Theorems · Definition · global analysis

HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented

{𝕜 : Type u_1} →
  [inst : NontriviallyNormedField 𝕜] →
    {E : Type u_2} →
      [inst_1 : NormedAddCommGroup E] →
        [inst_2 : NormedSpace 𝕜 E] →
          [CompleteSpace E] →
            {F : Type u_3} →
              [inst_4 : NormedAddCommGroup F] →
                [inst_5 : NormedSpace 𝕜 F] →
                  [CompleteSpace F] →
                    (f : E → F) →
                      (f' : E →L[𝕜] F) →
                        {a : E} →
                          HasStrictFDerivAt f f' a →
                            (↑f').range = ⊤ → (↑f').ker.ClosedComplemented → OpenPartialHomeomorph E (F × ↥(↑f').ker)

An open partial homeomorphism between E and F × f'.ker sending level surfaces of f to vertical subspaces.

Defined in

Mathlib.Analysis.Calculus.Implicit

Cited by

10 results in Mathlib

Foundations

Depth 185 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace