Homs of C^n vector bundles over the same base space

Here we show that Bundle.ContinuousLinearMap is a C^n vector bundle.

Note that we only do this for bundles of linear maps, not for bundles of arbitrary semilinear maps. To do it for semilinear maps, we would need to generalize ContinuousLinearMap.contMDiff (and ContinuousLinearMap.contDiff) to semilinear maps.

theorem contMDiffOn_continuousLinearMapCoordChange {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {n : WithTop ℕ∞} {E₁ : B → Type u_6} {E₂ : B → Type u_7} [NontriviallyNormedField 𝕜] [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → TopologicalSpace (E₂ x)] {EB : Type u_8} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_9} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] {e₁ e₁' : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ e₂' : Trivialization F₂ Bundle.TotalSpace.proj} [ContMDiffVectorBundle n F₁ E₁ IB] [ContMDiffVectorBundle n F₂ E₂ IB] [MemTrivializationAtlas e₁] [MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂] [MemTrivializationAtlas e₂'] : ContMDiffOn IB (modelWithCornersSelf 𝕜 ((F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂)) n (Pretrivialization.continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))

@[deprecated contMDiffOn_continuousLinearMapCoordChange (since := "2024-11-21")]

theorem smoothOn_continuousLinearMapCoordChange {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {n : WithTop ℕ∞} {E₁ : B → Type u_6} {E₂ : B → Type u_7} [NontriviallyNormedField 𝕜] [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → TopologicalSpace (E₂ x)] {EB : Type u_8} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_9} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] {e₁ e₁' : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ e₂' : Trivialization F₂ Bundle.TotalSpace.proj} [ContMDiffVectorBundle n F₁ E₁ IB] [ContMDiffVectorBundle n F₂ E₂ IB] [MemTrivializationAtlas e₁] [MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂] [MemTrivializationAtlas e₂'] : ContMDiffOn IB (modelWithCornersSelf 𝕜 ((F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂)) n (Pretrivialization.continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))

Alias of contMDiffOn_continuousLinearMapCoordChange.

theorem hom_chart {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {E₁ : B → Type u_6} {E₂ : B → Type u_7} [NontriviallyNormedField 𝕜] [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → TopologicalSpace (E₂ x)] {HB : Type u_9} [TopologicalSpace HB] [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] [∀ (x : B), TopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜 (E₂ x)] (y₀ y : Bundle.TotalSpace (F₁ →L[𝕜] F₂) (Bundle.ContinuousLinearMap (RingHom.id 𝕜) E₁ E₂)) : ↑(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)

theorem contMDiffAt_hom_bundle {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {M : Type u_5} {E₁ : B → Type u_6} {E₂ : B → Type u_7} [NontriviallyNormedField 𝕜] [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → TopologicalSpace (E₂ x)] {EB : Type u_8} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_9} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] {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] [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] [∀ (x : B), TopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜 (E₂ x)] (f : M → Bundle.TotalSpace (F₁ →L[𝕜] F₂) (Bundle.ContinuousLinearMap (RingHom.id 𝕜) E₁ E₂)) {x₀ : M} {n : ℕ∞} : ContMDiffAt IM (IB.prod (modelWithCornersSelf 𝕜 (F₁ →L[𝕜] F₂))) (↑n) f x₀ ↔ ContMDiffAt IM IB (↑n) (fun (x : M) => (f x).proj) x₀ ∧ ContMDiffAt IM (modelWithCornersSelf 𝕜 (F₁ →L[𝕜] F₂)) (↑n) (fun (x : M) => ContinuousLinearMap.inCoordinates F₁ E₁ F₂ E₂ (f x₀).proj (f x).proj (f x₀).proj (f x).proj (f x).snd) x₀

@[deprecated contMDiffAt_hom_bundle (since := "2024-11-21")]

theorem smoothAt_hom_bundle {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {M : Type u_5} {E₁ : B → Type u_6} {E₂ : B → Type u_7} [NontriviallyNormedField 𝕜] [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → TopologicalSpace (E₂ x)] {EB : Type u_8} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_9} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] {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] [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] [∀ (x : B), TopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜 (E₂ x)] (f : M → Bundle.TotalSpace (F₁ →L[𝕜] F₂) (Bundle.ContinuousLinearMap (RingHom.id 𝕜) E₁ E₂)) {x₀ : M} {n : ℕ∞} : ContMDiffAt IM (IB.prod (modelWithCornersSelf 𝕜 (F₁ →L[𝕜] F₂))) (↑n) f x₀ ↔ ContMDiffAt IM IB (↑n) (fun (x : M) => (f x).proj) x₀ ∧ ContMDiffAt IM (modelWithCornersSelf 𝕜 (F₁ →L[𝕜] F₂)) (↑n) (fun (x : M) => ContinuousLinearMap.inCoordinates F₁ E₁ F₂ E₂ (f x₀).proj (f x).proj (f x₀).proj (f x).proj (f x).snd) x₀

