-- Module: Mathlib.GroupTheory.FreeAbelianGroup
Goal:
α : Type u
G : Type u_1
β : Type v
γ : Type w
R : Type u_2
inst : Monoid α
inst_1 : Ring R
f : α →* R
x : FreeAbelianGroup α
L2 : α
ih :
  (((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) : AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
        (a : FreeAbelianGroup α) → R)
      (x * of L2) =
    HMul.hMul (α := R) (β := R)
      ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
              AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
            (a : FreeAbelianGroup α) → R)
          x :
        R)
      ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
              AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
            (a : FreeAbelianGroup α) → R)
          (of L2) :
        R)
neg :
  (((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) : AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
        (a : FreeAbelianGroup α) → R)
      (-(x * of L2)) =
    HMul.hMul (α := R) (β := R)
      ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
              AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
            (a : FreeAbelianGroup α) → R)
          x :
        R)
      ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
              AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
            (a : FreeAbelianGroup α) → R)
          (-of L2) :
        R)
neg₁ :
  -(((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) : AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
          (a : FreeAbelianGroup α) → R)
        (x * of L2) =
    HMul.hMul (α := R) (β := R)
      ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
              AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
            (a : FreeAbelianGroup α) → R)
          x :
        R)
      ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
              AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
            (a : FreeAbelianGroup α) → R)
          (-of L2) :
        R)
neg₂ :
  -(((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) : AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
          (a : FreeAbelianGroup α) → R)
        (x * of L2) =
    HMul.hMul (α := R) (β := R)
      ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
              AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
            (a : FreeAbelianGroup α) → R)
          x :
        R)
      (-(((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
                AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
              (a : FreeAbelianGroup α) → R)
            (of L2) :
        R)
neg₃ :
  -(((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) : AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
          (a : FreeAbelianGroup α) → R)
        (x * of L2) =
    -HMul.hMul (α := R) (β := R)
        ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
                AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
              (a : FreeAbelianGroup α) → R)
            x :
          R)
        ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
                AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
              (a : FreeAbelianGroup α) → R)
            (of L2) :
          R)
neg₄ neg₅ :
  -HMul.hMul (α := R) (β := R)
        ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
                AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
              (a : FreeAbelianGroup α) → R)
            x :
          R)
        ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
                AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
              (a : FreeAbelianGroup α) → R)
            (of L2) :
          R) =
    -HMul.hMul (α := R) (β := R)
        ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
                AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
              (a : FreeAbelianGroup α) → R)
            x :
          R)
        ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
                AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
              (a : FreeAbelianGroup α) → R)
            (of L2) :
          R)
⊢ (((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) : AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
        (a : FreeAbelianGroup α) → R)
      (x * -of L2) =
    HMul.hMul (α := R) (β := R)
      ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
              AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
            (a : FreeAbelianGroup α) → R)
          x :
        R)
      ((((lift : (a : α → R) → FreeAbelianGroup α →+ R) (⇑f : (a : α) → R) :
              AddMonoidHom.{u, u_2} (FreeAbelianGroup α) R) :
            (a : FreeAbelianGroup α) → R)
          (-of L2) :
        R)

Tactic:
rw [mul_neg, map_neg, map_neg, mul_neg, ih]

Premises used: map_neg, mul_neg