Theorems · Theorem · global analysis

ContMDiffOn.contMDiffOn_tangentMapWithin

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {m n : WithTop ℕ∞} {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] {E' : Type u_5} [inst_6 : NormedAddCommGroup E']
  [inst_7 : NormedSpace 𝕜 E'] {H' : Type u_6} [inst_8 : TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'}
  {M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] {f : M → M'} {s : Set M}
  [Is : IsManifold I 1 M] [I's : IsManifold I' 1 M'],
  ContMDiffOn I I' n f s →
    m + 1 ≤ n →
      UniqueMDiff[s] → ContMDiffOn I.tangent I'.tangent m (tangentMapWithin I I' f s) (Bundle.TotalSpace.proj ⁻¹' s)

If a function is C^n on a domain with unique derivatives, then its bundled derivative is C^m when m+1 ≤ n.

Defined in

Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Cited by

2 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifold