File: Mathlib/Topology/Spectral/Hom.lean

full_name: IsSpectralMap
code: /-- A function between topological spaces is spectral if it is continuous and the preimage of every
compact open set is compact open. -/
structure IsSpectralMap (f : α → β) extends Continuous f : Prop where
  /-- A function between topological spaces is spectral if it is continuous and the preimage of
   every compact open set is compact open. -/
  isCompact_preimage_of_isOpen ⦃s : Set β⦄ : IsOpen s → IsCompact s → IsCompact (f ⁻¹' s)
goal: Prop

full_name: IsSpectralMap.isCompact_preimage_of_isOpen
code: /-- A function between topological spaces is spectral if it is continuous and the preimage of every
compact open set is compact open. -/
structure IsSpectralMap (f : α → β) extends Continuous f : Prop where
  /-- A function between topological spaces is spectral if it is continuous and the preimage of
   every compact open set is compact open. -/
  isCompact_preimage_of_isOpen ⦃s : Set β⦄ : IsOpen s → IsCompact s → IsCompact (f ⁻¹' s)
goal: IsOpen s → IsCompact s → IsCompact (f ⁻¹' s)

full_name: IsCompact.preimage_of_isOpen
code: theorem IsCompact.preimage_of_isOpen (hf : IsSpectralMap f) (h₀ : IsCompact s) (h₁ : IsOpen s) :
    IsCompact (f ⁻¹' s) 
goal: IsCompact (f ⁻¹' s)

full_name: IsSpectralMap.continuous
code: theorem IsSpectralMap.continuous {f : α → β} (hf : IsSpectralMap f) : Continuous f 
goal: Continuous f

full_name: isSpectralMap_id
code: theorem isSpectralMap_id : IsSpectralMap (@id α) 
goal: IsSpectralMap id

full_name: IsSpectralMap.comp
code: theorem IsSpectralMap.comp {f : β → γ} {g : α → β} (hf : IsSpectralMap f) (hg : IsSpectralMap g) :
    IsSpectralMap (f ∘ g) 
goal: IsSpectralMap (f ∘ g)

full_name: SpectralMap.spectral'
code: /-- The type of spectral maps from `α` to `β`. -/
structure SpectralMap (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] where
  /-- function between topological spaces-/
  toFun : α → β
  /-- proof that `toFun` is a spectral map-/
  spectral' : IsSpectralMap toFun
goal: IsSpectralMap self.toFun

full_name: SpectralMap
code: /-- The type of spectral maps from `α` to `β`. -/
structure SpectralMap (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] where
  /-- function between topological spaces-/
  toFun : α → β
  /-- proof that `toFun` is a spectral map-/
  spectral' : IsSpectralMap toFun
goal: Type (max u_6 u_7)

full_name: SpectralMap.toFun
code: /-- The type of spectral maps from `α` to `β`. -/
structure SpectralMap (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] where
  /-- function between topological spaces-/
  toFun : α → β
  /-- proof that `toFun` is a spectral map-/
  spectral' : IsSpectralMap toFun
goal: α → β

full_name: SpectralMapClass.map_spectral
code: /-- `SpectralMapClass F α β` states that `F` is a type of spectral maps.

You should extend this class when you extend `SpectralMap`. -/
class SpectralMapClass (F α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
    [FunLike F α β] : Prop where
  /-- statement that `F` is a type of spectral maps-/
  map_spectral (f : F) : IsSpectralMap f
goal: IsSpectralMap ⇑f

full_name: SpectralMapClass
code: /-- `SpectralMapClass F α β` states that `F` is a type of spectral maps.

You should extend this class when you extend `SpectralMap`. -/
class SpectralMapClass (F α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
    [FunLike F α β] : Prop where
  /-- statement that `F` is a type of spectral maps-/
  map_spectral (f : F) : IsSpectralMap f
goal: Prop

full_name: SpectralMapClass.toContinuousMapClass
code: instance (priority := 100) SpectralMapClass.toContinuousMapClass [TopologicalSpace α]
    [TopologicalSpace β] [FunLike F α β] [SpectralMapClass F α β] : ContinuousMapClass F α β 
goal: ContinuousMapClass F α β

full_name: instCoeTCSpectralMapOfSpectralMapClass
code: instance [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [SpectralMapClass F α β] :
    CoeTC F (SpectralMap α β) 
goal: CoeTC F (SpectralMap α β)

full_name: SpectralMap.toContinuousMap
code: /-- Reinterpret a `SpectralMap` as a `ContinuousMap`. -/
def toContinuousMap (f : SpectralMap α β) : ContinuousMap α β 
goal: C(α, β)

full_name: SpectralMap.instFunLike
code: instance instFunLike : FunLike (SpectralMap α β) α β 
goal: FunLike (SpectralMap α β) α β

full_name: SpectralMap.instSpectralMapClass
code: instance : SpectralMapClass (SpectralMap α β) α β 
goal: SpectralMapClass (SpectralMap α β) α β

full_name: SpectralMap.toFun_eq_coe
code: @[simp]
theorem toFun_eq_coe {f : SpectralMap α β} : f.toFun = (f : α → β) 
goal: f.toFun = ⇑f

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

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

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

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

full_name: SpectralMap.id
code: /-- `id` as a `SpectralMap`. -/
protected def id : SpectralMap α α 
goal: SpectralMap α α

full_name: SpectralMap.instInhabited
code: instance : Inhabited (SpectralMap α α) 
goal: Inhabited (SpectralMap α α)

full_name: SpectralMap.coe_id
code: @[simp]
theorem coe_id : ⇑(SpectralMap.id α) = id 
goal: ⇑(SpectralMap.id α) = id

full_name: SpectralMap.id_apply
code: @[simp]
theorem id_apply (a : α) : SpectralMap.id α a = a 
goal: (SpectralMap.id α) a = a

full_name: SpectralMap.comp
code: /-- Composition of `SpectralMap`s as a `SpectralMap`. -/
def comp (f : SpectralMap β γ) (g : SpectralMap α β) : SpectralMap α γ 
goal: SpectralMap α γ

full_name: SpectralMap.coe_comp
code: @[simp]
theorem coe_comp (f : SpectralMap β γ) (g : SpectralMap α β) : (f.comp g : α → γ) = f ∘ g 
goal: ⇑(f.comp g) = ⇑f ∘ ⇑g

full_name: SpectralMap.comp_apply
code: @[simp]
theorem comp_apply (f : SpectralMap β γ) (g : SpectralMap α β) (a : α) : (f.comp g) a = f (g a) 
goal: (f.comp g) a = f (g a)

full_name: SpectralMap.coe_comp_continuousMap
code: theorem coe_comp_continuousMap (f : SpectralMap β γ) (g : SpectralMap α β) :
    f ∘ g = (f : ContinuousMap β γ) ∘ (g : ContinuousMap α β) 
goal: ⇑f ∘ ⇑g = ⇑↑f ∘ ⇑↑g

full_name: SpectralMap.coe_comp_continuousMap'
code: @[simp]
theorem coe_comp_continuousMap' (f : SpectralMap β γ) (g : SpectralMap α β) :
    (f.comp g : ContinuousMap α γ) = (f : ContinuousMap β γ).comp g 
goal: ↑(f.comp g) = (↑f).comp ↑g

full_name: SpectralMap.comp_assoc
code: @[simp]
theorem comp_assoc (f : SpectralMap γ δ) (g : SpectralMap β γ) (h : SpectralMap α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
goal: (f.comp g).comp h = f.comp (g.comp h)

full_name: SpectralMap.comp_id
code: @[simp]
theorem comp_id (f : SpectralMap α β) : f.comp (SpectralMap.id α) = f 
goal: f.comp (SpectralMap.id α) = f

full_name: SpectralMap.id_comp
code: @[simp]
theorem id_comp (f : SpectralMap α β) : (SpectralMap.id β).comp f = f 
goal: (SpectralMap.id β).comp f = f

full_name: SpectralMap.cancel_right
code: @[simp]
theorem cancel_right {g₁ g₂ : SpectralMap β γ} {f : SpectralMap α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂

full_name: SpectralMap.cancel_left
code: @[simp]
theorem cancel_left {g : SpectralMap β γ} {f₁ f₂ : SpectralMap α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