Theorems · Theorem · manifolds

SmoothPartitionOfUnity.contMDiffAt_finsum

∀ {ι : Type uι} {E : Type uE} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {F : Type uF}
  [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {H : Type uH} [inst_4 : TopologicalSpace H]
  {I : ModelWithCorners ℝ E H} {M : Type uM} [inst_5 : TopologicalSpace M] [inst_6 : ChartedSpace H M] {s : Set M}
  (f : SmoothPartitionOfUnity ι I M s) {n : ℕ∞} {x₀ : M} {g : ι → M → F},
  (∀ (i : ι), x₀ ∈ tsupport ⇑(f i) → ContMDiffAt I (modelWithCornersSelf ℝ F) (↑n) (g i) x₀) →
    ContMDiffAt I (modelWithCornersSelf ℝ F) (↑n) (fun x => ∑ᶠ (i : ι), (f i) x • g i x) x₀

Defined in

Mathlib.Geometry.Manifold.PartitionOfUnity

Cited by

1 results in Mathlib

Foundations

Depth 208 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpace