Theorems · Theorem · global analysis

extChartAt_preimage_mem_nhdsWithin

∀ {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : TopologicalSpace H] [inst_4 : TopologicalSpace M]
  {I : ModelWithCorners 𝕜 E H} {s t : Set M} [inst_5 : ChartedSpace H M] {x : M},
  t ∈ nhdsWithin x s →
    ↑(extChartAt I x).symm ⁻¹' t ∈ nhdsWithin (↑(extChartAt I x) x) (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I)

Technical lemma ensuring that the preimage under an extended chart of a neighborhood of the base point is a neighborhood of the preimage, within a set.

Defined in

Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Cited by

5 results in Mathlib

Foundations

Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpace