Theorems · Theorem · global analysis

ImplicitFunctionData.fderiv_implicitFunction_apply_eq_iff

∀ {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] {G : Type u_4} [inst_7 : NormedAddCommGroup G]
  [inst_8 : NormedSpace 𝕜 G] [inst_9 : CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G) {x : G} {y : E},
  (fderiv 𝕜 (φ.implicitFunction (φ.leftFun φ.pt)) (φ.rightFun φ.pt)) x = y ↔ φ.leftDeriv y = 0 ∧ φ.rightDeriv y = x

Defined in

Mathlib.Analysis.Calculus.Implicit

Cited by

3 results in Mathlib

Foundations

Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace