-- Lean 4 proof step from Mathlib.RingTheory.Algebraic.LinearIndependent (goal)
Goal:
F : Type u_1
E : Type u_2
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
x : E
H : Transcendental F x
rw rw₁ rw₂ :
  ∀ (H : ∀ (p : F[X]), Eq.{u_2 + 1} (α := E) ((aeval x : (a : F[X]) → E) p : E) (0 : E) → p = 0),
    LinearIndependent.{u_1, u_1, u_2} F (M := E) fun (a : F) ↦
      (HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) a : E))⁻¹
rw₃ :
  ∀ (H : ∀ (p : F[X]), Eq.{u_2 + 1} (α := E) ((aeval x : (a : F[X]) → E) p : E) (0 : E) → p = 0) (s : Finset F)
    (m : F → F)
    (hm : Eq.{u_2 + 1} (α := E) (∑ i ∈ s, m i • (HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) i : E))⁻¹) 0)
    (i : F) (hi : i ∈ s), m i = 0
rw₄ rw₅ :
  ∀ (H : ∀ (p : F[X]), Eq.{u_2 + 1} (α := E) ((aeval x : (a : F[X]) → E) p : E) (0 : E) → p = 0) (s : Finset F)
    (m : F → F)
    (hm : Eq.{u_2 + 1} (α := E) (∑ i ∈ s, m i • (HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) i : E))⁻¹) 0)
    (i : F) (hi : i ∈ s) (hnz : ∀ (a : F), HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) a : E) ≠ 0), m i = 0
rw₆ :
  ∀ (H : ∀ (p : F[X]), Eq.{u_2 + 1} (α := E) ((aeval x : (a : F[X]) → E) p : E) (0 : E) → p = 0) (s : Finset F)
    (m : F → F)
    (hm : Eq.{u_2 + 1} (α := E) (∑ i ∈ s, m i • (HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) i : E))⁻¹) 0)
    (i : F) (hi : i ∈ s) (hnz : ∀ (a : F), HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) a : E) ≠ 0),
    let b : E := ∏ j ∈ s, HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) j : E);
    ∀
      (h1 :
        ∀ i ∈ s,
          m i • (b * (HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) i : E))⁻¹) =
            HSMul.hSMul (β := E) (m i) (∏ j ∈ s.erase i, HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) j : E))),
      m i = 0
⊢ LinearIndependent.{u_1, u_1, u_2} F (M := E) fun (a : F) ↦
    (HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) a : E))⁻¹
have rw₇ : ∀ (H : ∀ (p : F[X]), Eq.{u_2 + 1} (α := E) ((aeval x : (a : F[X]) → E) p : E) (0 : E) → p = 0) (s : Finset F)
  (m : F → F) (i : F) (hi : i ∈ s) (hnz : ∀ (a : F), HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) a : E) ≠ 0),
  let b : E := ∏ j ∈ s, HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) j : E);
  ∀
    (h1 :
      ∀ i ∈ s,
        m i • (b * (HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) i : E))⁻¹) =
          HSMul.hSMul (β := E) (m i) (∏ j ∈ s.erase i, HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) j : E)))
    (hm : b * ∑ i ∈ s, m i • (HSub.hSub (β := E) x ((algebraMap F E : (a : F) → E) i : E))⁻¹ = b * 0), m i = 0
Tactic: classical
  rw [transcendental_iff] at H
  refine linearIndependent_iff'.2 fun s m hm i hi ↦ ?_
  have hnz (a : F) : x - algebraMap F E a ≠ 0 := fun h ↦
    X_sub_C_ne_zero a <| H (.X - .C a) (by simp [h])
  let b := s.prod fun j ↦ x - algebraMap F E j
  have h1 : ∀ i ∈ s, m i • (b * (x - algebraMap F E i)⁻¹) =
      m i • (s.erase i).prod fun j ↦ x - algebraMap F E j := fun i hi ↦ by
    simp_rw [b, ← s.prod_erase_mul _ hi, mul_inv_cancel_right₀ (hnz i)]
  replace hm := congr(b * $(hm))
  simp_rw [mul_zero, Finset.mul_sum, mul_smul_comm, Finset.sum_congr rfl h1] at hm
  let p : Polynomial F := s.sum fun i ↦ .C (m i) * (s.erase i).prod fun j ↦ .X - .C j
  replace hm := congr(Polynomial.aeval i $(H p (by simp_rw [← hm, p, map_sum, map_mul, map_prod,
    map_sub, aeval_X, aeval_C, Algebra.smul_def])))
  have h2 : ∀ j ∈ s.erase i, m j * ((s.erase j).prod fun x ↦ i - x) = 0 := fun j hj ↦ by
    have := Finset.mem_erase_of_ne_of_mem (Finset.ne_of_mem_erase hj).symm hi
    simp_rw [← (s.erase j).prod_erase_mul _ this, sub_self, mul_zero]
  simp_rw [map_zero, p, map_sum, map_mul, map_prod, map_sub, aeval_X,
    aeval_C, Algebra.algebraMap_self_apply, ← s.sum_erase_add _ hi,
    Finset.sum_eq_zero h2, zero_add] at hm
  exact eq_zero_of_ne_zero_of_mul_right_eq_zero (Finset.prod_ne_zero_iff.2 fun j hj ↦
    sub_ne_zero.2 (Finset.ne_of_mem_erase hj).symm) hm
Cleared hypotheses: H