Theorems · Theorem · several complex variables

HasFPowerSeriesWithinAt.comp

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {g : F → G} {f : E → F} {q : FormalMultilinearSeries 𝕜 F G}
  {p : FormalMultilinearSeries 𝕜 E F} {x : E} {t : Set F} {s : Set E},
  HasFPowerSeriesWithinAt g q t (f x) →
    HasFPowerSeriesWithinAt f p s x → Set.MapsTo f s t → HasFPowerSeriesWithinAt (g ∘ f) (q.comp p) s x

If two functions g and f have power series q and p respectively at f x and x, within two sets s and t such that f maps s to t, then g ∘ f admits the power series q.comp p at x within s.

Defined in

Mathlib.Analysis.Analytic.Composition

Cited by

2 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace