-- 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_1 :
  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
this : ‖I‖₊ = 1
rw :
  (↑‖(2⁻¹ * I) ^ nrComplexPlaces 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∞)
rw₁ :
  (↑(‖(2⁻¹ * I) ^ nrComplexPlaces 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∞)
rw₂ :
  (↑(‖2⁻¹ * I‖₊ ^ nrComplexPlaces 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∞)
rw₃ :
  (↑((nnnorm (E := ℂ) 2⁻¹ * ‖I‖₊) ^ nrComplexPlaces 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∞)
rw₄ :
  (↑((nnnorm (E := ℂ) 2⁻¹ * 1) ^ nrComplexPlaces 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∞)
rw₅ :
  (↑(nnnorm (E := ℂ) 2⁻¹ ^ nrComplexPlaces 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∞)
rw₆ :
  (↑((nnnorm (E := ℂ) 2)⁻¹ ^ nrComplexPlaces 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∞)
rw₇ :
  (↑((nnnorm (E := ℂ) 2)⁻¹ ^ nrComplexPlaces K) : ℝ≥0∞) * (↑‖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∞)
rw₈ :
  (↑(nnnorm (E := ℂ) 2)⁻¹ : ℝ≥0∞) ^ nrComplexPlaces 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∞)
rw₉ :
  (↑(norm (E := ℂ) 2).toNNReal⁻¹ : ℝ≥0∞) ^ nrComplexPlaces 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∞)
rw₁₀ :
  (↑(Real.toNNReal 2)⁻¹ : ℝ≥0∞) ^ nrComplexPlaces 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∞)
rw₁₁ :
  (↑2⁻¹ : ℝ≥0∞) ^ nrComplexPlaces 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∞)
rw₁₂ :
  (↑2 : ℝ≥0∞)⁻¹ ^ nrComplexPlaces 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∞)
rw₁₃ :
  HPow.hPow (α := ℝ≥0∞) 2⁻¹ (nrComplexPlaces 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∞)
rw₁₄ :
  HPow.hPow (α := ℝ≥0∞) 2⁻¹ (nrComplexPlaces 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∞)
rw₁₅ rw₁₆ :
  HPow.hPow (α := ℝ≥0∞) 2⁻¹ (nrComplexPlaces K) * (↑‖N.det‖₊ : ℝ≥0∞) =
    HPow.hPow (α := ℝ≥0∞) 2⁻¹ (Fintype.card.{u_1} { w : InfinitePlace K // w.IsComplex }) * (↑‖N.det‖₊ : ℝ≥0∞)
⊢ (↑‖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∞)

Tactic:
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]

Premises used: nnnorm_inv, ENNReal.coe_inv, NNReal.sqrt_sq, Real.toNNReal_ofNat, nnnorm_pow, NumberField.mixedEmbedding.det_matrixToStdBasis, two_ne_zero, ENNReal.coe_mul, RCLike.norm_two, ENNReal.coe_pow, nnnorm_mul, mul_one, ENNReal.coe_ofNat, norm_toNNReal