Theorems · Theorem · algebraic topology

continuousWithinAt_continuousAlternatingMap_bundle

∀ {𝕜 : Type u_1} {ι : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : Fintype ι] {B : Type u_3}
  [inst_2 : TopologicalSpace B] {F₁ : Type u_4} [inst_3 : NormedAddCommGroup F₁] [inst_4 : NormedSpace 𝕜 F₁]
  {E₁ : B → Type u_5} [inst_5 : (x : B) → AddCommGroup (E₁ x)] [inst_6 : (x : B) → Module 𝕜 (E₁ x)]
  [inst_7 : TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [inst_8 : (x : B) → TopologicalSpace (E₁ x)]
  [inst_9 : FiberBundle F₁ E₁] [inst_10 : VectorBundle 𝕜 F₁ E₁] {F₂ : Type u_6} [inst_11 : NormedAddCommGroup F₂]
  [inst_12 : NormedSpace 𝕜 F₂] {E₂ : B → Type u_7} [inst_13 : (x : B) → AddCommGroup (E₂ x)]
  [inst_14 : (x : B) → Module 𝕜 (E₂ x)] [inst_15 : TopologicalSpace (Bundle.TotalSpace F₂ E₂)]
  [inst_16 : (x : B) → TopologicalSpace (E₂ x)] [inst_17 : FiberBundle F₂ E₂] [inst_18 : VectorBundle 𝕜 F₂ E₂]
  [inst_19 : ∀ (x : B), IsTopologicalAddGroup (E₂ x)] [inst_20 : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)] {X : Type u_8}
  [inst_21 : TopologicalSpace X] {s : Set X} {x₀ : X}
  (f : X → Bundle.TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun x => E₁ x [⋀^ι]→L[𝕜] E₂ x),
  ContinuousWithinAt f s x₀ ↔
    ContinuousWithinAt (fun x => (f x).proj) s x₀ ∧
      ContinuousWithinAt
        (fun x => ContinuousAlternatingMap.inCoordinates F₁ F₂ (f x₀).proj (f x).proj (f x₀).proj (f x).proj (f x).snd)
        s x₀

Defined in

Mathlib.Topology.VectorBundle.ContinuousAlternatingMap

Cited by

0 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldFintypeTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceTopologicalSpaceFiberBundleVectorBundleNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceTopologicalSpaceFiberBundleVectorBundleIsTopologicalAddGroupContinuousSMulTopologicalSpace