Theorems · Theorem · global analysis

eventually_apply_eq_iff_implicitFunctionOfBivariate

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] [inst_1 : IsRCLikeNormedField 𝕜] {E₁ : Type u_2}
  [inst_2 : NormedAddCommGroup E₁] [inst_3 : NormedSpace 𝕜 E₁] [inst_4 : CompleteSpace E₁] {E₂ : Type u_3}
  [inst_5 : NormedAddCommGroup E₂] [inst_6 : NormedSpace 𝕜 E₂] [inst_7 : CompleteSpace E₂] {F : Type u_4}
  [inst_8 : NormedAddCommGroup F] [inst_9 : NormedSpace 𝕜 F] [inst_10 : CompleteSpace F] {u : E₁ × E₂} {f : E₁ → E₂ → F}
  {f₁ : E₁ → E₂ → E₁ →L[𝕜] F} {f₂ : E₁ → E₂ → E₂ →L[𝕜] F}
  (df₁ : ∀ᶠ (v : E₁ × E₂) in nhds u, HasFDerivAt (fun x => f x v.2) (f₁ v.1 v.2) v.1)
  (df₂ : ∀ᶠ (v : E₁ × E₂) in nhds u, HasFDerivAt (fun x => f v.1 x) (f₂ v.1 v.2) v.2) (cf₁ : ContinuousAt (↿f₁) u)
  (cf₂ : ContinuousAt (↿f₂) u) (if₂u : (f₂ u.1 u.2).IsInvertible),
  ∀ᶠ (v : E₁ × E₂) in nhds u, f v.1 v.2 = f u.1 u.2 ↔ implicitFunctionOfBivariate df₁ df₂ cf₁ cf₂ if₂u v.1 = v.2

Defined in

Mathlib.Analysis.Calculus.ImplicitFunction.Bivariate

Cited by

0 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldIsRCLikeNormedFieldNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace