File: Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.lean

full_name: ContMDiffSection.toFun
code: /-- Bundled `n` times continuously differentiable sections of a vector bundle. -/
structure ContMDiffSection where
  /-- the underlying function of this section -/
  protected toFun : ∀ x, V x
  /-- proof that this section is `C^n` -/
  protected contMDiff_toFun : ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x ↦
    TotalSpace.mk' F x (toFun x)
goal: V x

full_name: ContMDiffSection.contMDiff_toFun
code: /-- Bundled `n` times continuously differentiable sections of a vector bundle. -/
structure ContMDiffSection where
  /-- the underlying function of this section -/
  protected toFun : ∀ x, V x
  /-- proof that this section is `C^n` -/
  protected contMDiff_toFun : ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x ↦
    TotalSpace.mk' F x (toFun x)
goal: ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x => Bundle.TotalSpace.mk' F x (self.toFun x)

full_name: ContMDiffSection
code: /-- Bundled `n` times continuously differentiable sections of a vector bundle. -/
structure ContMDiffSection where
  /-- the underlying function of this section -/
  protected toFun : ∀ x, V x
  /-- proof that this section is `C^n` -/
  protected contMDiff_toFun : ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x ↦
    TotalSpace.mk' F x (toFun x)
goal: Type (max u_12 u_6)

full_name: SmoothSection
code: /-- Bundled smooth sections of a vector bundle. -/
abbrev SmoothSection 
goal: Type (max u_12 u_6)

full_name: Manifold._aux_Mathlib_Geometry_Manifold_VectorBundle_SmoothSection___unexpand_ContMDiffSection_1
code: @[inherit_doc] scoped[Manifold] notation "Cₛ^" n "⟮" I "; " F ", " V "⟯" => ContMDiffSection I F n V
goal: Lean.PrettyPrinter.Unexpander

full_name: Manifold.«_aux_Mathlib_Geometry_Manifold_VectorBundle_SmoothSection___macroRules_Manifold_termCₛ^_⟮_;_,_⟯_1»
code: @[inherit_doc] scoped[Manifold] notation "Cₛ^" n "⟮" I "; " F ", " V "⟯" => ContMDiffSection I F n V
goal: Lean.Macro

full_name: ContMDiffSection.instDFunLike
code: instance : DFunLike Cₛ^n⟮I; F, V⟯ M V 
goal: DFunLike Cₛ^n⟮I; F, V⟯ M V

full_name: ContMDiffSection.coeFn_mk
code: @[simp]
theorem coeFn_mk (s : ∀ x, V x)
    (hs : ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x => TotalSpace.mk x (s x)) :
    (mk s hs : ∀ x, V x) = s 
goal: ⇑{ toFun := s, contMDiff_toFun := hs } = s

full_name: ContMDiffSection.contMDiff
code: protected theorem contMDiff (s : Cₛ^n⟮I; F, V⟯) :
    ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x => TotalSpace.mk' F x (s x : V x) 
goal: ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x => Bundle.TotalSpace.mk' F x (s x)

full_name: ContMDiffSection.smooth
code: protected theorem smooth (s : Cₛ^∞⟮I; F, V⟯) :
    Smooth I (I.prod 𝓘(𝕜, F)) fun x => TotalSpace.mk' F x (s x : V x) 
goal: Smooth I (I.prod 𝓘(𝕜, F)) fun x => Bundle.TotalSpace.mk' F x (s x)

full_name: ContMDiffSection.mdifferentiable'
code: protected theorem mdifferentiable' (s : Cₛ^n⟮I; F, V⟯) (hn : 1 ≤ n) :
    MDifferentiable I (I.prod 𝓘(𝕜, F)) fun x => TotalSpace.mk' F x (s x : V x) 
goal: MDifferentiable I (I.prod 𝓘(𝕜, F)) fun x => Bundle.TotalSpace.mk' F x (s x)

full_name: ContMDiffSection.mdifferentiable
code: protected theorem mdifferentiable (s : Cₛ^∞⟮I; F, V⟯) :
    MDifferentiable I (I.prod 𝓘(𝕜, F)) fun x => TotalSpace.mk' F x (s x : V x) 
goal: MDifferentiable I (I.prod 𝓘(𝕜, F)) fun x => Bundle.TotalSpace.mk' F x (s x)

full_name: ContMDiffSection.mdifferentiableAt
code: protected theorem mdifferentiableAt (s : Cₛ^∞⟮I; F, V⟯) {x} :
    MDifferentiableAt I (I.prod 𝓘(𝕜, F)) (fun x => TotalSpace.mk' F x (s x : V x)) x 
goal: MDifferentiableAt I (I.prod 𝓘(𝕜, F)) (fun x => Bundle.TotalSpace.mk' F x (s x)) x

full_name: ContMDiffSection.coe_inj
code: theorem coe_inj ⦃s t : Cₛ^n⟮I; F, V⟯⦄ (h : (s : ∀ x, V x) = t) : s = t 
goal: s = t

full_name: ContMDiffSection.coe_injective
code: theorem coe_injective : Injective ((↑) : Cₛ^n⟮I; F, V⟯ → ∀ x, V x) 
goal: Function.Injective DFunLike.coe

full_name: ContMDiffSection.ext
code: @[ext]
theorem ext (h : ∀ x, s x = t x) : s = t 
goal: s = t

full_name: ContMDiffSection.instAdd
code: instance instAdd : Add Cₛ^n⟮I; F, V⟯ 
goal: Add Cₛ^n⟮I; F, V⟯

full_name: ContMDiffSection.coe_add
code: @[simp]
theorem coe_add (s t : Cₛ^n⟮I; F, V⟯) : ⇑(s + t) = ⇑s + t 
goal: ⇑(s + t) = ⇑s + ⇑t

full_name: ContMDiffSection.instSub
code: instance instSub : Sub Cₛ^n⟮I; F, V⟯ 
goal: Sub Cₛ^n⟮I; F, V⟯

full_name: ContMDiffSection.coe_sub
code: @[simp]
theorem coe_sub (s t : Cₛ^n⟮I; F, V⟯) : ⇑(s - t) = s - t 
goal: ⇑(s - t) = ⇑s - ⇑t

full_name: ContMDiffSection.instZero
code: instance instZero : Zero Cₛ^n⟮I; F, V⟯ 
goal: Zero Cₛ^n⟮I; F, V⟯

full_name: ContMDiffSection.inhabited
code: instance inhabited : Inhabited Cₛ^n⟮I; F, V⟯ 
goal: Inhabited Cₛ^n⟮I; F, V⟯

full_name: ContMDiffSection.coe_zero
code: @[simp]
theorem coe_zero : ⇑(0 : Cₛ^n⟮I; F, V⟯) = 0 
goal: ⇑0 = 0

full_name: ContMDiffSection.instSMul
code: instance instSMul : SMul 𝕜 Cₛ^n⟮I; F, V⟯ 
goal: SMul 𝕜 Cₛ^n⟮I; F, V⟯

full_name: ContMDiffSection.coe_smul
code: @[simp]
theorem coe_smul (r : 𝕜) (s : Cₛ^n⟮I; F, V⟯) : ⇑(r • s : Cₛ^n⟮I; F, V⟯) = r • ⇑s 
goal: ⇑(r • s) = r • ⇑s

full_name: ContMDiffSection.instNeg
code: instance instNeg : Neg Cₛ^n⟮I; F, V⟯ 
goal: Neg Cₛ^n⟮I; F, V⟯

full_name: ContMDiffSection.coe_neg
code: @[simp]
theorem coe_neg (s : Cₛ^n⟮I; F, V⟯) : ⇑(-s : Cₛ^n⟮I; F, V⟯) = -s 
goal: ⇑(-s) = -⇑s

full_name: ContMDiffSection.instNSMul
code: instance instNSMul : SMul ℕ Cₛ^n⟮I; F, V⟯ 
goal: SMul ℕ Cₛ^n⟮I; F, V⟯

full_name: ContMDiffSection.coe_nsmul
code: @[simp]
theorem coe_nsmul (s : Cₛ^n⟮I; F, V⟯) (k : ℕ) : ⇑(k • s : Cₛ^n⟮I; F, V⟯) = k • ⇑s 
goal: ⇑(k • s) = k • ⇑s

full_name: ContMDiffSection.instZSMul
code: instance instZSMul : SMul ℤ Cₛ^n⟮I; F, V⟯ 
goal: SMul ℤ Cₛ^n⟮I; F, V⟯

full_name: ContMDiffSection.coe_zsmul
code: @[simp]
theorem coe_zsmul (s : Cₛ^n⟮I; F, V⟯) (z : ℤ) : ⇑(z • s : Cₛ^n⟮I; F, V⟯) = z • ⇑s 
goal: ⇑(z • s) = z • ⇑s

full_name: ContMDiffSection.instAddCommGroup
code: instance instAddCommGroup : AddCommGroup Cₛ^n⟮I; F, V⟯ 
goal: AddCommGroup Cₛ^n⟮I; F, V⟯

full_name: ContMDiffSection.coeAddHom
code: /-- The additive morphism from smooth sections to dependent maps. -/
def coeAddHom : Cₛ^n⟮I; F, V⟯ →+ ∀ x, V x 
goal: Cₛ^n⟮I; F, V⟯ →+ (x : M) → V x

full_name: ContMDiffSection.instModule
code: instance instModule : Module 𝕜 Cₛ^n⟮I; F, V⟯ 
goal: Module 𝕜 Cₛ^n⟮I; F, V⟯