Theorems · Theorem · global analysis

IsLocalFrameOn.contMDiffAt_of_coeff_aux

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {F : Type u_5} [inst_6 : NormedAddCommGroup F]
  [inst_7 : NormedSpace 𝕜 F] {V : M → Type u_6} [inst_8 : TopologicalSpace (Bundle.TotalSpace F V)]
  [inst_9 : (x : M) → AddCommGroup (V x)] [inst_10 : (x : M) → Module 𝕜 (V x)]
  [inst_11 : (x : M) → TopologicalSpace (V x)] [inst_12 : FiberBundle F V] {ι : Type u_7} {s : ι → (x : M) → V x}
  {u : Set M} {n : WithTop ℕ∞} {x : M} {t : (x : M) → V x} (hs : IsLocalFrameOn I F n s u) [VectorBundle 𝕜 F V]
  [FiniteDimensional 𝕜 F],
  (∀ (i : ι), ContMDiffAt I (modelWithCornersSelf 𝕜 𝕜) n ((LinearMap.piApply (hs.coeff i)) t) x) →
    IsOpen u → x ∈ u → ContMDiffAt I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, t x⟩) x

Given a local frame s i on an open set u containing x, if a section t has C^k coefficients at x ∈ u w.r.t. s i, then t is C^n at x.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Cited by

0 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceFiberBundleVectorBundleFiniteDimensional