Theorems · Theorem · algebraic topology

ContMDiffWithinAt.clm_bundle_apply

∀ {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {M : Type u_6} [inst : NontriviallyNormedField 𝕜]
  {n : WithTop ℕ∞} {E₁ : B → Type u_7} [inst_1 : (x : B) → AddCommGroup (E₁ x)] [inst_2 : (x : B) → Module 𝕜 (E₁ x)]
  [inst_3 : NormedAddCommGroup F₁] [inst_4 : NormedSpace 𝕜 F₁] [inst_5 : TopologicalSpace (Bundle.TotalSpace F₁ E₁)]
  [inst_6 : (x : B) → TopologicalSpace (E₁ x)] {E₂ : B → Type u_8} [inst_7 : (x : B) → AddCommGroup (E₂ x)]
  [inst_8 : (x : B) → Module 𝕜 (E₂ x)] [inst_9 : NormedAddCommGroup F₂] [inst_10 : NormedSpace 𝕜 F₂]
  [inst_11 : TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [inst_12 : (x : B) → TopologicalSpace (E₂ x)] {EB : Type u_10}
  [inst_13 : NormedAddCommGroup EB] [inst_14 : NormedSpace 𝕜 EB] {HB : Type u_11} [inst_15 : TopologicalSpace HB]
  {IB : ModelWithCorners 𝕜 EB HB} [inst_16 : TopologicalSpace B] [inst_17 : ChartedSpace HB B] {EM : Type u_12}
  [inst_18 : NormedAddCommGroup EM] [inst_19 : NormedSpace 𝕜 EM] {HM : Type u_13} [inst_20 : TopologicalSpace HM]
  {IM : ModelWithCorners 𝕜 EM HM} [inst_21 : TopologicalSpace M] [inst_22 : ChartedSpace HM M]
  [inst_23 : FiberBundle F₁ E₁] [inst_24 : VectorBundle 𝕜 F₁ E₁] [inst_25 : FiberBundle F₂ E₂]
  [inst_26 : VectorBundle 𝕜 F₂ E₂] {b : M → B} {v : (x : M) → E₁ (b x)} {s : Set M} {x : M}
  [inst_27 : ∀ (x : B), IsTopologicalAddGroup (E₂ x)] [inst_28 : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)]
  {ϕ : (x : M) → E₁ (b x) →L[𝕜] E₂ (b x)},
  ContMDiffWithinAt IM (IB.prod ⋯) n (fun m => ⋯) s x →
    ContMDiffWithinAt IM ⋯ n (fun m => ⟨b m, v m⟩) s x → ContMDiffWithinAt IM ⋯ n ⋯ s x

Consider a C^n map v : M → E₁ to a vector bundle, over a base map b : M → B, and linear maps ϕ m : E₁ (b m) → E₂ (b m) depending smoothly on m. One can apply ϕ m to v m, and the resulting map is C^n. We give here a version of this statement within a set at a point.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.Hom

Cited by

3 results in Mathlib

Foundations

Depth 209 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceAddCommGroupModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceFiberBundleVectorBundleFiberBundleVectorBundleIsTopologicalAddGroupContinuousSMul