Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case hr.e_f.e_6.inl
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fa : ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)
f : ContinuousMultilinearMap 𝕜 (fun x ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)) F :=
  (ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F)).continuousMultilinearMapOption
Hf :
  HasStrictFDerivAt (fun x ↦ f fun x_1 ↦ x)
    ((f.linearDeriv fun x ↦ fa) ∘SL
      ContinuousLinearMap.pi fun x ↦ ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)))
    fa
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
g : (i : ι) → E i
j : ι
⊢ Function.update fa.2 j (g j) j = (Function.update (fun x ↦ fa) (some j) (0, g) (some j)).2 j

pp: case hr.e_f.e_6.inr
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fa : ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)
f : ContinuousMultilinearMap 𝕜 (fun x ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)) F :=
  (ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F)).continuousMultilinearMapOption
Hf :
  HasStrictFDerivAt (fun x ↦ f fun x_1 ↦ x)
    ((f.linearDeriv fun x ↦ fa) ∘SL
      ContinuousLinearMap.pi fun x ↦ ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)))
    fa
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
g : (i : ι) → E i
i j : ι
hij : j ≠ i
⊢ Function.update fa.2 i (g i) j = (Function.update (fun x ↦ fa) (some i) (0, g) (some j)).2 j

Tactic:
<;>

Goal state after:
case hr.e_f.e_6.inl
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fa : ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)
f : ContinuousMultilinearMap 𝕜 (fun x ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)) F :=
  (ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F)).continuousMultilinearMapOption
Hf :
  HasStrictFDerivAt (fun x ↦ f fun x_1 ↦ x)
    ((f.linearDeriv fun x ↦ fa) ∘SL
      ContinuousLinearMap.pi fun x ↦ ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)))
    fa
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
g : (i : ι) → E i
j : ι
⊢ Function.update fa.2 j (g j) j = (Function.update (fun x ↦ fa) (some j) (0, g) (some j)).2 j

case hr.e_f.e_6.inr
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fa : ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)
f : ContinuousMultilinearMap 𝕜 (fun x ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)) F :=
  (ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F)).continuousMultilinearMapOption
Hf :
  HasStrictFDerivAt (fun x ↦ f fun x_1 ↦ x)
    ((f.linearDeriv fun x ↦ fa) ∘SL
      ContinuousLinearMap.pi fun x ↦ ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)))
    fa
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
g : (i : ι) → E i
i j : ι
hij : j ≠ i
⊢ Function.update fa.2 i (g i) j = (Function.update (fun x ↦ fa) (some i) (0, g) (some j)).2 j