Theorems · Definition · global analysis

IsLocalFrameOn.fintypeOfFiniteDimensional

{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 : ι → (x : M) → V x} →
                                            {u : Set M} →
                                              {x : M} →
                                                {n : WithTop ℕ∞} →
                                                  [VectorBundle 𝕜 F V] →
                                                    [FiniteDimensional 𝕜 F] →
                                                      IsLocalFrameOn I F n s u → x ∈ u → Fintype ι

If {sᵢ} is a local frame on a vector bundle, F being finite-dimensional implies the indexing set being finite.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Cited by

6 results in Mathlib

Foundations

Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceFiberBundleVectorBundleFiniteDimensional