Theorem: SchwartzMap.hasFTaylorSeriesUpTo_integral_vectorFourierIntegrand
Proof state before:
case succ
E' : Type ?u.1044983
ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
R : Type u_4
M : ι → Type u_5
D : Type u_6
E : Type u_7
F : Type u_8
G : Type u_9
inst✝²² : DecidableEq ι
inst✝²¹ : Fintype ι
inst✝²⁰ : NontriviallyNormedField 𝕜
inst✝¹⁹ : (i : ι) → NormedAddCommGroup (M i)
inst✝¹⁸ : (i : ι) → NormedSpace ℝ (M i)
inst✝¹⁷ : NormedAddCommGroup D
inst✝¹⁶ : NormedSpace ℝ D
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : InnerProductSpace ℝ E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : InnerProductSpace ℝ E'
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℂ F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : MeasurableSpace E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : BorelSpace E
inst✝³ : MeasurableSpace E'
inst✝² : FiniteDimensional ℝ E'
inst✝¹ : BorelSpace E'
inst✝ : CompleteSpace F
f : 𝓢(E, F)
n : ℕ
IH :
  HasFTaylorSeriesUpTo (↑n) (fun ξ => ∫ (x : E), ((vectorFourierIntegrand ξ) f) x) fun ξ n =>
    (-(2 * ↑π * I)) ^ n •
      ∫ (x : E),
        ContinuousMultilinearMap.compContinuousLinearMap
          (ContinuousMultilinearMap.mkPiRing ℝ (Fin n) (((vectorFourierIntegrand ξ) f) x)) fun x_1 => (innerSL ℝ) x
⊢ HasFTaylorSeriesUpToOn (↑n + 1) (fun ξ => ∫ (x : E), ((vectorFourierIntegrand ξ) f) x)
    (fun ξ n =>
      (-(2 * ↑π * I)) ^ n •
        ∫ (x : E),
          ContinuousMultilinearMap.compContinuousLinearMap
            (ContinuousMultilinearMap.mkPiRing ℝ (Fin n) (((vectorFourierIntegrand ξ) f) x)) fun x_1 => (innerSL ℝ) x)
    Set.univ
Tactic:
rw [hasFTaylorSeriesUpToOn_succ_iff_left]
Proof state after:
case succ
E' : Type ?u.1044983
ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
R : Type u_4
M : ι → Type u_5
D : Type u_6
E : Type u_7
F : Type u_8
G : Type u_9
inst✝²² : DecidableEq ι
inst✝²¹ : Fintype ι
inst✝²⁰ : NontriviallyNormedField 𝕜
inst✝¹⁹ : (i : ι) → NormedAddCommGroup (M i)
inst✝¹⁸ : (i : ι) → NormedSpace ℝ (M i)
inst✝¹⁷ : NormedAddCommGroup D
inst✝¹⁶ : NormedSpace ℝ D
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : InnerProductSpace ℝ E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : InnerProductSpace ℝ E'
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℂ F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : MeasurableSpace E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : BorelSpace E
inst✝³ : MeasurableSpace E'
inst✝² : FiniteDimensional ℝ E'
inst✝¹ : BorelSpace E'
inst✝ : CompleteSpace F
f : 𝓢(E, F)
n : ℕ
IH :
  HasFTaylorSeriesUpTo (↑n) (fun ξ => ∫ (x : E), ((vectorFourierIntegrand ξ) f) x) fun ξ n =>
    (-(2 * ↑π * I)) ^ n •
      ∫ (x : E),
        ContinuousMultilinearMap.compContinuousLinearMap
          (ContinuousMultilinearMap.mkPiRing ℝ (Fin n) (((vectorFourierIntegrand ξ) f) x)) fun x_1 => (innerSL ℝ) x
⊢ HasFTaylorSeriesUpToOn (↑n) (fun ξ => ∫ (x : E), ((vectorFourierIntegrand ξ) f) x)
      (fun ξ n =>
        (-(2 * ↑π * I)) ^ n •
          ∫ (x : E),
            ContinuousMultilinearMap.compContinuousLinearMap
              (ContinuousMultilinearMap.mkPiRing ℝ (Fin n) (((vectorFourierIntegrand ξ) f) x)) fun x_1 => (innerSL ℝ) x)
      Set.univ ∧
    (∀ x ∈ Set.univ,
        HasFDerivWithinAt
          (fun y =>
            (-(2 * ↑π * I)) ^ n •
              ∫ (x : E),
                ContinuousMultilinearMap.compContinuousLinearMap
                  (ContinuousMultilinearMap.mkPiRing ℝ (Fin n) (((vectorFourierIntegrand y) f) x)) fun x_1 =>
                  (innerSL ℝ) x)
          (ContinuousMultilinearMap.curryLeft
            ((-(2 * ↑π * I)) ^ Nat.succ n •
              ∫ (x_1 : E),
                ContinuousMultilinearMap.compContinuousLinearMap
                  (ContinuousMultilinearMap.mkPiRing ℝ (Fin (Nat.succ n)) (((vectorFourierIntegrand x) f) x_1)) fun x =>
                  (innerSL ℝ) x_1))
          Set.univ x) ∧
      ContinuousOn
        (fun x =>
          (-(2 * ↑π * I)) ^ (n + 1) •
            ∫ (x_1 : E),
              ContinuousMultilinearMap.compContinuousLinearMap
                (ContinuousMultilinearMap.mkPiRing ℝ (Fin (n + 1)) (((vectorFourierIntegrand x) f) x_1)) fun x =>
                (innerSL ℝ) x_1)
        Set.univ