Theorems · Theorem · several complex variables

AnalyticWithinAt.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} {x : E} {t : Set F} {s : Set E},
  AnalyticWithinAt 𝕜 g t (f x) → AnalyticWithinAt 𝕜 f s x → Set.MapsTo f s t → AnalyticWithinAt 𝕜 (g ∘ f) s x

If two functions g and f are analytic respectively at f x and x, within two sets s and t such that f maps s to t, then g ∘ f is analytic at x within s.

Defined in

Mathlib.Analysis.Analytic.Composition

Cited by

9 results in Mathlib

Foundations

Depth 182 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace