-- Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  Basis.indexEquiv (canonicalEmbedding.latticeBasis K) (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := Equiv.trans (indexEquiv K) f.symm
M : Matrix (index K) (index K) ℝ :=
  Basis.toMatrix (mixedEmbedding.stdBasis K) (⇑(Basis.reindex (latticeBasis K) e.symm) : (a : index K) → mixedSpace K)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ
    (Function.comp (β := Free.ChooseBasisIndex ℤ (𝓞 K))
      (⇑(integralBasis K : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K) : (a : Free.ChooseBasisIndex ℤ (𝓞 K)) → K)
      (⇑f.symm : (a : K →+* ℂ) → Free.ChooseBasisIndex ℤ (𝓞 K)))
    RingHom.equivRatAlgHom
this :
  M.map (⇑ofRealHom : (a : ℝ) → ℂ) =
    HMul.hMul (α := Matrix (index K) (index K) ℂ) (matrixToStdBasis K)
      ((reindex (Equiv.symm (indexEquiv K)) (Equiv.symm (indexEquiv K)) :
            (a : Matrix (K →+* ℂ) (K →+* ℂ) ℂ) → Matrix (index K) (index K) ℂ)
          N)ᵀ
calc :
  ((volume : Measure (mixedSpace K)) : (a : Set (mixedSpace K)) → ℝ≥0∞)
      (fundamentalDomain (ι := Free.ChooseBasisIndex ℤ (𝓞 K)) (K := ℝ) (latticeBasis K)) =
    (↑‖det (n := index K)
            (Basis.toMatrix (mixedEmbedding.stdBasis K)
              (⇑(Basis.reindex (latticeBasis K) e.symm) : (a : index K) → mixedSpace K))‖₊ :
      ℝ≥0∞)
calc₁ :
  (↑‖det (n := index K)
            (Basis.toMatrix (mixedEmbedding.stdBasis K)
              (⇑(Basis.reindex (latticeBasis K) e.symm) : (a : index K) → mixedSpace K))‖₊ :
      ℝ≥0∞) =
    (↑‖det (n := index K) (matrixToStdBasis K) * N.det‖₊ : ℝ≥0∞)
calc₂ :
  (↑‖det (n := index K) (matrixToStdBasis K) * N.det‖₊ : ℝ≥0∞) =
    HPow.hPow (α := ℝ≥0∞) 2⁻¹ (Fintype.card.{u_1} { w : InfinitePlace K // w.IsComplex }) *
      (↑((sqrt : (a : ℝ≥0) → ℝ≥0) ‖HPow.hPow (β := ℕ) N.det 2‖₊) : ℝ≥0∞)
calc₃ :
  HPow.hPow (α := ℝ≥0∞) 2⁻¹ (Fintype.card.{u_1} { w : InfinitePlace K // w.IsComplex }) *
      (↑((sqrt : (a : ℝ≥0) → ℝ≥0) ‖HPow.hPow (β := ℕ) N.det 2‖₊) : ℝ≥0∞) =
    HPow.hPow (α := ℝ≥0∞) 2⁻¹ (Fintype.card.{u_1} { w : InfinitePlace K // w.IsComplex }) *
      (↑((sqrt : (a : ℝ≥0) → ℝ≥0) ‖discr K‖₊) : ℝ≥0∞)
⊢ ((volume : Measure (mixedSpace K)) : (a : Set (mixedSpace K)) → ℝ≥0∞)
      (fundamentalDomain (ι := Free.ChooseBasisIndex ℤ (𝓞 K)) (K := ℝ) (latticeBasis K)) =
    HPow.hPow (α := ℝ≥0∞) 2⁻¹ (nrComplexPlaces K) * (↑((sqrt : (a : ℝ≥0) → ℝ≥0) ‖discr K‖₊) : ℝ≥0∞)

Tactic:
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, ← ofRealHom_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_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, RCLike.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_intCast]

Premises used: NumberField.mixedEmbedding.latticeBasis, NNNorm.nnnorm, NumberField.InfinitePlace.IsComplex, NumberField.mixedEmbedding.stdBasis, Fintype.card, Subtype, Module.Basis.toMatrix, NNReal.sqrt, HPow.hPow, MeasureTheory.MeasureSpace.volume, NumberField.mixedEmbedding.matrixToStdBasis, Module.Basis.reindex, ZSpan.fundamentalDomain, NumberField.discr, Equiv.symm, OfNat.ofNat, DFunLike.coe, Matrix.det