Theorems · Theorem · global analysis

ImplicitFunctionData.contDiffAt_implicitFunction

∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {E₁ : Type u_2} [inst_1 : NormedAddCommGroup E₁] [inst_2 : NormedSpace 𝕜 E₁]
  [inst_3 : CompleteSpace E₁] {E₂ : Type u_3} [inst_4 : NormedAddCommGroup E₂] [inst_5 : NormedSpace 𝕜 E₂]
  [inst_6 : CompleteSpace E₂] {F : Type u_4} [inst_7 : NormedAddCommGroup F] [inst_8 : NormedSpace 𝕜 F]
  [inst_9 : CompleteSpace F] {φ : ImplicitFunctionData 𝕜 E₁ E₂ F} {n : WithTop ℕ∞},
  ContDiffAt 𝕜 n φ.leftFun φ.pt →
    ContDiffAt 𝕜 n φ.rightFun φ.pt → n ≠ 0 → ContDiffAt 𝕜 n (Function.uncurry φ.implicitFunction) (φ.prodFun φ.pt)

The implicit function defined by a $C^n$ implicit equation is $C^n$. This applies to the general form of the implicit function theorem.

Defined in

Mathlib.Analysis.Calculus.ImplicitContDiff

Cited by

1 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace