Theorems · Theorem · global analysis

MDifferentiableWithinAt.change_section_trivialization

∀ {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]
  [inst_16 : (x : B) → AddCommMonoid (E x)] [inst_17 : (x : B) → Module 𝕜 (E x)] [inst_18 : VectorBundle 𝕜 F E]
  [ContMDiffVectorBundle 1 F E IB] {e : Bundle.Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e]
  {e' : Bundle.Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e'] {f : M → Bundle.TotalSpace F E}
  {s : Set M} {x₀ : M},
  MDiffAt[s] (Bundle.TotalSpace.proj ∘ f) x₀ →
    (MDiffAt[s] fun x => (↑e (f x)).2) x₀ → f x₀ ∈ e.source → f x₀ ∈ e'.source → (MDiffAt[s] fun x => (↑e' (f x)).2) x₀

Defined in

Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Cited by

1 results in Mathlib

Foundations

Depth 211 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceFiberBundleAddCommMonoidModuleVectorBundleContMDiffVectorBundleMemTrivializationAtlasMemTrivializationAtlas