Theorems · Theorem · global analysis

HasStrictFDerivAt.map_implicitFunctionOfComplemented_eq

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : CompleteSpace E] {F : Type u_3} [inst_4 : NormedAddCommGroup F]
  [inst_5 : NormedSpace 𝕜 F] [inst_6 : CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E}
  (hf : HasStrictFDerivAt f f' a) (hf' : (↑f').range = ⊤) (hker : (↑f').ker.ClosedComplemented),
  ∀ᶠ (p : F × ↥(↑f').ker) in nhds (f a, 0),
    f (HasStrictFDerivAt.implicitFunctionOfComplemented f f' hf hf' hker p.1 p.2) = p.1

HasStrictFDerivAt.implicitFunctionOfComplemented sends (z, y) to a point in f ⁻¹' z.

Defined in

Mathlib.Analysis.Calculus.Implicit

Cited by

1 results in Mathlib

Foundations

Depth 189 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace