Theorems · Theorem · several complex variables

HasFiniteFPowerSeriesAt.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] {m n : ℕ} {g : F → G} {f : E → F}
  {q : FormalMultilinearSeries 𝕜 F G} {p : FormalMultilinearSeries 𝕜 E F} {x : E},
  HasFiniteFPowerSeriesAt g q (f x) m →
    HasFiniteFPowerSeriesAt f p x n → 0 < n → HasFiniteFPowerSeriesAt (g ∘ f) (q.comp p) x (m * n)

If two functions g and f have finite power series q and p respectively at f x and x, then g ∘ f admits the finite power series q.comp p at x.

Defined in

Mathlib.Analysis.Analytic.Composition

Cited by

1 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace