-- Module: Mathlib.Analysis.Calculus.ContDiff.Convolution
Goal:
𝕜 : Type u𝕜
E : Type uE
inst : NormedAddCommGroup E
inst_1 : RCLike 𝕜
inst_2 : NormedSpace 𝕜 E
G P : Type uP
inst_3 : MeasurableSpace G
μ : Measure G
inst_4 : NormedAddCommGroup G
inst_5 : BorelSpace G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup P
inst_8 : NormedSpace 𝕜 P
f : G → E
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hf : LocallyIntegrable f μ
n : ℕ
ih :
  ∀ {E' F : Type uP} [inst_14 : NormedAddCommGroup E'] [inst_15 : NormedAddCommGroup F] [inst_16 : NormedSpace 𝕜 E']
    [inst_17 : NormedSpace ℝ F] [inst_18 : NormedSpace 𝕜 F]
    (L : ContinuousLinearMap.{u𝕜, u𝕜, uE, uP} (RingHom.id 𝕜) E (E' →L[𝕜] F)) {g : P → G → E'}
    (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0)
    (hg : ContDiffOn 𝕜 (↑(↑n : ℕ∞) : WithTop ℕ∞) (↿g) (SProd.sprod (β := Set G) s univ)),
    ContDiffOn 𝕜 (F := F) (↑(↑n : ℕ∞) : WithTop ℕ∞)
      (fun (q : P × G) ↦ convolution.{u𝕜, uP, uE, uP, uP} (𝕜 := 𝕜) f (g q.1) L μ q.2) (SProd.sprod (β := Set G) s univ)
E' F : Type uP
inst_9 : NormedAddCommGroup E'
inst_10 : NormedAddCommGroup F
inst_11 : NormedSpace 𝕜 E'
inst_12 : NormedSpace ℝ F
inst_13 : NormedSpace 𝕜 F
L : ContinuousLinearMap.{u𝕜, u𝕜, uE, uP} (RingHom.id 𝕜) E (E' →L[𝕜] F)
g : P → G → E'
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hg : ContDiffOn 𝕜 (HAdd.hAdd (α := WithTop ℕ∞) (↑n : WithTop ℕ∞) 1) (↿g) (SProd.sprod (β := Set G) s univ)
f' : P → G → P × G →L[𝕜] F :=
  fun (p : P) (a : G) ↦
    convolution.{u𝕜, uP, uE, uP, uP} (𝕜 := 𝕜) (E' := P × G →L[𝕜] E') f (fun (x : G) ↦ fderiv 𝕜 (uncurry g) (p, x))
      (precompR (P × G) L) μ a
A :
  ∀ (q₀ : P × G),
    q₀.1 ∈ s →
      HasFDerivAt (fun (q : P × G) ↦ convolution.{u𝕜, uP, uE, uP, uP} (𝕜 := 𝕜) f (g q.1) L μ q.2) (f' q₀.1 q₀.2) q₀
succ :
  ∀
    (hg :
      DifferentiableOn 𝕜 (↿g) (SProd.sprod (β := Set G) s univ) ∧
        (Eq (α := WithTop ℕ∞) (↑n : WithTop ℕ∞) ⊤ → AnalyticOn 𝕜 (↿g) (SProd.sprod (β := Set G) s univ)) ∧
          ContDiffOn 𝕜 (F := P × G →L[𝕜] E') (↑n : WithTop ℕ∞) (fderiv 𝕜 ↿g) (SProd.sprod (β := Set G) s univ)),
    let f' : P → G → P × G →L[𝕜] F := fun (p : P) (a : G) ↦
      convolution.{u𝕜, uP, uE, uP, uP} (𝕜 := 𝕜) (E' := P × G →L[𝕜] E') f (fun (x : G) ↦ fderiv 𝕜 (uncurry g) (p, x))
        (precompR (P × G) L) μ a;
    ∀
      (A :
        ∀ (q₀ : P × G),
          q₀.1 ∈ s →
            HasFDerivAt (fun (q : P × G) ↦ convolution.{u𝕜, uP, uE, uP, uP} (𝕜 := 𝕜) f (g q.1) L μ q.2) (f' q₀.1 q₀.2)
              q₀),
      DifferentiableOn 𝕜 (F := F) (fun (q : P × G) ↦ convolution.{u𝕜, uP, uE, uP, uP} (𝕜 := 𝕜) f (g q.1) L μ q.2)
          (SProd.sprod (β := Set G) s univ) ∧
        (Eq (α := WithTop ℕ∞) (↑n : WithTop ℕ∞) ⊤ →
            AnalyticOn 𝕜 (F := F) (fun (q : P × G) ↦ convolution.{u𝕜, uP, uE, uP, uP} (𝕜 := 𝕜) f (g q.1) L μ q.2)
              (SProd.sprod (β := Set G) s univ)) ∧
          ContDiffOn 𝕜 (F := P × G →L[𝕜] F) (↑n : WithTop ℕ∞)
            (fderiv 𝕜 fun (q : P × G) ↦ convolution.{u𝕜, uP, uE, uP, uP} (𝕜 := 𝕜) f (g q.1) L μ q.2)
            (SProd.sprod (β := Set G) s univ)
⊢ ContDiffOn 𝕜 (F := F) (HAdd.hAdd (α := WithTop ℕ∞) (↑n : WithTop ℕ∞) 1)
    (fun (q : P × G) ↦ convolution.{u𝕜, uP, uE, uP, uP} (𝕜 := 𝕜) f (g q.1) L μ q.2) (SProd.sprod (β := Set G) s univ)

Tactic:
rw [contDiffOn_succ_iff_fderiv_of_isOpen (hs.prod (@isOpen_univ G _))] at hg ⊢

Premises used: contDiffOn_succ_iff_fderiv_of_isOpen, isOpen_univ, IsOpen.prod