Theorems · Theorem · real analysis

ContinuousLinearMap.hasDerivWithinAt_of_bilinear

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] {E : Type w} [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace 𝕜 E] {G : Type u_1}
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {x : 𝕜} {s : Set 𝕜} {B : E →L[𝕜] F →L[𝕜] G} {u : 𝕜 → E}
  {v : 𝕜 → F} {u' : E} {v' : F},
  HasDerivWithinAt u u' s x →
    HasDerivWithinAt v v' s x → HasDerivWithinAt (fun x => (B (u x)) (v x)) ((B (u x)) v' + (B u') (v x)) s x

Defined in

Mathlib.Analysis.Calculus.Deriv.Mul

Cited by

1 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace