-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
rw :
  |(LinearMap.det : (a : (InfinitePlace K → ℝ) →ₗ[ℝ] InfinitePlace K → ℝ) → ℝ)
        (↑(↑(ContinuousLinearEquiv.symm (Basis.equivFunL (completeBasis K))) :
              (InfinitePlace K → ℝ) →L[ℝ] realSpace K) :
          (InfinitePlace K → ℝ) →ₗ[ℝ] InfinitePlace K → ℝ)| =
    (↑(finrank ℚ K) : ℝ) * regulator K
rw₁ :
  |Matrix.det.{0, u_1} (n := InfinitePlace K)
        ((LinearMap.toMatrix (completeBasis K) (completeBasis K) :
            (a : realSpace K →ₗ[ℝ] realSpace K) → Matrix (InfinitePlace K) (InfinitePlace K) ℝ)
          (↑(↑(ContinuousLinearEquiv.symm (Basis.equivFunL (completeBasis K))) :
                (InfinitePlace K → ℝ) →L[ℝ] realSpace K) :
            (InfinitePlace K → ℝ) →ₗ[ℝ] InfinitePlace K → ℝ))| =
    (↑(finrank ℚ K) : ℝ) * regulator K
rw₂ :
  |Matrix.det.{0, u_1} (n := InfinitePlace K)
        (Matrix.transpose
          ((LinearMap.toMatrix (completeBasis K) (completeBasis K) :
              (a : realSpace K →ₗ[ℝ] realSpace K) → Matrix (InfinitePlace K) (InfinitePlace K) ℝ)
            (↑(↑(ContinuousLinearEquiv.symm (Basis.equivFunL (completeBasis K))) :
                  (InfinitePlace K → ℝ) →L[ℝ] realSpace K) :
              (InfinitePlace K → ℝ) →ₗ[ℝ] InfinitePlace K → ℝ)))| =
    (↑(finrank ℚ K) : ℝ) * regulator K
rw₃ :
  |Matrix.det.{0, u_1} (n := InfinitePlace K)
        (Matrix.transpose
          ((LinearMap.toMatrix (completeBasis K) (completeBasis K) :
              (a : realSpace K →ₗ[ℝ] realSpace K) → Matrix (InfinitePlace K) (InfinitePlace K) ℝ)
            (↑(↑(ContinuousLinearEquiv.symm (Basis.equivFunL (completeBasis K))) :
                  (InfinitePlace K → ℝ) →L[ℝ] realSpace K) :
              (InfinitePlace K → ℝ) →ₗ[ℝ] InfinitePlace K → ℝ)))| =
    (↑(finrank ℚ K) : ℝ) * regOfFamily (fundSystem K)
⊢ |ContinuousLinearMap.det (M := InfinitePlace K → ℝ)
        (↑(ContinuousLinearEquiv.symm (Basis.equivFunL (completeBasis K))) : (InfinitePlace K → ℝ) →L[ℝ] realSpace K)| =
    (↑(finrank ℚ K) : ℝ) * regulator K
have rw₄ : Eq (α := ℝ)
  |Matrix.det.{0, u_1} (n := InfinitePlace K)
      (Matrix.transpose
        ((LinearMap.toMatrix (completeBasis K) (completeBasis K) :
            (a : realSpace K →ₗ[ℝ] realSpace K) → Matrix (InfinitePlace K) (InfinitePlace K) ℝ)
          (↑(↑(ContinuousLinearEquiv.symm (Basis.equivFunL (completeBasis K))) :
                (InfinitePlace K → ℝ) →L[ℝ] realSpace K) :
            (InfinitePlace K → ℝ) →ₗ[ℝ] InfinitePlace K → ℝ)))|
  |Matrix.det (n := InfinitePlace K)
      ((Matrix.of : (a : InfinitePlace K → InfinitePlace K → ℝ) → Matrix (InfinitePlace K) (InfinitePlace K) ℝ)
          fun (i w : InfinitePlace K) ↦
          if h : i = w₀ then (↑w.mult : ℝ)
          else
            (↑w.mult : ℝ) *
              Real.log
                ((w : (a : K) → ℝ)
                  ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K)
                      (↑(fundSystem K
                            (((Equiv.symm equivFinRank : { w : InfinitePlace K // w ≠ w₀ } ≃ Fin (rank K)) :
                                (a : { w : InfinitePlace K // w ≠ w₀ }) → Fin (rank K))
                              (Subtype.mk (p := fun (w : InfinitePlace K) ↦ w ≠ w₀) i h))) :
                        𝓞 K) :
                    K)) :
        Matrix (InfinitePlace K) (InfinitePlace K) ℝ)|
Tactic: classical
  rw [ContinuousLinearMap.det, ← LinearMap.det_toMatrix (completeBasis K), ← Matrix.det_transpose,
    regulator_eq_regOfFamily_fundSystem, finrank_mul_regOfFamily_eq_det _ w₀ equivFinRank.symm]
  congr 2 with w i
  rw [Matrix.transpose_apply, LinearMap.toMatrix_apply, Matrix.of_apply, ← Basis.equivFunL_apply,
    ContinuousLinearMap.coe_coe, ContinuousLinearEquiv.coe_apply,
    (completeBasis K).equivFunL.apply_symm_apply]
  split_ifs with hw
  · rw [hw, completeBasis_apply_of_eq]
  · simp_rw [completeBasis_apply_of_ne K ⟨w, hw⟩, expMap_symm_apply, normAtAllPlaces_mixedEmbedding]