Theorems · Theorem · global analysis

HasStrictFDerivAt.eventually_apply_eq_iff_implicitFunctionOfProdDomain

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 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} {f'u : E₁ × E₂ →L[𝕜] F}
  (dfu : HasStrictFDerivAt f f'u u) (if₂u : (f'u ∘SL ContinuousLinearMap.inr 𝕜 E₁ E₂).IsInvertible),
  ∀ᶠ (v : E₁ × E₂) in nhds u, f v = f u ↔ dfu.implicitFunctionOfProdDomain if₂u v.1 = v.2

Defined in

Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain

Cited by

4 results in Mathlib

Foundations

Depth 190 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace