__1»
code: local notation α " →co " β => CocompactMap α β
goal: Lean.Macro

full_name: ZeroAtInftyContinuousMap.comp
code: /-- Composition of a continuous function vanishing at infinity with a cocompact map yields another
continuous function vanishing at infinity. -/
def comp (f : C₀(γ, δ)) (g : β →co γ) : C₀(β, δ) 
goal: β →C₀ δ

full_name: ZeroAtInftyContinuousMap.coe_comp_to_continuous_fun
code: @[simp]
theorem coe_comp_to_continuous_fun (f : C₀(γ, δ)) (g : β →co γ) : ((f.comp g) : β → δ) = f ∘ g 
goal: ⇑(f.comp g) = ⇑f ∘ ⇑g

full_name: ZeroAtInftyContinuousMap.comp_id
code: @[simp]
theorem comp_id (f : C₀(γ, δ)) : f.comp (CocompactMap.id γ) = f 
goal: f.comp (CocompactMap.id γ) = f

full_name: ZeroAtInftyContinuousMap.comp_assoc
code: @[simp]
theorem comp_assoc (f : C₀(γ, δ)) (g : β →co γ) (h : α →co β) :
    (f.comp g).comp h = f.comp (g.comp h) 
goal: (f.comp g).comp h = f.comp (g.comp h)

full_name: ZeroAtInftyContinuousMap.zero_comp
code: @[simp]
theorem zero_comp (g : β →co γ) : (0 : C₀(γ, δ)).comp g = 0 
goal: ZeroAtInftyContinuousMap.comp 0 g = 0

full_name: ZeroAtInftyContinuousMap.compAddMonoidHom
code: /-- Composition as an additive monoid homomorphism. -/
def compAddMonoidHom [AddMonoid δ] [ContinuousAdd δ] (g : β →co γ) : C₀(γ, δ) →+ C₀(β, δ) 
goal: (γ →C₀ δ) →+ β →C₀ δ

full_name: ZeroAtInftyContinuousMap.compMulHom
code: /-- Composition as a semigroup homomorphism. -/
def compMulHom [MulZeroClass δ] [ContinuousMul δ] (g : β →co γ) : C₀(γ, δ) →ₙ* C₀(β, δ) 
goal: (γ →C₀ δ) →ₙ* β →C₀ δ

full_name: ZeroAtInftyContinuousMap.compLinearMap
code: /-- Composition as a linear map. -/
def compLinearMap [AddCommMonoid δ] [ContinuousAdd δ] {R : Type*} [Semiring R] [Module R δ]
    [ContinuousConstSMul R δ] (g : β →co γ) : C₀(γ, δ) →ₗ[R] C₀(β, δ) 
goal: (γ →C₀ δ) →ₗ[R] β →C₀ δ

full_name: ZeroAtInftyContinuousMap.compNonUnitalAlgHom
code: /-- Composition as a non-unital algebra homomorphism. -/
def compNonUnitalAlgHom {R : Type*} [Semiring R] [NonUnitalNonAssocSemiring δ]
    [TopologicalSemiring δ] [Module R δ] [ContinuousConstSMul R δ] (g : β →co γ) :
    C₀(γ, δ) →ₙₐ[R] C₀(β, δ) 
goal: (γ →C₀ δ) →ₙₐ[R] β →C₀ δ