/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
    (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖)
    (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m := by
  -- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
  -- block, but I think it's more legible this way. We start with preliminaries about the integrand.
  let e : C(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩
  have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by
    have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x))
    intro K g
    simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
  have eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight n) = e := by
    intro n; ext1 x
    have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
    simpa only [mul_one] using this.int_mul n x
  -- Now the main argument. First unwind some definitions.
  calc
    fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
        ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, f.comp (ContinuousMap.addRight n)) x := by
      simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
        coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
    -- Transform sum in C(ℝ, ℂ) evaluated at x into pointwise sum of values.
    _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by
      simp_rw [coe_mul, Pi.mul_apply,
        ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
    -- Swap sum and integral.
    _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
      refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
      convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1
      exact funext fun n => neK _ _
    _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by
      simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢
      simp_rw [eadd]
    -- Rearrange sum of interval integrals into an integral over `ℝ`.
    _ = ∫ x, e x * f x := by
      suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
      apply integrable_of_summable_norm_Icc
      convert hf ⟨Icc 0 1, isCompact_Icc⟩ using 1
      simp_rw [mul_comp] at eadd ⊢
      simp_rw [eadd]
      exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _
    -- Minor tidying to finish
    _ = 𝓕 f m := by
      rw [fourierIntegral_eq_integral_exp_smul]
      congr 1 with x : 1
      rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
      congr 2
      push_cast
      

Proof state:
case e_f.h.e_a.e_z
f : C(ℝ, ℂ)
hf :
  ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e
x : ℝ
⊢ 2 * ↑π * I * -↑m * ↑x / 1 = -2 * ↑π * ↑x * ↑m * I

Next tactic:
ring