-- Module: Mathlib.Analysis.Fourier.FourierTransform
Goal:
𝕜 : Type u_1
inst : CommRing 𝕜
V : Type u_2
inst_1 : AddCommGroup V
inst_2 : Module 𝕜 V
inst_3 : MeasurableSpace V
W : Type u_3
inst_4 : AddCommGroup W
inst_5 : Module 𝕜 W
E : Type u_4
F : Type u_5
G : Type u_6
inst_6 : NormedAddCommGroup E
inst_7 : NormedSpace ℂ E
inst_8 : NormedAddCommGroup F
inst_9 : NormedSpace ℂ F
inst_10 : NormedAddCommGroup G
inst_11 : NormedSpace ℂ G
inst_12 : TopologicalSpace 𝕜
inst_13 : IsTopologicalRing 𝕜
inst_14 : TopologicalSpace V
inst_15 : BorelSpace V
inst_16 : TopologicalSpace W
inst_17 : MeasurableSpace W
inst_18 : BorelSpace W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜
ν : Measure W
inst_19 : SigmaFinite μ
inst_20 : SigmaFinite ν
inst_21 : SecondCountableTopology V
inst_22 : CompleteSpace E
inst_23 : CompleteSpace F
f : V → E
g : W → F
M : E →L⋆[ℂ] F →L[ℂ] G
he : Continuous[inst_12, _] (⇑e : (a : 𝕜) → 𝕊)
hL :
  Continuous[instTopologicalSpaceProd, inst_12] fun (p : V × W) ↦
    (((L : (a : V) → W →ₗ[𝕜] 𝕜) p.1 : W →ₗ[𝕜] 𝕜) : (a : W) → 𝕜) p.2
hf : Integrable f μ
hg : Integrable g ν
hG : CompleteSpace G
e₁ :
  Eq.{u_6 + 1} (α := G)
    (∫ (ξ : W),
      (((M.flip : (a : F) → E →L⋆[ℂ] G) (g ξ) : E →L⋆[ℂ] G) : (a : E) → G)
        (∫ (x : V),
          HSMul.hSMul (α := 𝕊) ((e : (a : 𝕜) → 𝕊) (-(((L : (a : V) → W →ₗ[𝕜] 𝕜) x : W →ₗ[𝕜] 𝕜) : (a : W) → 𝕜) ξ) : 𝕊)
            (f x) ∂μ) ∂ν)
    (∫ (ξ : W),
      ∫ (x : V),
        (((M.flip : (a : F) → E →L⋆[ℂ] G) (g ξ) : E →L⋆[ℂ] G) : (a : E) → G)
          (HSMul.hSMul (α := 𝕊) ((e : (a : 𝕜) → 𝕊) (-(((L : (a : V) → W →ₗ[𝕜] 𝕜) x : W →ₗ[𝕜] 𝕜) : (a : W) → 𝕜) ξ) : 𝕊)
            (f x)) ∂μ ∂ν)
e₂ :
  Eq.{u_6 + 1} (α := G)
    (∫ (x : V),
      ∫ (ξ : W),
        (((M.flip : (a : F) → E →L⋆[ℂ] G) (g ξ) : E →L⋆[ℂ] G) : (a : E) → G)
          (HSMul.hSMul (α := 𝕊) ((e : (a : 𝕜) → 𝕊) (-(((L : (a : V) → W →ₗ[𝕜] 𝕜) x : W →ₗ[𝕜] 𝕜) : (a : W) → 𝕜) ξ) : 𝕊)
            (f x)) ∂ν ∂μ)
    (∫ (x : V),
      ∫ (ξ : W),
        (((M : (a : E) → F →L[ℂ] G) (f x) : F →L[ℂ] G) : (a : F) → G)
          (HSMul.hSMul (α := 𝕊)
            ((e : (a : 𝕜) → 𝕊) ((((L.flip : (a : W) → V →ₗ[𝕜] 𝕜) ξ : V →ₗ[𝕜] 𝕜) : (a : V) → 𝕜) x) : 𝕊) (g ξ)) ∂ν ∂μ)
e₃ :
  Eq.{u_6 + 1} (α := G)
    (∫ (x : V),
      ∫ (ξ : W),
        (((M : (a : E) → F →L[ℂ] G) (f x) : F →L[ℂ] G) : (a : F) → G)
          (HSMul.hSMul (α := 𝕊)
            ((e : (a : 𝕜) → 𝕊) ((((L.flip : (a : W) → V →ₗ[𝕜] 𝕜) ξ : V →ₗ[𝕜] 𝕜) : (a : V) → 𝕜) x) : 𝕊) (g ξ)) ∂ν ∂μ)
    (∫ (x : V),
      ∫ (ξ : W),
        (((M : (a : E) → F →L[ℂ] G) (f x) : F →L[ℂ] G) : (a : F) → G)
          (HSMul.hSMul (α := 𝕊)
            ((e : (a : 𝕜) → 𝕊) (-((-(L.flip : (a : W) → V →ₗ[𝕜] 𝕜) ξ : V →ₗ[𝕜] 𝕜) : (a : V) → 𝕜) x) : 𝕊) (g ξ)) ∂ν ∂μ)
e₄ :
  Eq.{u_6 + 1} (α := G)
    (∫ (x : V),
      ∫ (ξ : W),
        (((M : (a : E) → F →L[ℂ] G) (f x) : F →L[ℂ] G) : (a : F) → G)
          (HSMul.hSMul (α := 𝕊)
            ((e : (a : 𝕜) → 𝕊) (-((-(L.flip : (a : W) → V →ₗ[𝕜] 𝕜) ξ : V →ₗ[𝕜] 𝕜) : (a : V) → 𝕜) x) : 𝕊) (g ξ)) ∂ν ∂μ)
    (∫ (x : V),
      (((M : (a : E) → F →L[ℂ] G) (f x) : F →L[ℂ] G) : (a : F) → G)
        (∫ (ξ : W),
          HSMul.hSMul (α := 𝕊)
            ((e : (a : 𝕜) → 𝕊) (-((-(L.flip : (a : W) → V →ₗ[𝕜] 𝕜) ξ : V →ₗ[𝕜] 𝕜) : (a : V) → 𝕜) x) : 𝕊) (g ξ) ∂ν) ∂μ)
⊢ Eq.{u_6 + 1} (α := G)
    (∫ (ξ : W), (((M : (a : E) → F →L[ℂ] G) (fourierIntegral e μ L f ξ) : F →L[ℂ] G) : (a : F) → G) (g ξ) ∂ν)
    (∫ (x : V), (((M : (a : E) → F →L[ℂ] G) (f x) : F →L[ℂ] G) : (a : F) → G) (fourierIntegral e ν (-L.flip) g x) ∂μ)

Tactic:
calc
  ∫ ξ, M.flip (g ξ) (∫ x, e (-L x ξ) • f x ∂μ) ∂ν
    = ∫ ξ, (∫ x, M.flip (g ξ) (e (-L x ξ) • f x) ∂μ) ∂ν := by
    congr with ξ
    apply (ContinuousLinearMap.integral_comp_commSL RCLike.conj_smul _ _).symm
    exact (fourierIntegral_convergent_iff he hL _).2 hf
  _ = ∫ x, (∫ ξ, M.flip (g ξ) (e (-L x ξ) • f x) ∂ν) ∂μ :=
    integral_fourierIntegral_swap M.flip he hL hf hg
  _ = ∫ x, (∫ ξ, M (f x) (e (L.flip ξ x) • g ξ) ∂ν) ∂μ := by
    congr with x
    congr with ξ
    rw [← smul_one_smul ℂ _ (f x), ← smul_one_smul ℂ _ (g ξ)]
    simp only [map_smulₛₗ, ContinuousLinearMap.flip_apply, LinearMap.flip_apply, RingHom.id_apply,
      Circle.smul_def, smul_eq_mul, mul_one, ← Circle.coe_inv_eq_conj, AddChar.map_neg_eq_inv,
      inv_inv]
  _ = ∫ x, (∫ ξ, M (f x) (e (-(-L.flip ξ) x) • g ξ) ∂ν) ∂μ := by
    simp only [LinearMap.flip_apply, ContinuousLinearMap.map_smul_of_tower, LinearMap.neg_apply,
      neg_neg]
  _ = ∫ x, M (f x) (∫ ξ, e (-(-L.flip ξ) x) • g ξ ∂ν) ∂μ := by
    congr with x
    apply ContinuousLinearMap.integral_comp_comm
    have hLflip : Continuous fun (p : W × V) => (-L.flip p.1) p.2 :=
      (continuous_neg.comp hL).comp continuous_swap
    exact (fourierIntegral_convergent_iff (L := -L.flip) he hLflip x).2 hg

Premises used: VectorFourier.integral_fourierIntegral_swap, ContinuousLinearMap.flip