theorem _root_.NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis :
    volume (fundamentalDomain (latticeBasis K)) =
      (2 : ℝ≥0∞)⁻¹ ^ NrComplexPlaces K * sqrt ‖discr K‖₊ := by
  let f : Module.Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
    (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ _)
  let e : (index K) ≃ Module.Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
  let M := (mixedEmbedding.stdBasis K).toMatrix ((latticeBasis K).reindex e.symm)
  let N := Algebra.embeddingsMatrixReindex ℚ ℂ (integralBasis K ∘ f.symm)
    RingHom.equivRatAlgHom
  suffices M.map Complex.ofReal = (matrixToStdBasis K) *
      (Matrix.reindex (indexEquiv K).symm (indexEquiv K).symm N).transpose by
    calc volume (fundamentalDomain (latticeBasis K))
      _ = ‖((mixedEmbedding.stdBasis K).toMatrix ((latticeBasis K).reindex e.symm)).det‖₊ := by
        rw [← fundamentalDomain_reindex _ e.symm, ← norm_toNNReal, measure_fundamentalDomain
          ((latticeBasis K).reindex e.symm), volume_fundamentalDomain_stdBasis, mul_one]
        rfl
      _ = ‖(matrixToStdBasis K).det * N.det‖₊ := by
        rw [← nnnorm_real, ← ofReal_eq_coe, RingHom.map_det, RingHom.mapMatrix_apply, this,
          det_mul, det_transpose, det_reindex_self]
      _ = (2 : ℝ≥0∞)⁻¹ ^ Fintype.card {w : InfinitePlace K // IsComplex w} * sqrt ‖N.det ^ 2‖₊ := by
        have : ‖Complex.I‖₊ = 1 := by rw [← norm_toNNReal, norm_eq_abs, abs_I, Real.toNNReal_one]
        rw [det_matrixToStdBasis, nnnorm_mul, nnnorm_pow, nnnorm_mul, this, mul_one, nnnorm_inv,
          coe_mul, ENNReal.coe_pow, ← norm_toNNReal, IsROrC.norm_two, Real.toNNReal_ofNat,
          coe_inv two_ne_zero, coe_ofNat, nnnorm_pow, NNReal.sqrt_sq]
      _ = (2 : ℝ≥0∞)⁻¹ ^ Fintype.card { w // IsComplex w } * NNReal.sqrt ‖discr K‖₊ := by
        rw [← Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two, Algebra.discr_reindex,
          ← coe_discr, map_intCast, ← Complex.nnnorm_int]
  ext : 2
  dsimp only
  

Proof state:
case a.h
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Module.Free.ChooseBasisIndex ℤ ↥(𝓞 K) ≃ (K →+* ℂ) :=
  Basis.indexEquiv (canonicalEmbedding.latticeBasis K) (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Module.Free.ChooseBasisIndex ℤ ↥(𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := Basis.toMatrix (mixedEmbedding.stdBasis K) ⇑(Basis.reindex (latticeBasis K) e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) RingHom.equivRatAlgHom
i✝ x✝ : index K
⊢ Matrix.map
      (Basis.toMatrix (mixedEmbedding.stdBasis K)
        ⇑(Basis.reindex (latticeBasis K)
            ((indexEquiv K).trans
                (Basis.indexEquiv (canonicalEmbedding.latticeBasis K) (Pi.basisFun ℂ (K →+* ℂ))).symm).symm))
      (⇑ofReal) i✝ x✝ =
    (matrixToStdBasis K *
        ((reindex (indexEquiv K).symm (indexEquiv K).symm)
            (Algebra.embeddingsMatrixReindex ℚ ℂ
              (⇑(integralBasis K) ∘
                ⇑(Basis.indexEquiv (canonicalEmbedding.latticeBasis K) (Pi.basisFun ℂ (K →+* ℂ))).symm)
              RingHom.equivRatAlgHom))ᵀ)
      i✝ x✝

Next tactic:
rw [Matrix.map_apply, Basis.toMatrix_apply, Basis.coe_reindex, Function.comp_apply,
    Equiv.symm_symm, latticeBasis_apply, ← commMap_canonical_eq_mixed, Complex.ofReal_eq_coe,
    stdBasis_repr_eq_matrixToStdBasis_mul K _ (fun _ => rfl)]