Theorems · Theorem · global analysis

ImplicitFunctionData.hasStrictFDerivAt_implicitFunction

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : CompleteSpace E] {F : Type u_3} [inst_4 : NormedAddCommGroup F]
  [inst_5 : NormedSpace 𝕜 F] [inst_6 : CompleteSpace F] {G : Type u_4} [inst_7 : NormedAddCommGroup G]
  [inst_8 : NormedSpace 𝕜 G] [inst_9 : CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G) (g'inv : G →L[𝕜] E),
  φ.rightDeriv ∘SL g'inv = ContinuousLinearMap.id 𝕜 G →
    φ.leftDeriv ∘SL g'inv = 0 → HasStrictFDerivAt (φ.implicitFunction (φ.leftFun φ.pt)) g'inv (φ.rightFun φ.pt)

Defined in

Mathlib.Analysis.Calculus.Implicit

Cited by

3 results in Mathlib

Foundations

Depth 189 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace