Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case e'_12
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 := ⋯
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
⊢ apply 𝕜 E F fa.2 ∘SL ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i) +
      fa.1.linearDeriv fa.2 ∘SL ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i) =
    (f.linearDeriv fun x ↦ fa) ∘SL
      ContinuousLinearMap.pi fun x ↦ ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))

Tactic:
ext g

Goal state after:
case e'_12.hl
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 : ContinuousMultilinearMap 𝕜 E F
⊢ ((apply 𝕜 E F fa.2 ∘SL ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i) +
          fa.1.linearDeriv fa.2 ∘SL ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i)) ∘SL
        ContinuousLinearMap.inl 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i))
      g =
    (((f.linearDeriv fun x ↦ fa) ∘SL
          ContinuousLinearMap.pi fun x ↦
            ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))) ∘SL
        ContinuousLinearMap.inl 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i))
      g

case hr
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
⊢ ((apply 𝕜 E F fa.2 ∘SL ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i) +
          fa.1.linearDeriv fa.2 ∘SL ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i)) ∘SL
        ContinuousLinearMap.inr 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i))
      g =
    (((f.linearDeriv fun x ↦ fa) ∘SL
          ContinuousLinearMap.pi fun x ↦
            ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))) ∘SL
        ContinuousLinearMap.inr 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i))
      g