/-- `charpoly f` is the characteristic polynomial of the matrix of `f` in any basis. -/
@[simp]
theorem charpoly_toMatrix {ι : Type w} [DecidableEq ι] [Fintype ι] (b : Basis ι R M) :
    (toMatrix b b f).charpoly = f.charpoly := by
  let A := toMatrix b b f
  let b' := chooseBasis R M
  let ι' := ChooseBasisIndex R M
  let A' := toMatrix b' b' f
  let e := Basis.indexEquiv b b'
  let φ := reindexLinearEquiv R R e e
  let φ₁ := reindexLinearEquiv R R e (Equiv.refl ι')
  let φ₂ := reindexLinearEquiv R R (Equiv.refl ι') (Equiv.refl ι')
  let φ₃ := reindexLinearEquiv R R (Equiv.refl ι') e
  let P := b.toMatrix b'
  let Q := b'.toMatrix b
  have hPQ : C.mapMatrix (φ₁ P) * C.mapMatrix (φ₃ Q) = 1 := by
    rw [RingHom.mapMatrix_apply, RingHom.mapMatrix_apply, ← Matrix.map_mul,
      reindexLinearEquiv_mul R R, Basis.toMatrix_mul_toMatrix_flip,
      reindexLinearEquiv_one, ← RingHom.mapMatrix_apply, RingHom.map_one]
  

Proof state:
R : Type u
M : Type v
inst✝⁷ : CommRing R
inst✝⁶ : Nontrivial R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module.Free R M
inst✝² : Module.Finite R M
f : M →ₗ[R] M
ι : Type w
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι R M
A : Matrix ι ι R := (toMatrix b b) f
b' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M
ι' : Type v := ChooseBasisIndex R M
A' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f
e : ι ≃ ChooseBasisIndex R M := Basis.indexEquiv b b'
φ : Matrix ι ι R ≃ₗ[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e
φ₁ : Matrix ι ι' R ≃ₗ[R] Matrix (ChooseBasisIndex R M) ι' R := reindexLinearEquiv R R e (Equiv.refl ι')
φ₂ : Matrix ι' ι' R ≃ₗ[R] Matrix ι' ι' R := reindexLinearEquiv R R (Equiv.refl ι') (Equiv.refl ι')
φ₃ : Matrix ι' ι R ≃ₗ[R] Matrix ι' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl ι') e
P : Matrix ι (ChooseBasisIndex R M) R := Basis.toMatrix b ⇑b'
Q : Matrix (ChooseBasisIndex R M) ι R := Basis.toMatrix b' ⇑b
hPQ : (RingHom.mapMatrix C) (φ₁ P) * (RingHom.mapMatrix C) (φ₃ Q) = 1
⊢ Matrix.charpoly ((toMatrix b b) f) = charpoly f

Next tactic:
calc
    A.charpoly = (reindex e e A).charpoly := (charpoly_reindex _ _).symm
    _ = det (scalar ι' X - C.mapMatrix (φ A)) := rfl
    _ = det (scalar ι' X - C.mapMatrix (φ (P * A' * Q))) := by
      rw [basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix]
    _ = det (scalar ι' X - C.mapMatrix (φ₁ P * φ₂ A' * φ₃ Q)) := by
      rw [reindexLinearEquiv_mul, reindexLinearEquiv_mul]
    _ = det (scalar ι' X - C.mapMatrix (φ₁ P) * C.mapMatrix A' * C.mapMatrix (φ₃ Q)) := by simp
    _ = det (scalar ι' X * C.mapMatrix (φ₁ P) * C.mapMatrix (φ₃ Q) -
          C.mapMatrix (φ₁ P) * C.mapMatrix A' * C.mapMatrix (φ₃ Q)) := by
      rw [Matrix.mul_assoc ((scalar ι') X), hPQ, Matrix.mul_one]
    _ = det (C.mapMatrix (φ₁ P) * scalar ι' X * C.mapMatrix (φ₃ Q) -
          C.mapMatrix (φ₁ P) * C.mapMatrix A' * C.mapMatrix (φ₃ Q)) := by
      rw [scalar_commute _ commute_X]
    _ = det (C.mapMatrix (φ₁ P) * (scalar ι' X - C.mapMatrix A') * C.mapMatrix (φ₃ Q)) := by
      rw [← Matrix.sub_mul, ← Matrix.mul_sub]
    _ = det (C.mapMatrix (φ₁ P)) * det (scalar ι' X - C.mapMatrix A') * det (C.mapMatrix (φ₃ Q)) :=
      by rw [det_mul, det_mul]
    _ = det (C.mapMatrix (φ₁ P)) * det (C.mapMatrix (φ₃ Q)) * det (scalar ι' X - C.mapMatrix A') :=
      by ring
    _ = det (scalar ι' X - C.mapMatrix A') := by
      rw [← det_mul, hPQ, det_one, one_mul]
    _ = f.charpoly := rfl