Theorem: LieAlgebra.ExtendScalars.bracket_leibniz_lie
File: Mathlib/Algebra/Lie/BaseChange.lean

tactic: simp only [bracket_def]
state_before: R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
⊢ ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆

tactic: refine' x.induction_on _ _ _
state_before: R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
⊢ ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) x) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) x) y)) z +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) x) z)

tactic: simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
state_before: case refine'_1
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
⊢ ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) 0) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) 0) y)) z +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) 0) z)

tactic: intro a₁ l₁
state_before: case refine'_2
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
⊢ ∀ (x : A) (y_1 : L),
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (x ⊗ₜ[R] y_1)) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (x ⊗ₜ[R] y_1)) y)) z +
        ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (x ⊗ₜ[R] y_1)) z)

tactic: refine' y.induction_on _ _ _
state_before: case refine'_2
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
⊢ ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) y)) z +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) z)

tactic: simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
state_before: case refine'_2.refine'_1
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
⊢ ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) 0) z) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) 0)) z +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) 0) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) z)

tactic: intro a₂ l₂
state_before: case refine'_2.refine'_2
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
⊢ ∀ (x : A) (y : L),
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
        (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (x ⊗ₜ[R] y)) z) =
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
              (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (x ⊗ₜ[R] y)))
          z +
        ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (x ⊗ₜ[R] y))
          (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) z)

tactic: refine' z.induction_on _ _ _
state_before: case refine'_2.refine'_2
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
a₂ : A
l₂ : L
⊢ ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
      (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂)) z) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (a₂ ⊗ₜ[R] l₂)))
        z +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂))
        (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) z)

tactic: rw [LinearMap.map_zero, LinearMap.map_zero, LinearMap.map_zero, LinearMap.map_zero,
  add_zero]
state_before: case refine'_2.refine'_2.refine'_1
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
a₂ : A
l₂ : L
⊢ ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
      (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂)) 0) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (a₂ ⊗ₜ[R] l₂)))
        0 +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂))
        (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) 0)

tactic: intro a₃ l₃
state_before: case refine'_2.refine'_2.refine'_2
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
a₂ : A
l₂ : L
⊢ ∀ (x : A) (y : L),
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
        (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂)) (x ⊗ₜ[R] y)) =
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
              (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (a₂ ⊗ₜ[R] l₂)))
          (x ⊗ₜ[R] y) +
        ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂))
          (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (x ⊗ₜ[R] y))

tactic: simp only [bracket'_tmul]
state_before: case refine'_2.refine'_2.refine'_2
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
a₂ : A
l₂ : L
a₃ : A
l₃ : L
⊢ ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
      (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂)) (a₃ ⊗ₜ[R] l₃)) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (a₂ ⊗ₜ[R] l₂)))
        (a₃ ⊗ₜ[R] l₃) +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂))
        (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (a₃ ⊗ₜ[R] l₃))

tactic: rw [mul_left_comm a₂ a₁ a₃, mul_assoc, leibniz_lie, TensorProduct.tmul_add]
state_before: case refine'_2.refine'_2.refine'_2
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
a₂ : A
l₂ : L
a₃ : A
l₃ : L
⊢ (a₁ * (a₂ * a₃)) ⊗ₜ[R] ⁅l₁, ⁅l₂, l₃⁆⁆ = (a₁ * a₂ * a₃) ⊗ₜ[R] ⁅⁅l₁, l₂⁆, l₃⁆ + (a₂ * (a₁ * a₃)) ⊗ₜ[R] ⁅l₂, ⁅l₁, l₃⁆⁆

tactic: intro u₁ u₂ h₁ h₂
state_before: case refine'_2.refine'_2.refine'_3
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
a₂ : A
l₂ : L
⊢ ∀ (x y : A ⊗[R] L),
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
          (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂)) x) =
        ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
                (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (a₂ ⊗ₜ[R] l₂)))
            x +
          ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂))
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) x) →
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂)) y) =
          ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
                  (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (a₂ ⊗ₜ[R] l₂)))
              y +
            ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂))
              (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) y) →
        ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂)) (x + y)) =
          ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
                  (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (a₂ ⊗ₜ[R] l₂)))
              (x + y) +
            ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂))
              (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (x + y))

tactic: rw [map_add, map_add, map_add, map_add, map_add, h₁, h₂, add_add_add_comm]
state_before: case refine'_2.refine'_2.refine'_3
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
a₂ : A
l₂ : L
u₁ u₂ : A ⊗[R] L
h₁ :
  ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
      (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂)) u₁) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (a₂ ⊗ₜ[R] l₂)))
        u₁ +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂))
        (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) u₁)
h₂ :
  ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
      (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂)) u₂) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (a₂ ⊗ₜ[R] l₂)))
        u₂ +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂))
        (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) u₂)
⊢ ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
      (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂)) (u₁ + u₂)) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (a₂ ⊗ₜ[R] l₂)))
        (u₁ + u₂) +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₂ ⊗ₜ[R] l₂))
        (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (u₁ + u₂))

tactic: intro u₁ u₂ h₁ h₂
state_before: case refine'_2.refine'_3
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
⊢ ∀ (x y : A ⊗[R] L),
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) x) z) =
        ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) x))
            z +
          ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) x)
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) z) →
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =
          ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) y))
              z +
            ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y)
              (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) z) →
        ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (x + y)) z) =
          ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
                  (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (x + y)))
              z +
            ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (x + y))
              (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) z)

tactic: rw [map_add, LinearMap.add_apply, LinearMap.add_apply, map_add, map_add, map_add,
  LinearMap.add_apply, h₁, h₂, add_add_add_comm]
state_before: case refine'_2.refine'_3
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
a₁ : A
l₁ : L
u₁ u₂ : A ⊗[R] L
h₁ :
  ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) u₁) z) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) u₁)) z +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) u₁) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) z)
h₂ :
  ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) u₂) z) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) u₂)) z +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) u₂) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) z)
⊢ ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁))
      (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (u₁ + u₂)) z) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L)
            (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) (u₁ + u₂)))
        z +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (u₁ + u₂))
        (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (a₁ ⊗ₜ[R] l₁)) z)

tactic: intro u₁ u₂ h₁ h₂
state_before: case refine'_3
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : A ⊗[R] L
⊢ ∀ (x y_1 : A ⊗[R] L),
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) x) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =
        ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) x) y)) z +
          ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) x) z) →
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y_1) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =
          ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y_1) y)) z +
            ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y_1) z) →
        ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (x + y_1)) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =
          ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (x + y_1)) y)) z +
            ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (x + y_1)) z)

tactic: rw [map_add, LinearMap.add_apply, LinearMap.add_apply, LinearMap.add_apply, map_add, map_add,
  LinearMap.add_apply, h₁, h₂, add_add_add_comm]
state_before: case refine'_3
R : Type u
A : Type w
L : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z u₁ u₂ : A ⊗[R] L
h₁ :
  ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) u₁) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) u₁) y)) z +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) u₁) z)
h₂ :
  ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) u₂) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) u₂) y)) z +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) u₂) z)
⊢ ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (u₁ + u₂)) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =
    ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (u₁ + u₂)) y)) z +
      ↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) y) (↑(↑(LieAlgebra.ExtendScalars.bracket' R A L) (u₁ + u₂)) z)