Theorems · Theorem · global analysis

MDifferentiableAt.coordChange

∀ {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} [MemTrivializationAtlas e] [MemTrivializationAtlas e']
  [inst_20 : VectorBundle 𝕜 F E] [ContMDiffVectorBundle 1 F E IB] {f : M → B} {g : M → F} {x : M},
  MDiffAt f x → MDiffAt g x → f x ∈ e.baseSet → f x ∈ e'.baseSet → (MDiffAt fun y => e.coordChange e' (f y) (g y)) x

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

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceFiberBundleAddCommMonoidModuleMemTrivializationAtlasMemTrivializationAtlasVectorBundleContMDiffVectorBundle