-- Module: Mathlib.Data.Matrix.Invertible
Goal:
m : Type u_1
n : Type u_2
α : Type u_3
inst : Fintype n
inst_1 : DecidableEq n
inst_2 : Fintype m
inst_3 : DecidableEq m
inst_4 : Ring α
A : Matrix n n α
U : Matrix n m α
C : Matrix m m α
V : Matrix m n α
inst_5 : Invertible A
inst_6 : Invertible C
inst_7 :
  Invertible
    (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
      V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
simp_rw :
  (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ -
        @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
              @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                  V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                _ *
            V *
          @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _) *
      (A + U * C * V) =
    @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * A -
        @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                  (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                    V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                  _ *
              V *
            @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ *
          A +
      (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U * C * V -
        @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                    @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                      (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                        V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                      _ *
                  V *
                @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ *
              U *
            C *
          V)
simp_rw₁ :
  (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ -
          @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                  (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                    V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                  _ *
              V *
            @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _) *
        A +
      (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ -
          @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                  (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                    V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                  _ *
              V *
            @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _) *
        (U * C * V) =
    @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * A -
        @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                  (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                    V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                  _ *
              V *
            @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ *
          A +
      (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U * C * V -
        @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                    @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                      (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                        V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                      _ *
                  V *
                @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ *
              U *
            C *
          V)
simp_rw₂ :
  @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * A -
        @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                  (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                    V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                  _ *
              V *
            @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ *
          A +
      (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * (U * C * V) -
        @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                  (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                    V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                  _ *
              V *
            @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ *
          (U * C * V)) =
    @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * A -
        @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                  (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                    V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                  _ *
              V *
            @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ *
          A +
      (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U * C * V -
        @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                    @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                      (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                        V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                      _ *
                  V *
                @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ *
              U *
            C *
          V)
⊢ (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ -
        @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
              @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                  V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                _ *
            V *
          @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _) *
      (A + U * C * V) =
    @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * A -
          @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                  @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                    (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                      V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                    _ *
                V *
              @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ *
            A +
        @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U * C * V -
      @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U *
                  @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one
                    (@Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one C _ +
                      V * @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ * U)
                    _ *
                V *
              @Invertible.invOf _ instMulOfFintypeOfAddCommMonoid one A _ *
            U *
          C *
        V

Tactic:
simp_rw [add_sub_assoc, _root_.mul_add, _root_.sub_mul, Matrix.mul_assoc]

Premises used: add_sub_assoc, Matrix.mul_assoc, sub_mul, mul_add