Theorems · Theorem · global analysis

IsLocalFrameOn.coeff_eq_of_eq

∀ {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] {ι : Type u_7} {s s' : ι → (x : M) → V x}
  {u : Set M} {n : WithTop ℕ∞} {x : M} (hs : IsLocalFrameOn I F n s u) (hs' : IsLocalFrameOn I F n s' u),
  (∀ (i : ι), s i x = s' i x) → ∀ {t : (x : M) → V x} (i : ι), (hs.coeff i x) (t x) = (hs'.coeff i x) (t x)

If s and s' are local frames which are equal at x, a section t has equal frame coefficients in them.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Cited by

0 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceFiberBundle