Theorems · Definition · global analysis

implicitFunctionOfBivariate

{𝕜 : Type u_1} →
  [inst : NontriviallyNormedField 𝕜] →
    [IsRCLikeNormedField 𝕜] →
      {E₁ : Type u_2} →
        [inst_2 : NormedAddCommGroup E₁] →
          [inst_3 : NormedSpace 𝕜 E₁] →
            [CompleteSpace E₁] →
              {E₂ : Type u_3} →
                [inst_5 : NormedAddCommGroup E₂] →
                  [inst_6 : NormedSpace 𝕜 E₂] →
                    [CompleteSpace E₂] →
                      {F : Type u_4} →
                        [inst_8 : NormedAddCommGroup F] →
                          [inst_9 : NormedSpace 𝕜 F] →
                            [CompleteSpace F] →
                              {u : E₁ × E₂} →
                                {f : E₁ → E₂ → F} →
                                  {f₁ : E₁ → E₂ → E₁ →L[𝕜] F} →
                                    {f₂ : E₁ → E₂ → E₂ →L[𝕜] F} →
                                      (∀ᶠ (v : E₁ × E₂) in nhds u, HasFDerivAt (fun x => f x v.2) (f₁ v.1 v.2) v.1) →
                                        (∀ᶠ (v : E₁ × E₂) in nhds u, HasFDerivAt (fun x => f v.1 x) (f₂ v.1 v.2) v.2) →
                                          ContinuousAt (↿f₁) u →
                                            ContinuousAt (↿f₂) u → (f₂ u.1 u.2).IsInvertible → E₁ → E₂

Implicit function ψ : E₁ → E₂ associated with the (curried) bivariate function f : E₁ → E₂ → F at u : E₁ × E₂.

Defined in

Mathlib.Analysis.Calculus.ImplicitFunction.Bivariate

Cited by

6 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldIsRCLikeNormedFieldNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace