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

full_name: _aux_Mathlib_Geometry_Manifold_VectorBundle_Hom___unexpand_Bundle_TotalSpace_1
code: local notation "LE₁E₂" => TotalSpace (F₁ →L[𝕜] F₂) (Bundle.ContinuousLinearMap (RingHom.id 𝕜) E₁ E₂)
goal: Lean.PrettyPrinter.Unexpander

full_name: _aux_Mathlib_Geometry_Manifold_VectorBundle_Hom___macroRules_termLE₁E₂_1
code: local notation "LE₁E₂" => TotalSpace (F₁ →L[𝕜] F₂) (Bundle.ContinuousLinearMap (RingHom.id 𝕜) E₁ E₂)
goal: Lean.Macro

full_name: smoothOn_continuousLinearMapCoordChange
code: theorem smoothOn_continuousLinearMapCoordChange
    [SmoothVectorBundle F₁ E₁ IB] [SmoothVectorBundle F₂ E₂ IB] [MemTrivializationAtlas e₁]
    [MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂] [MemTrivializationAtlas e₂'] :
    SmoothOn IB 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂)
      (continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂')
      (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) 
goal: SmoothOn IB 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂)
  (Pretrivialization.continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂')
  (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))

full_name: hom_chart
code: theorem hom_chart (y₀ y : LE₁E₂) :
    chartAt (ModelProd HB (F₁ →L[𝕜] F₂)) y₀ y =
      (chartAt HB y₀.1 y.1, inCoordinates F₁ E₁ F₂ E₂ y₀.1 y.1 y₀.1 y.1 y.2) 
goal: ↑(chartAt (ModelProd HB (F₁ →L[𝕜] F₂)) y₀) y =
  (↑(chartAt HB y₀.proj) y.proj, ContinuousLinearMap.inCoordinates F₁ E₁ F₂ E₂ y₀.proj y.proj y₀.proj y.proj y.snd)

full_name: contMDiffAt_hom_bundle
code: theorem contMDiffAt_hom_bundle (f : M → LE₁E₂) {x₀ : M} {n : ℕ∞} :
    ContMDiffAt IM (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂)) n f x₀ ↔
      ContMDiffAt IM IB n (fun x => (f x).1) x₀ ∧
        ContMDiffAt IM 𝓘(𝕜, F₁ →L[𝕜] F₂) n
          (fun x => inCoordinates F₁ E₁ F₂ E₂ (f x₀).1 (f x).1 (f x₀).1 (f x).1 (f x).2) x₀ 
goal: ContMDiffAt IM (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂)) n f x₀ ↔
  ContMDiffAt IM IB n (fun x => (f x).proj) x₀ ∧
    ContMDiffAt IM 𝓘(𝕜, F₁ →L[𝕜] F₂) n
      (fun x => ContinuousLinearMap.inCoordinates F₁ E₁ F₂ E₂ (f x₀).proj (f x).proj (f x₀).proj (f x).proj (f x).snd)
      x₀

full_name: smoothAt_hom_bundle
code: theorem smoothAt_hom_bundle (f : M → LE₁E₂) {x₀ : M} :
    SmoothAt IM (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂)) f x₀ ↔
      SmoothAt IM IB (fun x => (f x).1) x₀ ∧
        SmoothAt IM 𝓘(𝕜, F₁ →L[𝕜] F₂)
          (fun x => inCoordinates F₁ E₁ F₂ E₂ (f x₀).1 (f x).1 (f x₀).1 (f x).1 (f x).2) x₀ 
goal: SmoothAt IM (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂)) f x₀ ↔
  SmoothAt IM IB (fun x => (f x).proj) x₀ ∧
    SmoothAt IM 𝓘(𝕜, F₁ →L[𝕜] F₂)
      (fun x => ContinuousLinearMap.inCoordinates F₁ E₁ F₂ E₂ (f x₀).proj (f x).proj (f x₀).proj (f x).proj (f x).snd)
      x₀

full_name: Bundle.ContinuousLinearMap.vectorPrebundle.isSmooth
code: instance Bundle.ContinuousLinearMap.vectorPrebundle.isSmooth :
    (Bundle.ContinuousLinearMap.vectorPrebundle (RingHom.id 𝕜) F₁ E₁ F₂ E₂).IsSmooth IB 
goal: VectorPrebundle.IsSmooth IB (Bundle.ContinuousLinearMap.vectorPrebundle (RingHom.id 𝕜) F₁ E₁ F₂ E₂)

full_name: SmoothVectorBundle.continuousLinearMap
code: instance SmoothVectorBundle.continuousLinearMap :
    SmoothVectorBundle (F₁ →L[𝕜] F₂) (Bundle.ContinuousLinearMap (RingHom.id 𝕜) E₁ E₂) IB 
goal: SmoothVectorBundle (F₁ →L[𝕜] F₂) (Bundle.ContinuousLinearMap (RingHom.id 𝕜) E₁ E₂) IB