ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Bundle.contMDiffWithinAt_totalSpace
theorem contMDiffWithinAt_totalSpace (f : M → TotalSpace F E) {s : Set M} {x₀ : M} :
    ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F)) n f s x₀ ↔
      ContMDiffWithinAt IM IB n (fun x => (f x).proj) s x₀ ∧
      ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun x ↦ (trivializationAt F E (f x₀).proj (f x)).2) s x₀ 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} {EM : Type u_10} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_11} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] {n : ℕ∞} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] (f : M → Bundle.TotalSpace F E) {s : Set M} {x₀ : M}
Goal: ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F)) n f s x₀ ↔
  ContMDiffWithinAt IM IB n (fun x => (f x).proj) s x₀ ∧
    ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun x => (↑(trivializationAt F E (f x₀).proj) (f x)).2) s x₀
-- Bundle.contMDiffAt_totalSpace
theorem contMDiffAt_totalSpace (f : M → TotalSpace F E) (x₀ : M) :
    ContMDiffAt IM (IB.prod 𝓘(𝕜, F)) n f x₀ ↔
      ContMDiffAt IM IB n (fun x => (f x).proj) x₀ ∧
        ContMDiffAt IM 𝓘(𝕜, F) n (fun x => (trivializationAt F E (f x₀).proj (f x)).2) x₀ 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} {EM : Type u_10} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_11} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] {n : ℕ∞} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] (f : M → Bundle.TotalSpace F E) (x₀ : M)
Goal: ContMDiffAt IM (IB.prod 𝓘(𝕜, F)) n f x₀ ↔
  ContMDiffAt IM IB n (fun x => (f x).proj) x₀ ∧
    ContMDiffAt IM 𝓘(𝕜, F) n (fun x => (↑(trivializationAt F E (f x₀).proj) (f x)).2) x₀
-- Bundle.contMDiffAt_section
theorem contMDiffAt_section (s : ∀ x, E x) (x₀ : B) :
    ContMDiffAt IB (IB.prod 𝓘(𝕜, F)) n (fun x => TotalSpace.mk' F x (s x)) x₀ ↔
      ContMDiffAt IB 𝓘(𝕜, F) n (fun x ↦ (trivializationAt F E x₀ ⟨x, s x⟩).2) x₀ 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} {n : ℕ∞} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] (s : (x : B) → E x) (x₀ : B)
Goal: ContMDiffAt IB (IB.prod 𝓘(𝕜, F)) n (fun x => Bundle.TotalSpace.mk' F x (s x)) x₀ ↔
  ContMDiffAt IB 𝓘(𝕜, F) n (fun x => (↑(trivializationAt F E x₀) { proj := x, snd := s x }).2) x₀
-- Bundle.contMDiff_proj
theorem contMDiff_proj : ContMDiff (IB.prod 𝓘(𝕜, F)) IB n (π F E) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} (E : B → Type u_6) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} {n : ℕ∞} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E]
Goal: ContMDiff (IB.prod 𝓘(𝕜, F)) IB n Bundle.TotalSpace.proj
-- Bundle.smooth_proj
theorem smooth_proj : Smooth (IB.prod 𝓘(𝕜, F)) IB (π F E) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} (E : B → Type u_6) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E]
Goal: Smooth (IB.prod 𝓘(𝕜, F)) IB Bundle.TotalSpace.proj
-- Bundle.contMDiffOn_proj
theorem contMDiffOn_proj {s : Set (TotalSpace F E)} :
    ContMDiffOn (IB.prod 𝓘(𝕜, F)) IB n (π F E) s 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} (E : B → Type u_6) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} {n : ℕ∞} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] {s : Set (Bundle.TotalSpace F E)}
Goal: ContMDiffOn (IB.prod 𝓘(𝕜, F)) IB n Bundle.TotalSpace.proj s
-- Bundle.smoothOn_proj
theorem smoothOn_proj {s : Set (TotalSpace F E)} : SmoothOn (IB.prod 𝓘(𝕜, F)) IB (π F E) s 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} (E : B → Type u_6) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] {s : Set (Bundle.TotalSpace F E)}
Goal: SmoothOn (IB.prod 𝓘(𝕜, F)) IB Bundle.TotalSpace.proj s
-- Bundle.contMDiffAt_proj
theorem contMDiffAt_proj {p : TotalSpace F E} : ContMDiffAt (IB.prod 𝓘(𝕜, F)) IB n (π F E) p 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} (E : B → Type u_6) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} {n : ℕ∞} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] {p : Bundle.TotalSpace F E}
Goal: ContMDiffAt (IB.prod 𝓘(𝕜, F)) IB n Bundle.TotalSpace.proj p
-- Bundle.smoothAt_proj
theorem smoothAt_proj {p : TotalSpace F E} : SmoothAt (IB.prod 𝓘(𝕜, F)) IB (π F E) p 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} (E : B → Type u_6) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] {p : Bundle.TotalSpace F E}
Goal: SmoothAt (IB.prod 𝓘(𝕜, F)) IB Bundle.TotalSpace.proj p
-- Bundle.contMDiffWithinAt_proj
theorem contMDiffWithinAt_proj {s : Set (TotalSpace F E)} {p : TotalSpace F E} :
    ContMDiffWithinAt (IB.prod 𝓘(𝕜, F)) IB n (π F E) s p 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} (E : B → Type u_6) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} {n : ℕ∞} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] {s : Set (Bundle.TotalSpace F E)} {p : Bundle.TotalSpace F E}
Goal: ContMDiffWithinAt (IB.prod 𝓘(𝕜, F)) IB n Bundle.TotalSpace.proj s p
-- Bundle.smoothWithinAt_proj
theorem smoothWithinAt_proj {s : Set (TotalSpace F E)} {p : TotalSpace F E} :
    SmoothWithinAt (IB.prod 𝓘(𝕜, F)) IB (π F E) s p 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} (E : B → Type u_6) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] {s : Set (Bundle.TotalSpace F E)} {p : Bundle.TotalSpace F E}
Goal: SmoothWithinAt (IB.prod 𝓘(𝕜, F)) IB Bundle.TotalSpace.proj s p
-- Bundle.smooth_zeroSection
theorem smooth_zeroSection : Smooth IB (IB.prod 𝓘(𝕜, F)) (zeroSection F E) 
Arguments: (𝕜 : Type u_1) {B : Type u_2} {F : Type u_4} (E : B → Type u_6) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E]
Goal: Smooth IB (IB.prod 𝓘(𝕜, F)) (Bundle.zeroSection F E)
-- SmoothVectorBundle.smoothOn_coordChangeL
class SmoothVectorBundle : Prop where
  protected smoothOn_coordChangeL :
    ∀ (e e' : Trivialization F (π F E)) [MemTrivializationAtlas e] [MemTrivializationAtlas e'],
      SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun b : B => (e.coordChangeL 𝕜 e' b : F →L[𝕜] F))
        (e.baseSet ∩ e'.baseSet)
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [self : SmoothVectorBundle F E IB] (e e' : Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e] [MemTrivializationAtlas e']
Goal: SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun b => ↑(Trivialization.coordChangeL 𝕜 e e' b)) (e.baseSet ∩ e'.baseSet)
-- smoothOn_coordChangeL
theorem smoothOn_coordChangeL :
    SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun b : B => (e.coordChangeL 𝕜 e' b : F →L[𝕜] F))
      (e.baseSet ∩ e'.baseSet) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] (e e' : Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e] [MemTrivializationAtlas e']
Goal: SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun b => ↑(Trivialization.coordChangeL 𝕜 e e' b)) (e.baseSet ∩ e'.baseSet)
-- smoothOn_symm_coordChangeL
theorem smoothOn_symm_coordChangeL :
    SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun b : B => ((e.coordChangeL 𝕜 e' b).symm : F →L[𝕜] F))
      (e.baseSet ∩ e'.baseSet) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] (e e' : Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e] [MemTrivializationAtlas e']
Goal: SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun b => ↑(Trivialization.coordChangeL 𝕜 e e' b).symm) (e.baseSet ∩ e'.baseSet)
-- contMDiffOn_coordChangeL
theorem contMDiffOn_coordChangeL :
    ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun b : B => (e.coordChangeL 𝕜 e' b : F →L[𝕜] F))
      (e.baseSet ∩ e'.baseSet) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] {n : ℕ∞} [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] (e e' : Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e] [MemTrivializationAtlas e']
Goal: ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun b => ↑(Trivialization.coordChangeL 𝕜 e e' b)) (e.baseSet ∩ e'.baseSet)
-- contMDiffOn_symm_coordChangeL
theorem contMDiffOn_symm_coordChangeL :
    ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun b : B => ((e.coordChangeL 𝕜 e' b).symm : F →L[𝕜] F))
      (e.baseSet ∩ e'.baseSet) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] {n : ℕ∞} [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] (e e' : Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e] [MemTrivializationAtlas e']
Goal: ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun b => ↑(Trivialization.coordChangeL 𝕜 e e' b).symm) (e.baseSet ∩ e'.baseSet)
-- contMDiffAt_coordChangeL
theorem contMDiffAt_coordChangeL {x : B} (h : x ∈ e.baseSet) (h' : x ∈ e'.baseSet) :
    ContMDiffAt IB 𝓘(𝕜, F →L[𝕜] F) n (fun b : B => (e.coordChangeL 𝕜 e' b : F →L[𝕜] F)) x 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] {n : ℕ∞} [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] {e e' : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] [MemTrivializationAtlas e'] {x : B} (h : x ∈ e.baseSet) (h' : x ∈ e'.baseSet)
Goal: ContMDiffAt IB 𝓘(𝕜, F →L[𝕜] F) n (fun b => ↑(Trivialization.coordChangeL 𝕜 e e' b)) x
-- smoothAt_coordChangeL
theorem smoothAt_coordChangeL {x : B} (h : x ∈ e.baseSet) (h' : x ∈ e'.baseSet) :
    SmoothAt IB 𝓘(𝕜, F →L[𝕜] F) (fun b : B => (e.coordChangeL 𝕜 e' b : F →L[𝕜] F)) x 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] {e e' : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] [MemTrivializationAtlas e'] {x : B} (h : x ∈ e.baseSet) (h' : x ∈ e'.baseSet)
Goal: SmoothAt IB 𝓘(𝕜, F →L[𝕜] F) (fun b => ↑(Trivialization.coordChangeL 𝕜 e e' b)) x
-- ContMDiffWithinAt.coordChangeL
protected theorem ContMDiffWithinAt.coordChangeL
    (hf : ContMDiffWithinAt IM IB n f s x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet) :
    ContMDiffWithinAt IM 𝓘(𝕜, F →L[𝕜] F) n (fun y ↦ (e.coordChangeL 𝕜 e' (f y) : F →L[𝕜] F)) s x 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] {EM : Type u_9} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_10} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] {n : ℕ∞} [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] {e e' : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] [MemTrivializationAtlas e'] {s : Set M} {f : M → B} {x : M} (hf : ContMDiffWithinAt IM IB n f s x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet)
Goal: ContMDiffWithinAt IM 𝓘(𝕜, F →L[𝕜] F) n (fun y => ↑(Trivialization.coordChangeL 𝕜 e e' (f y))) s x
-- ContMDiffAt.coordChangeL
protected nonrec theorem ContMDiffAt.coordChangeL
    (hf : ContMDiffAt IM IB n f x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet) :
    ContMDiffAt IM 𝓘(𝕜, F →L[𝕜] F) n (fun y ↦ (e.coordChangeL 𝕜 e' (f y) : F →L[𝕜] F)) x 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] {EM : Type u_9} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_10} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] {n : ℕ∞} [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] {e e' : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] [MemTrivializationAtlas e'] {f : M → B} {x : M} (hf : ContMDiffAt IM IB n f x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet)
Goal: ContMDiffAt IM 𝓘(𝕜, F →L[𝕜] F) n (fun y => ↑(Trivialization.coordChangeL 𝕜 e e' (f y))) x
-- ContMDiffOn.coordChangeL
protected theorem ContMDiffOn.coordChangeL
    (hf : ContMDiffOn IM IB n f s) (he : MapsTo f s e.baseSet) (he' : MapsTo f s e'.baseSet) :
    ContMDiffOn IM 𝓘(𝕜, F →L[𝕜] F) n (fun y ↦ (e.coordChangeL 𝕜 e' (f y) : F →L[𝕜] F)) s 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] {EM : Type u_9} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_10} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] {n : ℕ∞} [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] {e e' : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] [MemTrivializationAtlas e'] {s : Set M} {f : M → B} (hf : ContMDiffOn IM IB n f s) (he : Set.MapsTo f s e.baseSet) (he' : Set.MapsTo f s e'.baseSet)
Goal: ContMDiffOn IM 𝓘(𝕜, F →L[𝕜] F) n (fun y => ↑(Trivialization.coordChangeL 𝕜 e e' (f y))) s
-- ContMDiff.coordChangeL
protected theorem ContMDiff.coordChangeL
    (hf : ContMDiff IM IB n f) (he : ∀ x, f x ∈ e.baseSet) (he' : ∀ x, f x ∈ e'.baseSet) :
    ContMDiff IM 𝓘(𝕜, F →L[𝕜] F) n (fun y ↦ (e.coordChangeL 𝕜 e' (f y) : F →L[𝕜] F)) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] {EM : Type u_9} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_10} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] {n : ℕ∞} [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] {e e' : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] [MemTrivializationAtlas e'] {f : M → B} (hf : ContMDiff IM IB n f) (he : ∀ (x : M), f x ∈ e.baseSet) (he' : ∀ (x : M), f x ∈ e'.baseSet)
Goal: ContMDiff IM 𝓘(𝕜, F →L[𝕜] F) n fun y => ↑(Trivialization.coordChangeL 𝕜 e e' (f y))
-- SmoothWithinAt.coordChangeL
protected nonrec theorem SmoothWithinAt.coordChangeL
    (hf : SmoothWithinAt IM IB f s x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet) :
    SmoothWithinAt IM 𝓘(𝕜, F →L[𝕜] F) (fun y ↦ (e.coordChangeL 𝕜 e' (f y) : F →L[𝕜] F)) s x 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] {EM : Type u_9} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_10} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] {e e' : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] [MemTrivializationAtlas e'] {s : Set M} {f : M → B} {x : M} (hf : SmoothWithinAt IM IB f s x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet)
Goal: SmoothWithinAt IM 𝓘(𝕜, F →L[𝕜] F) (fun y => ↑(Trivialization.coordChangeL 𝕜 e e' (f y))) s x
-- SmoothAt.coordChangeL
protected nonrec theorem SmoothAt.coordChangeL
    (hf : SmoothAt IM IB f x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet) :
    SmoothAt IM 𝓘(𝕜, F →L[𝕜] F) (fun y ↦ (e.coordChangeL 𝕜 e' (f y) : F →L[𝕜] F)) x 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] {EM : Type u_9} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_10} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] {e e' : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] [MemTrivializationAtlas e'] {f : M → B} {x : M} (hf : SmoothAt IM IB f x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet)
Goal: SmoothAt IM 𝓘(𝕜, F →L[𝕜] F) (fun y => ↑(Trivialization.coordChangeL 𝕜 e e' (f y))) x