Theorems · Theorem · algebraic topology

Bundle.contMDiffWithinAt_totalSpace

∀ {n : WithTop ℕ∞} {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6}
  [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 F]
  [inst_3 : TopologicalSpace (Bundle.TotalSpace F E)] [inst_4 : (x : B) → TopologicalSpace (E x)] {EB : Type u_7}
  [inst_5 : NormedAddCommGroup EB] [inst_6 : NormedSpace 𝕜 EB] {HB : Type u_8} [inst_7 : TopologicalSpace HB]
  {IB : ModelWithCorners 𝕜 EB HB} {EM : Type u_10} [inst_8 : NormedAddCommGroup EM] [inst_9 : NormedSpace 𝕜 EM]
  {HM : Type u_11} [inst_10 : TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [inst_11 : TopologicalSpace M]
  [inst_12 : ChartedSpace HM M] [inst_13 : TopologicalSpace B] [inst_14 : ChartedSpace HB B] [inst_15 : FiberBundle F E]
  {f : M → Bundle.TotalSpace F E} {s : Set M} {x₀ : M},
  ContMDiffWithinAt IM (IB.prod (modelWithCornersSelf 𝕜 F)) n f s x₀ ↔
    ContMDiffWithinAt IM IB n (fun x => (f x).proj) s x₀ ∧
      ContMDiffWithinAt IM (modelWithCornersSelf 𝕜 F) n (fun x => (↑(trivializationAt F E (f x₀).proj) (f x)).2) s x₀

Characterization of C^n functions into a vector bundle. Version at a point within a set.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.Basic

Cited by

5 results in Mathlib

Foundations

Depth 206 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceFiberBundle