Theorems · Theorem · global analysis

TensorialAt.sum

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {F : Type u_5} [inst_6 : NormedAddCommGroup F]
  [inst_7 : NormedSpace 𝕜 F] {V : M → Type u_6} [inst_8 : TopologicalSpace (Bundle.TotalSpace F V)]
  [inst_9 : (x : M) → AddCommGroup (V x)] [inst_10 : (x : M) → Module 𝕜 (V x)]
  [inst_11 : (x : M) → TopologicalSpace (V x)] [inst_12 : FiberBundle F V] {A : Type u_9} [inst_13 : AddCommGroup A]
  [inst_14 : Module 𝕜 A] {Φ : ((x : M) → V x) → A} {x : M} [VectorBundle 𝕜 F V],
  TensorialAt I F Φ x →
    ∀ {ι : Type u_10} {s : Finset ι} (σ : ι → (x : M) → V x),
      (∀ i ∈ s, MDiffAt (T% σ) x) → (Φ fun x' => ∑ i ∈ s, σ i x') = ∑ i ∈ s, Φ (σ i)

A tensorial operation on sections of a vector bundle respects sums (since it respects binary addition).

Defined in

Mathlib.Geometry.Manifold.VectorBundle.Tensoriality

Cited by

1 results in Mathlib

Foundations

Depth 214 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceFiberBundleAddCommGroupModuleVectorBundle