Theorems · Theorem · global analysis

contMDiffWithinAt_iff_of_mem_source

∀ {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} {x x' : M}
  {y : M'} {n : WithTop ℕ∞} [IsManifold I n M] [IsManifold I' n M'],
  x' ∈ (chartAt H x).source →
    f x' ∈ (chartAt H' y).source →
      (ContMDiffWithinAt I I' n f s x' ↔
        ContinuousWithinAt f s x' ∧
          ContDiffWithinAt 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
            (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I) (↑(extChartAt I x) x'))

One can reformulate being C^n within a set at a point as continuity within this set at this point, and being C^n in any chart containing that point.

Defined in

Mathlib.Geometry.Manifold.ContMDiff.Defs

Cited by

4 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifold