Theorems · Theorem · global analysis

MDifferentiableWithinAt.mpullbackWithin_vectorField_inter

∀ {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] {H' : Type u_5} [inst_6 : TopologicalSpace H']
  {E' : Type u_6} [inst_7 : NormedAddCommGroup E'] [inst_8 : NormedSpace 𝕜 E'] {I' : ModelWithCorners 𝕜 E' H'}
  {M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] {f : M → M'} {s : Set M} {x₀ : M}
  {V : (x : M') → TangentSpace I' x} {n : WithTop ℕ∞} {t : Set M'} [inst_11 : IsManifold I 2 M]
  [inst_12 : IsManifold I' 2 M'] [CompleteSpace E],
  MDiffAt[t] (T% V) (f x₀) →
    ContMDiffWithinAt I I' n f s x₀ →
      (mfderiv[s] f x₀).IsInvertible →
        x₀ ∈ s →
          UniqueMDiff[s] → 2 ≤ n → (MDiffAt[s ∩ f ⁻¹' t] fun x => ⟨x, VectorField.mpullbackWithin I I' f V s x⟩) x₀

The pullback of a differentiable vector field by a C^n function with 2 ≤ n is differentiable. Version within a set at a point.

Defined in

Mathlib.Geometry.Manifold.VectorField.Pullback

Cited by

3 results in Mathlib

Foundations

Depth 222 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifoldCompleteSpace