Theorems · Theorem · algebraic topology

ContMDiffWithinAt.coordChange

∀ {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] {EB : Type u_7} [inst_1 : NormedAddCommGroup EB] [inst_2 : NormedSpace 𝕜 EB]
  {HB : Type u_8} [inst_3 : TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [inst_4 : TopologicalSpace B]
  [inst_5 : ChartedSpace HB B] {EM : Type u_9} [inst_6 : NormedAddCommGroup EM] [inst_7 : NormedSpace 𝕜 EM]
  {HM : Type u_10} [inst_8 : TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [inst_9 : TopologicalSpace M]
  [inst_10 : ChartedSpace HM M] [inst_11 : (x : B) → AddCommMonoid (E x)] [inst_12 : (x : B) → Module 𝕜 (E x)]
  [inst_13 : NormedAddCommGroup F] [inst_14 : NormedSpace 𝕜 F] [inst_15 : TopologicalSpace (Bundle.TotalSpace F E)]
  [inst_16 : (x : B) → TopologicalSpace (E x)] [inst_17 : FiberBundle F E] [inst_18 : VectorBundle 𝕜 F E]
  [ContMDiffVectorBundle n F E IB] {e e' : Bundle.Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e]
  [MemTrivializationAtlas e'] {s : Set M} {f : M → B} {g : M → F} {x : M},
  ContMDiffWithinAt IM IB n f s x →
    ContMDiffWithinAt IM (modelWithCornersSelf 𝕜 F) n g s x →
      f x ∈ e.baseSet →
        f x ∈ e'.baseSet → ContMDiffWithinAt IM (modelWithCornersSelf 𝕜 F) n (fun y => e.coordChange e' (f y) (g y)) s x

Defined in

Mathlib.Geometry.Manifold.VectorBundle.Basic

Cited by

3 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceAddCommMonoidModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceFiberBundleVectorBundleContMDiffVectorBundleMemTrivializationAtlasMemTrivializationAtlas