Theorems · Theorem · algebraic topology

Bundle.Pretrivialization.continuousAlternatingMapCoordChange_apply

∀ {𝕜 : Type u_1} {ι : Type u_2} [inst : NontriviallyNormedField 𝕜] {B : Type u_3} [inst_1 : TopologicalSpace B]
  {F₁ : Type u_4} [inst_2 : NormedAddCommGroup F₁] [inst_3 : NormedSpace 𝕜 F₁] {E₁ : B → Type u_5}
  [inst_4 : (x : B) → AddCommGroup (E₁ x)] [inst_5 : (x : B) → Module 𝕜 (E₁ x)]
  [inst_6 : TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [inst_7 : NormedAddCommGroup F₂]
  [inst_8 : NormedSpace 𝕜 F₂] {E₂ : B → Type u_7} [inst_9 : (x : B) → AddCommGroup (E₂ x)]
  [inst_10 : (x : B) → Module 𝕜 (E₂ x)] [inst_11 : TopologicalSpace (Bundle.TotalSpace F₂ E₂)]
  [inst_12 : (x : B) → TopologicalSpace (E₁ x)] [inst_13 : FiberBundle F₁ E₁]
  [inst_14 : (x : B) → TopologicalSpace (E₂ x)] [inst_15 : FiberBundle F₂ E₂]
  {e₁ e₁' : Bundle.Trivialization F₁ Bundle.TotalSpace.proj} {e₂ e₂' : Bundle.Trivialization F₂ Bundle.TotalSpace.proj}
  [inst_16 : Bundle.Trivialization.IsLinear 𝕜 e₁] [inst_17 : Bundle.Trivialization.IsLinear 𝕜 e₁']
  [inst_18 : Bundle.Trivialization.IsLinear 𝕜 e₂] [inst_19 : Bundle.Trivialization.IsLinear 𝕜 e₂'],
  ∀ b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet),
    ∀ (L : F₁ [⋀^ι]→L[𝕜] F₂),
      (Bundle.Pretrivialization.continuousAlternatingMapCoordChange 𝕜 ι e₁ e₁' e₂ e₂' b) L =
        (↑(Bundle.Pretrivialization.continuousAlternatingMap 𝕜 ι e₁' e₂')
            ⟨b, (Bundle.Pretrivialization.continuousAlternatingMap 𝕜 ι e₁ e₂).symm b L⟩).2

Defined in

Mathlib.Topology.VectorBundle.ContinuousAlternatingMap

Cited by

0 results in Mathlib

Foundations

Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceTopologicalSpaceFiberBundleTopologicalSpaceFiberBundleBundle.Trivialization.IsLinearBundle.Trivialization.IsLinearBundle.Trivialization.IsLinearBundle.Trivialization.IsLinear