Theorems · Theorem · global analysis

ContMDiffWithinAt.comp

∀ {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'] {E'' : Type u_8} [inst_9 : NormedAddCommGroup E''] [inst_10 : NormedSpace 𝕜 E'']
  {H'' : Type u_9} [inst_11 : TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10}
  [inst_12 : TopologicalSpace M''] [inst_13 : ChartedSpace H M] [inst_14 : ChartedSpace H' M']
  [inst_15 : ChartedSpace H'' M''] {f : M → M'} {s : Set M} {n : WithTop ℕ∞} {t : Set M'} {g : M' → M''} (x : M),
  ContMDiffWithinAt I' I'' n g t (f x) →
    ContMDiffWithinAt I I' n f s x → Set.MapsTo f s t → ContMDiffWithinAt I I'' n (g ∘ f) s x

The composition of C^n functions within domains at points is C^n.

Defined in

Mathlib.Geometry.Manifold.ContMDiff.Basic

Cited by

18 results in Mathlib

Foundations

Depth 193 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceChartedSpaceChartedSpace