Theorems · Theorem · algebraic topology

mdifferentiableOn_continuousLinearMapCoordChange

∀ {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {E₁ : B → Type u_6} {E₂ : B → Type u_7}
  [inst : NontriviallyNormedField 𝕜] [inst_1 : (x : B) → AddCommGroup (E₁ x)] [inst_2 : (x : B) → Module 𝕜 (E₁ x)]
  [inst_3 : NormedAddCommGroup F₁] [inst_4 : NormedSpace 𝕜 F₁] [inst_5 : TopologicalSpace (Bundle.TotalSpace F₁ E₁)]
  [inst_6 : (x : B) → TopologicalSpace (E₁ x)] [inst_7 : (x : B) → AddCommGroup (E₂ x)]
  [inst_8 : (x : B) → Module 𝕜 (E₂ x)] [inst_9 : NormedAddCommGroup F₂] [inst_10 : NormedSpace 𝕜 F₂]
  [inst_11 : TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [inst_12 : (x : B) → TopologicalSpace (E₂ x)] {EB : Type u_8}
  [inst_13 : NormedAddCommGroup EB] [inst_14 : NormedSpace 𝕜 EB] {HB : Type u_9} [inst_15 : TopologicalSpace HB]
  {IB : ModelWithCorners 𝕜 EB HB} [inst_16 : TopologicalSpace B] [inst_17 : ChartedSpace HB B]
  [inst_18 : FiberBundle F₁ E₁] [inst_19 : VectorBundle 𝕜 F₁ E₁] [inst_20 : FiberBundle F₂ E₂]
  [inst_21 : VectorBundle 𝕜 F₂ E₂] {e₁ e₁' : Bundle.Trivialization F₁ Bundle.TotalSpace.proj}
  {e₂ e₂' : Bundle.Trivialization F₂ Bundle.TotalSpace.proj} [ContMDiffVectorBundle 1 F₁ E₁ IB]
  [ContMDiffVectorBundle 1 F₂ E₂ IB] [inst_24 : MemTrivializationAtlas e₁] [inst_25 : MemTrivializationAtlas e₁']
  [inst_26 : MemTrivializationAtlas e₂] [inst_27 : MemTrivializationAtlas e₂'],
  MDiff[e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)]
    (Bundle.Pretrivialization.continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂')

Defined in

Mathlib.Geometry.Manifold.VectorBundle.Hom

Cited by

0 results in Mathlib

Foundations

Depth 207 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceAddCommGroupModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceFiberBundleVectorBundleFiberBundleVectorBundleContMDiffVectorBundleContMDiffVectorBundleMemTrivializationAtlasMemTrivializationAtlasMemTrivializationAtlasMemTrivializationAtlas