Theorems · Theorem · global analysis

UniqueMDiffWithinAt.image_denseRange

∀ {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'] {s : Set M} {x : M},
  UniqueMDiffAt[s] x →
    ∀ {f : M → M'} {f' : E →L[𝕜] E'}, HasMFDerivAt[s] f x f' → DenseRange ⇑f' → UniqueMDiffAt[f '' s] (f x)

If s has the unique differential property at x, f is differentiable within s at x and its derivative has dense range, then f '' s has the unique differential property at f x.

Defined in

Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential

Cited by

2 results in Mathlib

Foundations

Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpace