-- Module: Mathlib.Algebra.Lie.Abelian
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
s : Set L
h : ∀ x ∈ s, ∀ y ∈ s, ⁅x, y⁆ = 0
x_1 x_2 : ↥(lieSpan R L s)
x y : L
hy : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) y
w : L
hw : w ∈ s
refine_2 :
  ∀ (u : L) (hu : u ∈ s),
    Eq (α := ↥(lieSpan R L s))
      (Bracket.bracket (L := ↥(lieSpan R L s))
        (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
        (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) u ⋯))
      0
refine_2₁ :
  Eq (α := ↥(lieSpan R L s))
    (Bracket.bracket (L := ↥(lieSpan R L s))
      (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
      (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) 0 ⋯))
    0
refine_2₂ :
  ∀ (u v : L) (hx : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) u)
    (hy : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) v)
    (hu :
      Bracket.bracket (L := ↥(lieSpan R L s))
          (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
          (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) u hx) =
        0)
    (hv :
      Bracket.bracket (L := ↥(lieSpan R L s))
          (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
          (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) v hy) =
        0),
    Bracket.bracket (L := ↥(lieSpan R L s))
        (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
        (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) (u + v) ⋯) =
      0
refine_2₃ :
  ∀ (t : R) (u : L) (hx : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) u)
    (hu :
      Bracket.bracket (L := ↥(lieSpan R L s))
          (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
          (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) u hx) =
        0),
    Bracket.bracket (L := ↥(lieSpan R L s))
        (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
        (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) (t • u) ⋯) =
      0
refine_2₄ :
  ∀ (u v : L) (hx : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) u)
    (hy : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) v)
    (hu :
      Bracket.bracket (L := ↥(lieSpan R L s))
          (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
          (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) u hx) =
        0)
    (hv :
      Bracket.bracket (L := ↥(lieSpan R L s))
          (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
          (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) v hy) =
        0),
    Bracket.bracket (L := ↥(lieSpan R L s))
        (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
        (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) ⁅u, v⁆ ⋯) =
      0
⊢ Bracket.bracket (L := ↥(lieSpan R L s))
      (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
      (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy) =
    0

Tactic:
induction hy using lieSpan_induction with
      | mem u hu => simpa [Subtype.ext_iff] using h w hw u hu
      | zero => simp [Subtype.ext_iff]
      | add u v _ _ hu hv =>
        simp only [Subtype.ext_iff, coe_bracket, ZeroMemClass.coe_zero, lie_add] at hu hv ⊢
        simp [hu, hv]
      | smul t u _ hu =>
        simp only [Subtype.ext_iff, coe_bracket, ZeroMemClass.coe_zero] at hu
        simp [Subtype.ext_iff, hu]
      | lie u v _ _ hu hv =>
        simp only [Subtype.ext_iff, coe_bracket, ZeroMemClass.coe_zero] at hu hv ⊢
        rw [leibniz_lie]
        simp [hu, hv]

Premises used: leibniz_lie, Subtype.ext_iff, LieSubalgebra.coe_bracket, ZeroMemClass.coe_zero, lie_add