Theorems · Theorem · global analysis

ContMDiffOn.iUnion_of_isOpen

∀ {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] {E' : Type u_5} [inst_5 : NormedAddCommGroup E'] [inst_6 : NormedSpace 𝕜 E']
  {H' : Type u_6} [inst_7 : TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7}
  [inst_8 : TopologicalSpace M'] [inst_9 : ChartedSpace H M] [inst_10 : ChartedSpace H' M'] {f : M → M'}
  {n : WithTop ℕ∞} {ι : Type u_11} {s : ι → Set M},
  (∀ (i : ι), ContMDiffOn I I' n f (s i)) → (∀ (i : ι), IsOpen (s i)) → ContMDiffOn I I' n f (⋃ i, s i)

If a function is C^k on open sets s i, it is C^k on their union

Defined in

Mathlib.Geometry.Manifold.ContMDiff.Basic

Cited by

2 results in Mathlib

Foundations

Depth 207 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceChartedSpace