Theorems · Theorem · algebraic topology

Bundle.Trivialization.continuousLinearMap_apply

∀ {𝕜₁ : Type u_1} [inst : NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [inst_1 : NontriviallyNormedField 𝕜₂]
  (σ : 𝕜₁ →+* 𝕜₂) {B : Type u_3} {F₁ : Type u_4} [inst_2 : NormedAddCommGroup F₁] [inst_3 : NormedSpace 𝕜₁ F₁]
  {E₁ : B → Type u_5} [inst_4 : (x : B) → AddCommGroup (E₁ x)] [inst_5 : (x : B) → Module 𝕜₁ (E₁ x)]
  [inst_6 : TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [inst_7 : NormedAddCommGroup F₂]
  [inst_8 : NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [inst_9 : (x : B) → AddCommGroup (E₂ x)]
  [inst_10 : (x : B) → Module 𝕜₂ (E₂ x)] [inst_11 : TopologicalSpace (Bundle.TotalSpace F₂ E₂)]
  [inst_12 : TopologicalSpace B] {e₁ : Bundle.Trivialization F₁ Bundle.TotalSpace.proj}
  {e₂ : Bundle.Trivialization F₂ Bundle.TotalSpace.proj} [inst_13 : (x : B) → TopologicalSpace (E₁ x)]
  [inst_14 : FiberBundle F₁ E₁] [inst_15 : VectorBundle 𝕜₁ F₁ E₁] [inst_16 : (x : B) → TopologicalSpace (E₂ x)]
  [inst_17 : FiberBundle F₂ E₂] [inst_18 : VectorBundle 𝕜₂ F₂ E₂] [inst_19 : ∀ (x : B), IsTopologicalAddGroup (E₂ x)]
  [inst_20 : ∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)] [inst_21 : RingHomIsometric σ] [he₁ : MemTrivializationAtlas e₁]
  [he₂ : MemTrivializationAtlas e₂] (p : Bundle.TotalSpace (F₁ →SL[σ] F₂) fun x => E₁ x →SL[σ] E₂ x),
  ↑(Bundle.Trivialization.continuousLinearMap σ e₁ e₂) p =
    (p.proj,
      Bundle.Trivialization.continuousLinearMapAt 𝕜₂ e₂ p.proj ∘SL p.snd ∘SL Bundle.Trivialization.symmL 𝕜₁ e₁ p.proj)

Defined in

Mathlib.Topology.VectorBundle.Hom

Cited by

0 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceTopologicalSpaceTopologicalSpaceFiberBundleVectorBundleTopologicalSpaceFiberBundleVectorBundleIsTopologicalAddGroupContinuousSMulRingHomIsometricMemTrivializationAtlasMemTrivializationAtlas