Theorems · Theorem · global analysis

MDifferentiableOn.coordChangeL

∀ {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)]
  {e e' : Bundle.Trivialization F Bundle.TotalSpace.proj} [inst_18 : MemTrivializationAtlas e]
  [inst_19 : MemTrivializationAtlas e'] [inst_20 : VectorBundle 𝕜 F E] [ContMDiffVectorBundle 1 F E IB] {s : Set M}
  {f : M → B},
  MDiff[s] f →
    Set.MapsTo f s e.baseSet →
      Set.MapsTo f s e'.baseSet → MDiff[s] fun y => ↑(Bundle.Trivialization.coordChangeL 𝕜 e e' (f y))

Defined in

Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Cited by

0 results in Mathlib

Foundations

Depth 210 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceFiberBundleAddCommMonoidModuleMemTrivializationAtlasMemTrivializationAtlasVectorBundleContMDiffVectorBundle