Theorems · Theorem · global analysis

VectorField.contMDiffWithinAt_mpullbackWithin_extChartAt_symm

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {H : Type u_2} [inst_1 : TopologicalSpace H] {E : Type u_3}
  [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {s : Set M} {x : M} {m n : WithTop ℕ∞} [CompleteSpace E]
  [inst_7 : IsManifold I 1 M] {V : (x : M) → TangentSpace I x},
  ContMDiffWithinAt I I.tangent m (fun x => ⟨x, V x⟩) s x →
    UniqueMDiff[s] →
      x ∈ s →
        m + 1 ≤ n →
          ContMDiffWithinAt (modelWithCornersSelf 𝕜 E) (modelWithCornersSelf 𝕜 E).tangent m
            (fun x_1 =>
              ⟨x_1,
                VectorField.mpullbackWithin (modelWithCornersSelf 𝕜 E) I (↑(extChartAt I x).symm) V (Set.range ↑I) x_1⟩)
            ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)

Defined in

Mathlib.Geometry.Manifold.VectorField.Pullback

Cited by

3 results in Mathlib

Foundations

Depth 226 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceCompleteSpaceIsManifold