ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SmoothOn.coordChangeL
protected nonrec theorem SmoothOn.coordChangeL
    (hf : SmoothOn IM IB f s) (he : MapsTo f s e.baseSet) (he' : MapsTo f s e'.baseSet) :
    SmoothOn IM 𝓘(𝕜, F →L[𝕜] F) (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] [(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 : SmoothOn IM IB f s) (he : Set.MapsTo f s e.baseSet) (he' : Set.MapsTo f s e'.baseSet)
Goal: SmoothOn IM 𝓘(𝕜, F →L[𝕜] F) (fun y => ↑(Trivialization.coordChangeL 𝕜 e e' (f y))) s
-- Smooth.coordChangeL
protected nonrec theorem Smooth.coordChangeL
    (hf : Smooth IM IB f) (he : ∀ x, f x ∈ e.baseSet) (he' : ∀ x, f x ∈ e'.baseSet) :
    Smooth IM 𝓘(𝕜, F →L[𝕜] F) (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] [(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 : Smooth IM IB f) (he : ∀ (x : M), f x ∈ e.baseSet) (he' : ∀ (x : M), f x ∈ e'.baseSet)
Goal: Smooth IM 𝓘(𝕜, F →L[𝕜] F) fun y => ↑(Trivialization.coordChangeL 𝕜 e e' (f y))
-- ContMDiffWithinAt.coordChange
protected theorem ContMDiffWithinAt.coordChange
    (hf : ContMDiffWithinAt IM IB n f s x) (hg : ContMDiffWithinAt IM 𝓘(𝕜, F) n g s x)
    (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet) :
    ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y ↦ e.coordChange e' (f y) (g y)) 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} {g : M → F} {x : M} (hf : ContMDiffWithinAt IM IB n f s x) (hg : ContMDiffWithinAt IM 𝓘(𝕜, F) n g s x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet)
Goal: ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y => e.coordChange e' (f y) (g y)) s x
-- ContMDiffAt.coordChange
protected nonrec theorem ContMDiffAt.coordChange
    (hf : ContMDiffAt IM IB n f x) (hg : ContMDiffAt IM 𝓘(𝕜, F) n g x) (he : f x ∈ e.baseSet)
    (he' : f x ∈ e'.baseSet) :
    ContMDiffAt IM 𝓘(𝕜, F) n (fun y ↦ e.coordChange e' (f y) (g y)) 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} {g : M → F} {x : M} (hf : ContMDiffAt IM IB n f x) (hg : ContMDiffAt IM 𝓘(𝕜, F) n g x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet)
Goal: ContMDiffAt IM 𝓘(𝕜, F) n (fun y => e.coordChange e' (f y) (g y)) x
-- ContMDiffOn.coordChange
protected theorem ContMDiffOn.coordChange (hf : ContMDiffOn IM IB n f s)
    (hg : ContMDiffOn IM 𝓘(𝕜, F) n g s) (he : MapsTo f s e.baseSet) (he' : MapsTo f s e'.baseSet) :
    ContMDiffOn IM 𝓘(𝕜, F) n (fun y ↦ e.coordChange e' (f y) (g y)) 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} {g : M → F} (hf : ContMDiffOn IM IB n f s) (hg : ContMDiffOn IM 𝓘(𝕜, F) n g s) (he : Set.MapsTo f s e.baseSet) (he' : Set.MapsTo f s e'.baseSet)
Goal: ContMDiffOn IM 𝓘(𝕜, F) n (fun y => e.coordChange e' (f y) (g y)) s
-- ContMDiff.coordChange
protected theorem ContMDiff.coordChange (hf : ContMDiff IM IB n f)
    (hg : ContMDiff IM 𝓘(𝕜, F) n g) (he : ∀ x, f x ∈ e.baseSet) (he' : ∀ x, f x ∈ e'.baseSet) :
    ContMDiff IM 𝓘(𝕜, F) n (fun y ↦ e.coordChange e' (f y) (g y)) 
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} {g : M → F} (hf : ContMDiff IM IB n f) (hg : ContMDiff IM 𝓘(𝕜, F) n g) (he : ∀ (x : M), f x ∈ e.baseSet) (he' : ∀ (x : M), f x ∈ e'.baseSet)
Goal: ContMDiff IM 𝓘(𝕜, F) n fun y => e.coordChange e' (f y) (g y)
-- SmoothWithinAt.coordChange
protected nonrec theorem SmoothWithinAt.coordChange
    (hf : SmoothWithinAt IM IB f s x) (hg : SmoothWithinAt IM 𝓘(𝕜, F) g s x)
    (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet) :
    SmoothWithinAt IM 𝓘(𝕜, F) (fun y ↦ e.coordChange e' (f y) (g y)) 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} {g : M → F} {x : M} (hf : SmoothWithinAt IM IB f s x) (hg : SmoothWithinAt IM 𝓘(𝕜, F) g s x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet)
Goal: SmoothWithinAt IM 𝓘(𝕜, F) (fun y => e.coordChange e' (f y) (g y)) s x
-- SmoothAt.coordChange
protected nonrec theorem SmoothAt.coordChange (hf : SmoothAt IM IB f x)
    (hg : SmoothAt IM 𝓘(𝕜, F) g x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet) :
    SmoothAt IM 𝓘(𝕜, F) (fun y ↦ e.coordChange e' (f y) (g y)) 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} {g : M → F} {x : M} (hf : SmoothAt IM IB f x) (hg : SmoothAt IM 𝓘(𝕜, F) g x) (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet)
Goal: SmoothAt IM 𝓘(𝕜, F) (fun y => e.coordChange e' (f y) (g y)) x
-- SmoothOn.coordChange
protected nonrec theorem SmoothOn.coordChange (hf : SmoothOn IM IB f s)
    (hg : SmoothOn IM 𝓘(𝕜, F) g s) (he : MapsTo f s e.baseSet) (he' : MapsTo f s e'.baseSet) :
    SmoothOn IM 𝓘(𝕜, F) (fun y ↦ e.coordChange e' (f y) (g y)) 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] [(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} {g : M → F} (hf : SmoothOn IM IB f s) (hg : SmoothOn IM 𝓘(𝕜, F) g s) (he : Set.MapsTo f s e.baseSet) (he' : Set.MapsTo f s e'.baseSet)
Goal: SmoothOn IM 𝓘(𝕜, F) (fun y => e.coordChange e' (f y) (g y)) s
-- Smooth.coordChange
protected theorem Smooth.coordChange (hf : Smooth IM IB f)
    (hg : Smooth IM 𝓘(𝕜, F) g) (he : ∀ x, f x ∈ e.baseSet) (he' : ∀ x, f x ∈ e'.baseSet) :
    Smooth IM 𝓘(𝕜, F) (fun y ↦ e.coordChange e' (f y) (g y)) 
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} {g : M → F} (hf : Smooth IM IB f) (hg : Smooth IM 𝓘(𝕜, F) g) (he : ∀ (x : M), f x ∈ e.baseSet) (he' : ∀ (x : M), f x ∈ e'.baseSet)
Goal: Smooth IM 𝓘(𝕜, F) fun y => e.coordChange e' (f y) (g y)
-- Trivialization.contMDiffOn_symm_trans
theorem Trivialization.contMDiffOn_symm_trans :
    ContMDiffOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) n
      (e.toPartialHomeomorph.symm ≫ₕ e'.toPartialHomeomorph) (e.target ∩ e'.target) 
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.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) n (↑(e.symm ≫ₕ e'.toPartialHomeomorph)) (e.target ∩ e'.target)
-- ContMDiffWithinAt.change_section_trivialization
theorem ContMDiffWithinAt.change_section_trivialization {f : M → TotalSpace F E}
    (hp : ContMDiffWithinAt IM IB n (π F E ∘ f) s x)
    (hf : ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y ↦ (e (f y)).2) s x)
    (he : f x ∈ e.source) (he' : f x ∈ e'.source) :
    ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y ↦ (e' (f y)).2) 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} {x : M} {f : M → Bundle.TotalSpace F E} (hp : ContMDiffWithinAt IM IB n (Bundle.TotalSpace.proj ∘ f) s x) (hf : ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y => (↑e (f y)).2) s x) (he : f x ∈ e.source) (he' : f x ∈ e'.source)
Goal: ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y => (↑e' (f y)).2) s x
-- Trivialization.contMDiffWithinAt_snd_comp_iff₂
theorem Trivialization.contMDiffWithinAt_snd_comp_iff₂ {f : M → TotalSpace F E}
    (hp : ContMDiffWithinAt IM IB n (π F E ∘ f) s x)
    (he : f x ∈ e.source) (he' : f x ∈ e'.source) :
    ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y ↦ (e (f y)).2) s x ↔
      ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y ↦ (e' (f y)).2) 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} {x : M} {f : M → Bundle.TotalSpace F E} (hp : ContMDiffWithinAt IM IB n (Bundle.TotalSpace.proj ∘ f) s x) (he : f x ∈ e.source) (he' : f x ∈ e'.source)
Goal: ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y => (↑e (f y)).2) s x ↔ ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y => (↑e' (f y)).2) s x
-- Trivialization.contMDiffWithinAt_iff
theorem Trivialization.contMDiffWithinAt_iff {f : M → TotalSpace F E} {s : Set M} {x₀ : M}
    (he : f x₀ ∈ e.source) :
    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 ↦ (e (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] {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 : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {f : M → Bundle.TotalSpace F E} {s : Set M} {x₀ : M} (he : f x₀ ∈ e.source)
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 => (↑e (f x)).2) s x₀
-- Trivialization.contMDiffAt_iff
theorem Trivialization.contMDiffAt_iff {f : M → TotalSpace F E} {x₀ : M} (he : f x₀ ∈ e.source) :
    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 ↦ (e (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] {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 : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {f : M → Bundle.TotalSpace F E} {x₀ : M} (he : f x₀ ∈ e.source)
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 => (↑e (f x)).2) x₀
-- Trivialization.contMDiffOn_iff
theorem Trivialization.contMDiffOn_iff {f : M → TotalSpace F E} {s : Set M}
    (he : MapsTo f s e.source) :
    ContMDiffOn IM (IB.prod 𝓘(𝕜, F)) n f s ↔
      ContMDiffOn IM IB n (fun x => (f x).proj) s ∧
      ContMDiffOn IM 𝓘(𝕜, F) n (fun x ↦ (e (f x)).2) 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 : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {f : M → Bundle.TotalSpace F E} {s : Set M} (he : Set.MapsTo f s e.source)
Goal: ContMDiffOn IM (IB.prod 𝓘(𝕜, F)) n f s ↔
  ContMDiffOn IM IB n (fun x => (f x).proj) s ∧ ContMDiffOn IM 𝓘(𝕜, F) n (fun x => (↑e (f x)).2) s
-- Trivialization.contMDiff_iff
theorem Trivialization.contMDiff_iff {f : M → TotalSpace F E} (he : ∀ x, f x ∈ e.source) :
    ContMDiff IM (IB.prod 𝓘(𝕜, F)) n f ↔
      ContMDiff IM IB n (fun x => (f x).proj) ∧
      ContMDiff IM 𝓘(𝕜, F) n (fun x ↦ (e (f x)).2) 
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 : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {f : M → Bundle.TotalSpace F E} (he : ∀ (x : M), f x ∈ e.source)
Goal: ContMDiff IM (IB.prod 𝓘(𝕜, F)) n f ↔
  (ContMDiff IM IB n fun x => (f x).proj) ∧ ContMDiff IM 𝓘(𝕜, F) n fun x => (↑e (f x)).2
-- Trivialization.smoothWithinAt_iff
theorem Trivialization.smoothWithinAt_iff {f : M → TotalSpace F E} {s : Set M} {x₀ : M}
    (he : f x₀ ∈ e.source) :
    SmoothWithinAt IM (IB.prod 𝓘(𝕜, F)) f s x₀ ↔
      SmoothWithinAt IM IB (fun x => (f x).proj) s x₀ ∧
      SmoothWithinAt IM 𝓘(𝕜, F) (fun x ↦ (e (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] {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 : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {f : M → Bundle.TotalSpace F E} {s : Set M} {x₀ : M} (he : f x₀ ∈ e.source)
Goal: SmoothWithinAt IM (IB.prod 𝓘(𝕜, F)) f s x₀ ↔
  SmoothWithinAt IM IB (fun x => (f x).proj) s x₀ ∧ SmoothWithinAt IM 𝓘(𝕜, F) (fun x => (↑e (f x)).2) s x₀