Theorems · Theorem · real analysis

derivWithin.scomp

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [inst_3 : NontriviallyNormedField 𝕜']
  [inst_4 : NormedAlgebra 𝕜 𝕜'] [inst_5 : NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {t' : Set 𝕜'} {h : 𝕜 → 𝕜'}
  {g₁ : 𝕜' → F},
  DifferentiableWithinAt 𝕜' g₁ t' (h x) →
    DifferentiableWithinAt 𝕜 h s x →
      Set.MapsTo h s t' → derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x)

Defined in

Mathlib.Analysis.Calculus.Deriv.Comp

Cited by

2 results in Mathlib

Foundations

Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNontriviallyNormedFieldNormedAlgebraNormedSpaceIsScalarTower