-- Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
B : ℝ
hB : 0 < B
_ :
  (@integral _ _ _ _ MeasureSpace.toMeasurableSpace volume fun (x : mixedSpace K) ↦ rexp (-convexBodySumFun x)) =
    HMul.hMul (α := ℝ)
      (@integral _ _ _ _ MeasureSpace.toMeasurableSpace volume fun (x : { w : InfinitePlace K // w.IsReal } → ℝ) ↦
        ∏ w : { w : InfinitePlace K // w.IsReal }, rexp (-‖x w‖))
      (@integral _ _ _ _ MeasureSpace.toMeasurableSpace volume fun (x : { w : InfinitePlace K // w.IsComplex } → ℂ) ↦
        ∏ w : { w : InfinitePlace K // w.IsComplex }, rexp (-2 * ‖x w‖))
_₁ :
  HMul.hMul (α := ℝ)
      (@integral _ _ _ _ MeasureSpace.toMeasurableSpace volume fun (x : { w : InfinitePlace K // w.IsReal } → ℝ) ↦
        ∏ w : { w : InfinitePlace K // w.IsReal }, rexp (-‖x w‖))
      (@integral _ _ _ _ MeasureSpace.toMeasurableSpace volume fun (x : { w : InfinitePlace K // w.IsComplex } → ℂ) ↦
        ∏ w : { w : InfinitePlace K // w.IsComplex }, rexp (-2 * ‖x w‖)) =
    HPow.hPow (α := ℝ) (@integral _ _ _ _ MeasureSpace.toMeasurableSpace volume fun (x : ℝ) ↦ rexp (-|x|))
        (nrRealPlaces K) *
      HPow.hPow (α := ℝ) (@integral _ _ _ _ MeasureSpace.toMeasurableSpace volume fun (x : ℂ) ↦ rexp (-2 * ‖x‖))
        (nrComplexPlaces K)
_₂ :
  HPow.hPow (α := ℝ) (@integral _ _ _ _ MeasureSpace.toMeasurableSpace volume fun (x : ℝ) ↦ rexp (-|x|))
        (nrRealPlaces K) *
      HPow.hPow (α := ℝ) (@integral _ _ _ _ MeasureSpace.toMeasurableSpace volume fun (x : ℂ) ↦ rexp (-2 * ‖x‖))
        (nrComplexPlaces K) =
    (2 * Gamma (1 / 1 + 1)) ^ nrRealPlaces K *
      (π * HPow.hPow (α := ℝ) 2 (-2 / 1) * Gamma (2 / 1 + 1)) ^ nrComplexPlaces K
_₃ :
  (2 * Gamma (1 / 1 + 1)) ^ nrRealPlaces K *
      (π * HPow.hPow (α := ℝ) 2 (-2 / 1) * Gamma (2 / 1 + 1)) ^ nrComplexPlaces K =
    HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (π / 2) ^ nrComplexPlaces K
⊢ (@integral _ _ _ _ MeasureSpace.toMeasurableSpace volume fun (x : mixedSpace K) ↦ rexp (-convexBodySumFun x)) =
    HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (π / 2) ^ nrComplexPlaces K

Tactic:
calc
      _ = (∫ x : {w : InfinitePlace K // IsReal w} → ℝ, ∏ w, exp (-‖x w‖)) *
              (∫ x : {w : InfinitePlace K // IsComplex w} → ℂ, ∏ w, exp (-2 * ‖x w‖)) := by
        simp_rw [convexBodySumFun_apply', neg_add, ← neg_mul, Finset.mul_sum,
          ← Finset.sum_neg_distrib, exp_add, exp_sum, ← integral_prod_mul, volume_eq_prod]
      _ = (∫ x : ℝ, exp (-|x|)) ^ nrRealPlaces K *
              (∫ x : ℂ, Real.exp (-2 * ‖x‖)) ^ nrComplexPlaces K := by
        rw [integral_fintype_prod_volume_eq_pow (fun x => exp (-‖x‖)),
          integral_fintype_prod_volume_eq_pow (fun x => exp (-2 * ‖x‖))]
        simp_rw [norm_eq_abs]
      _ = (2 * Gamma (1 / 1 + 1)) ^ nrRealPlaces K *
              (π * (2 : ℝ) ^ (-(2 : ℝ) / 1) * Gamma (2 / 1 + 1)) ^ nrComplexPlaces K := by
        rw [integral_comp_abs (f := fun x => exp (-x)), ← integral_exp_neg_rpow zero_lt_one,
          ← Complex.integral_exp_neg_mul_rpow le_rfl zero_lt_two]
        simp_rw [Real.rpow_one]
      _ = (2 : ℝ) ^ nrRealPlaces K * (π / 2) ^ nrComplexPlaces K := by
        simp_rw [div_one, one_add_one_eq_two, Gamma_add_one two_ne_zero, Gamma_two, mul_one,
          mul_assoc, ← Real.rpow_add_one two_ne_zero, show (-2 : ℝ) + 1 = -1 by norm_num,
          Real.rpow_neg_one, div_eq_mul_inv]

Premises used: Norm.norm, Real.pi, HMul.hMul, Finset.univ, abs, HDiv.hDiv, Real.exp, NumberField.InfinitePlace.nrRealPlaces, Finset.prod, HPow.hPow, MeasureTheory.MeasureSpace.volume, HAdd.hAdd, MeasureTheory.integral, Real.Gamma, NumberField.InfinitePlace.nrComplexPlaces, OfNat.ofNat, Neg.neg