Theorems · Definition · global analysis

ContDiffAt.implicitFunction

{𝕜 : Type u_1} →
  [inst : RCLike 𝕜] →
    {E₁ : Type u_2} →
      [inst_1 : NormedAddCommGroup E₁] →
        [inst_2 : NormedSpace 𝕜 E₁] →
          [CompleteSpace E₁] →
            {E₂ : Type u_3} →
              [inst_4 : NormedAddCommGroup E₂] →
                [inst_5 : NormedSpace 𝕜 E₂] →
                  [CompleteSpace E₂] →
                    {F : Type u_4} →
                      [inst_7 : NormedAddCommGroup F] →
                        [inst_8 : NormedSpace 𝕜 F] →
                          [CompleteSpace F] →
                            {u : E₁ × E₂} →
                              {f : E₁ × E₂ → F} →
                                {n : WithTop ℕ∞} →
                                  ContDiffAt 𝕜 n f u →
                                    n ≠ 0 → (fderiv 𝕜 f u ∘SL ContinuousLinearMap.inr 𝕜 E₁ E₂).IsInvertible → E₁ → E₂

Implicit function ψ defined by f (x, ψ x) = f u.

Defined in

Mathlib.Analysis.Calculus.ImplicitContDiff

Cited by

10 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace