Theorems · Theorem · algebraic topology

contMDiffWithinAt_hom_bundle

∀ {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {M : Type u_5} {n : WithTop ℕ∞} {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] {EM : Type u_10} [inst_18 : NormedAddCommGroup EM]
  [inst_19 : NormedSpace 𝕜 EM] {HM : Type u_11} [inst_20 : TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM}
  [inst_21 : TopologicalSpace M] [inst_22 : ChartedSpace HM M] [inst_23 : FiberBundle F₁ E₁]
  [inst_24 : VectorBundle 𝕜 F₁ E₁] [inst_25 : FiberBundle F₂ E₂] [inst_26 : VectorBundle 𝕜 F₂ E₂]
  [inst_27 : ∀ (x : B), IsTopologicalAddGroup (E₂ x)] [inst_28 : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)]
  (f : M → Bundle.TotalSpace (F₁ →L[𝕜] F₂) fun b => E₁ b →L[𝕜] E₂ b) {s : Set M} {x₀ : M},
  ContMDiffWithinAt IM (IB.prod (modelWithCornersSelf 𝕜 (F₁ →L[𝕜] F₂))) n f s x₀ ↔
    ContMDiffWithinAt IM IB n (fun x => (f x).proj) s x₀ ∧
      ContMDiffWithinAt IM (modelWithCornersSelf 𝕜 (F₁ →L[𝕜] F₂)) n
        (fun x => ContinuousLinearMap.inCoordinates F₁ E₁ F₂ E₂ ⋯ ⋯ ⋯ ⋯ ⋯) s x₀

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

NontriviallyNormedFieldAddCommGroupModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceAddCommGroupModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceFiberBundleVectorBundleFiberBundleVectorBundleIsTopologicalAddGroupContinuousSMul