Theorems · Theorem · global analysis

mdifferentiableOn_iff_of_mem_maximalAtlas

∀ {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] {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}
  {e : OpenPartialHomeomorph M H} {e' : OpenPartialHomeomorph M' H'},
  e ∈ IsManifold.maximalAtlas I 1 M →
    e' ∈ IsManifold.maximalAtlas I' 1 M' →
      s ⊆ e.source →
        Set.MapsTo f s e'.source →
          (MDiff[s] f ↔
            ContinuousOn f s ∧ DifferentiableOn 𝕜 (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s))

Defined in

Mathlib.Geometry.Manifold.MFDeriv.Basic

Cited by

2 results in Mathlib

Foundations

Depth 206 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpace