-- 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
rw :
  HPow.hPow (α := ℝ≥0∞) 2⁻¹ (Fintype.card.{u_1} { w : InfinitePlace K // w.IsComplex }) *
      (↑((sqrt : (a : ℝ≥0) → ℝ≥0)
            (nnnorm (E := ℂ)
              ((algebraMap ℚ ℂ : (a : ℚ) → ℂ)
                  (Algebra.discr (ι := K →+* ℂ) ℚ (B := K)
                    (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)))) :
                ℂ))) :
        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∞)
rw₁ :
  HPow.hPow (α := ℝ≥0∞) 2⁻¹ (Fintype.card.{u_1} { w : InfinitePlace K // w.IsComplex }) *
      (↑((sqrt : (a : ℝ≥0) → ℝ≥0)
            (nnnorm (E := ℂ)
              ((algebraMap ℚ ℂ : (a : ℚ) → ℂ)
                  (Algebra.discr (ι := Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ (B := K)
                    (⇑(integralBasis K : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K) :
                      (a : Free.ChooseBasisIndex ℤ (𝓞 K)) → K)) :
                ℂ))) :
        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∞)
rw₂ :
  HPow.hPow (α := ℝ≥0∞) 2⁻¹ (Fintype.card.{u_1} { w : InfinitePlace K // w.IsComplex }) *
      (↑((sqrt : (a : ℝ≥0) → ℝ≥0) (nnnorm (E := ℂ) ((algebraMap ℚ ℂ : (a : ℚ) → ℂ) (↑(discr K) : ℚ) : ℂ))) : 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∞)
rw₃ :
  HPow.hPow (α := ℝ≥0∞) 2⁻¹ (Fintype.card.{u_1} { w : InfinitePlace K // w.IsComplex }) *
      (↑((sqrt : (a : ℝ≥0) → ℝ≥0) (nnnorm (E := ℂ) (↑(discr K) : ℂ))) : 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∞)
rw₄ rw₅ :
  HPow.hPow (α := ℝ≥0∞) 2⁻¹ (Fintype.card.{u_1} { w : InfinitePlace K // w.IsComplex }) *
      (↑((sqrt : (a : ℝ≥0) → ℝ≥0) (nnnorm (E := ℂ) (↑(discr K) : ℂ))) : ℝ≥0∞) =
    HPow.hPow (α := ℝ≥0∞) 2⁻¹ (Fintype.card.{u_1} { w : InfinitePlace K // w.IsComplex }) *
      (↑((sqrt : (a : ℝ≥0) → ℝ≥0) (nnnorm (E := ℂ) (↑(discr K) : ℂ))) : 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∞) =
    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∞)

Tactic:
rw [← Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two, Algebra.discr_reindex,
          ← coe_discr, map_intCast, ← Complex.nnnorm_intCast]

Premises used: map_intCast, Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two, Complex.nnnorm_intCast, NumberField.coe_discr, Algebra.discr_reindex