Alias of contMDiffAt_hom_bundle.

instance Bundle.ContinuousLinearMap.vectorPrebundle.isContMDiff {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {n : WithTop ℕ∞} {E₁ : B → Type u_6} {E₂ : B → Type u_7} [NontriviallyNormedField 𝕜] [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (TotalSpace F₁ E₁)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (TotalSpace F₂ E₂)] [(x : B) → TopologicalSpace (E₂ x)] {EB : Type u_8} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_9} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] [∀ (x : B), TopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜 (E₂ x)] [ContMDiffVectorBundle n F₁ E₁ IB] [ContMDiffVectorBundle n F₂ E₂ IB] : VectorPrebundle.IsContMDiff IB (vectorPrebundle (RingHom.id 𝕜) F₁ E₁ F₂ E₂) n

@[deprecated Bundle.ContinuousLinearMap.vectorPrebundle.isContMDiff (since := "2025-01-09")]

theorem Bundle.ContinuousLinearMap.vectorPrebundle.isSmooth {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {n : WithTop ℕ∞} {E₁ : B → Type u_6} {E₂ : B → Type u_7} [NontriviallyNormedField 𝕜] [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (TotalSpace F₁ E₁)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (TotalSpace F₂ E₂)] [(x : B) → TopologicalSpace (E₂ x)] {EB : Type u_8} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_9} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] [∀ (x : B), TopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜 (E₂ x)] [ContMDiffVectorBundle n F₁ E₁ IB] [ContMDiffVectorBundle n F₂ E₂ IB] : VectorPrebundle.IsContMDiff IB (vectorPrebundle (RingHom.id 𝕜) F₁ E₁ F₂ E₂) n

Alias of Bundle.ContinuousLinearMap.vectorPrebundle.isContMDiff.

instance ContMDiffVectorBundle.continuousLinearMap {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {n : WithTop ℕ∞} {E₁ : B → Type u_6} {E₂ : B → Type u_7} [NontriviallyNormedField 𝕜] [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → TopologicalSpace (E₂ x)] {EB : Type u_8} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_9} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] [∀ (x : B), TopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜 (E₂ x)] [ContMDiffVectorBundle n F₁ E₁ IB] [ContMDiffVectorBundle n F₂ E₂ IB] : ContMDiffVectorBundle n (F₁ →L[𝕜] F₂) (Bundle.ContinuousLinearMap (RingHom.id 𝕜) E₁ E₂) IB

theorem ContMDiffWithinAt.clm_apply_of_inCoordinates {𝕜 : Type u_1} {F₁ : Type u_2} {F₂ : Type u_3} {B₁ : Type u_4} {B₂ : Type u_5} {M : Type u_6} {E₁ : B₁ → Type u_7} {E₂ : B₂ → Type u_8} [NontriviallyNormedField 𝕜] [(x : B₁) → AddCommGroup (E₁ x)] [(x : B₁) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [(x : B₁) → TopologicalSpace (E₁ x)] [(x : B₂) → AddCommGroup (E₂ x)] [(x : B₂) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B₂) → TopologicalSpace (E₂ x)] {EB₁ : Type u_9} [NormedAddCommGroup EB₁] [NormedSpace 𝕜 EB₁] {HB₁ : Type u_10} [TopologicalSpace HB₁] {IB₁ : ModelWithCorners 𝕜 EB₁ HB₁} [TopologicalSpace B₁] [ChartedSpace HB₁ B₁] {EB₂ : Type u_11} [NormedAddCommGroup EB₂] [NormedSpace 𝕜 EB₂] {HB₂ : Type u_12} [TopologicalSpace HB₂] {IB₂ : ModelWithCorners 𝕜 EB₂ HB₂} [TopologicalSpace B₂] [ChartedSpace HB₂ B₂] {EM : Type u_13} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_14} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] {n : WithTop ℕ∞} [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] {b₁ : M → B₁} {b₂ : M → B₂} {m₀ : M} {ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)} {v : (m : M) → E₁ (b₁ m)} {s : Set M} (hϕ : ContMDiffWithinAt IM (modelWithCornersSelf 𝕜 (F₁ →L[𝕜] F₂)) n (fun (m : M) => ContinuousLinearMap.inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) s m₀) (hv : ContMDiffWithinAt IM (IB₁.prod (modelWithCornersSelf 𝕜 F₁)) n (fun (m : M) => { proj := b₁ m, snd := v m }) s m₀) (hb₂ : ContMDiffWithinAt IM IB₂ n b₂ s m₀) : ContMDiffWithinAt IM (IB₂.prod (modelWithCornersSelf 𝕜 F₂)) n (fun (m : M) => { proj := b₂ m, snd := (ϕ m) (v m) }) s m₀

Consider a C^n map v : M → E₁ to a vector bundle, over a basemap b₁ : M → B₁, and another basemap b₂ : M → B₂. Given linear maps ϕ m : E₁ (b₁ m) → E₂ (b₂ m) depending smoothly on m, one can apply ϕ m to g m, and the resulting map is C^n.

Note that the smoothness of ϕ can not be always be stated as smoothness of a map into a manifold, as the pullback bundles b₁ *ᵖ E₁ and b₂ *ᵖ E₂ only make sense when b₁ and b₂ are globally smooth, but we want to apply this lemma with only local information. Therefore, we formulate it using smoothness of ϕ read in coordinates.

Version for ContMDiffWithinAt. We also give a version for ContMDiffAt, but no version for ContMDiffOn or ContMDiff as our assumption, written in coordinates, only makes sense around a point.

theorem ContMDiffAt.clm_apply_of_inCoordinates {𝕜 : Type u_1} {F₁ : Type u_2} {F₂ : Type u_3} {B₁ : Type u_4} {B₂ : Type u_5} {M : Type u_6} {E₁ : B₁ → Type u_7} {E₂ : B₂ → Type u_8} [NontriviallyNormedField 𝕜] [(x : B₁) → AddCommGroup (E₁ x)] [(x : B₁) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [(x : B₁) → TopologicalSpace (E₁ x)] [(x : B₂) → AddCommGroup (E₂ x)] [(x : B₂) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B₂) → TopologicalSpace (E₂ x)] {EB₁ : Type u_9} [NormedAddCommGroup EB₁] [NormedSpace 𝕜 EB₁] {HB₁ : Type u_10} [TopologicalSpace HB₁] {IB₁ : ModelWithCorners 𝕜 EB₁ HB₁} [TopologicalSpace B₁] [ChartedSpace HB₁ B₁] {EB₂ : Type u_11} [NormedAddCommGroup EB₂] [NormedSpace 𝕜 EB₂] {HB₂ : Type u_12} [TopologicalSpace HB₂] {IB₂ : ModelWithCorners 𝕜 EB₂ HB₂} [TopologicalSpace B₂] [ChartedSpace HB₂ B₂] {EM : Type u_13} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_14} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] {n : WithTop ℕ∞} [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] {b₁ : M → B₁} {b₂ : M → B₂} {m₀ : M} {ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)} {v : (m : M) → E₁ (b₁ m)} (hϕ : ContMDiffAt IM (modelWithCornersSelf 𝕜 (F₁ →L[𝕜] F₂)) n (fun (m : M) => ContinuousLinearMap.inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀) (hv : ContMDiffAt IM (IB₁.prod (modelWithCornersSelf 𝕜 F₁)) n (fun (m : M) => { proj := b₁ m, snd := v m }) m₀) (hb₂ : ContMDiffAt IM IB₂ n b₂ m₀) : ContMDiffAt IM (IB₂.prod (modelWithCornersSelf 𝕜 F₂)) n (fun (m : M) => { proj := b₂ m, snd := (ϕ m) (v m) }) m₀

Consider a C^n map v : M → E₁ to a vector bundle, over a basemap b₁ : M → B₁, and another basemap b₂ : M → B₂. Given linear maps ϕ m : E₁ (b₁ m) → E₂ (b₂ m) depending smoothly on m, one can apply ϕ m to g m, and the resulting map is C^n.

Note that the smoothness of ϕ can not be always be stated as smoothness of a map into a manifold, as the pullback bundles b₁ *ᵖ E₁ and b₂ *ᵖ E₂ only make sense when b₁ and b₂ are globally smooth, but we want to apply this lemma with only local information. Therefore, we formulate it using smoothness of ϕ read in coordinates.

Version for ContMDiffAt. We also give a version for ContMDiffWithinAt, but no version for ContMDiffOn or ContMDiff as our assumption, written in coordinates, only makes sense around a point.