Theorems · Theorem · global analysis

fderivWithin_extChartAt_comp_extChartAt_symm_range

∀ {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] {x : M},
  fderivWithin 𝕜 (↑(extChartAt I x) ∘ ↑(extChartAt I x).symm) (Set.range ↑I) (↑(extChartAt I x) x) =
    ContinuousLinearMap.id 𝕜 E

The fderivWithin of the round-trip composition (extChartAt I x) ∘ (extChartAt I x).symm at the chart point in range I equals the identity.

Defined in

Mathlib.Geometry.Manifold.MFDeriv.Atlas

Cited by

0 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpace