Theorems · Theorem · global analysis

ContDiffAt.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] {u : E₁ × E₂} {f : E₁ × E₂ → F} {n : WithTop ℕ∞} (cdf : ContDiffAt 𝕜 n f u) (pn : n ≠ 0)
  (if₂ : (fderiv 𝕜 f u ∘SL ContinuousLinearMap.inr 𝕜 E₁ E₂).IsInvertible),
  ContDiffAt 𝕜 n (cdf.implicitFunction pn if₂) u.1

If the implicit equation f is $C^n$ at (u₁, u₂), then its implicit function ψ around u₁ is also $C^n$ at u₁.

Defined in

Mathlib.Analysis.Calculus.ImplicitContDiff

Cited by

1 results in Mathlib

Foundations

Depth 206 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace