Theorems · Theorem · global analysis

HasStrictFDerivAt.map_implicitFunction_eq

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] [inst_1 : CompleteSpace 𝕜] {E : Type u_2}
  [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] [inst_4 : CompleteSpace E] {F : Type u_3}
  [inst_5 : NormedAddCommGroup F] [inst_6 : NormedSpace 𝕜 F] [inst_7 : FiniteDimensional 𝕜 F] {f : E → F}
  {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : (↑f').range = ⊤),
  ∀ᶠ (p : F × ↥(↑f').ker) in nhds (f a, 0), f (HasStrictFDerivAt.implicitFunction f f' hf hf' p.1 p.2) = p.1

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

Defined in

Mathlib.Analysis.Calculus.Implicit

Cited by

0 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceFiniteDimensional