Theorems · Theorem · global analysis

implicitFunctionOfBivariate.congr_simp

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] [inst_1 : IsRCLikeNormedField 𝕜] {E₁ : Type u_2}
  [inst_2 : NormedAddCommGroup E₁] [inst_3 : NormedSpace 𝕜 E₁] [inst_4 : CompleteSpace E₁] {E₂ : Type u_3}
  [inst_5 : NormedAddCommGroup E₂] [inst_6 : NormedSpace 𝕜 E₂] [inst_7 : CompleteSpace E₂] {F : Type u_4}
  [inst_8 : NormedAddCommGroup F] [inst_9 : NormedSpace 𝕜 F] [inst_10 : CompleteSpace F] {u u_1 : E₁ × E₂}
  (e_u : u = u_1) {f f_1 : E₁ → E₂ → F} (e_f : f = f_1) {f₁ f₁_1 : E₁ → E₂ → E₁ →L[𝕜] F} (e_f₁ : f₁ = f₁_1)
  {f₂ f₂_1 : E₁ → E₂ → E₂ →L[𝕜] F} (e_f₂ : f₂ = f₂_1)
  (df₁ : ∀ᶠ (v : E₁ × E₂) in nhds u, HasFDerivAt (fun x => f x v.2) (f₁ v.1 v.2) v.1)
  (df₂ : ∀ᶠ (v : E₁ × E₂) in nhds u, HasFDerivAt (fun x => f v.1 x) (f₂ v.1 v.2) v.2) (cf₁ : ContinuousAt (↿f₁) u)
  (cf₂ : ContinuousAt (↿f₂) u) (if₂u : (f₂ u.1 u.2).IsInvertible) (a a_1 : E₁),
  a = a_1 → implicitFunctionOfBivariate df₁ df₂ cf₁ cf₂ if₂u a = implicitFunctionOfBivariate ⋯ ⋯ ⋯ ⋯ ⋯ a_1

Defined in

Mathlib.Analysis.Calculus.ImplicitFunction.Bivariate

Cited by

0 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldIsRCLikeNormedFieldNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace