-- Module: Mathlib.LinearAlgebra.AffineSpace.Midpoint
Goal:
R : Type u_1
V : Type u_2
P : Type u_4
inst : Ring R
inst_1 : Invertible (α := R) 2
inst_2 : AddCommGroup V
inst_3 : Module R V
inst_4 : AddTorsor V P
p₁ p₂ p : P
rw :
  midpoint R p₁ p₂ -ᵥ p =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _)
        (p₁ -ᵥ p₂ - (p -ᵥ p₂)) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p)
rw₁ :
  midpoint R p₁ p₂ -ᵥ p =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p₂) -
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₂) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p)
rw₂ :
  midpoint R p₁ p₂ -ᵥ p =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p₂) +
        -HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₂) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p)
rw₃ :
  midpoint R p₁ p₂ -ᵥ p =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p₂) +
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _)
          (-(p -ᵥ p₂)) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p)
rw₄ :
  midpoint R p₁ p₂ -ᵥ p =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p₂) +
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p)
rw₅ :
  midpoint R p₁ p₂ -ᵥ p =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p₂) +
      (HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p) +
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p))
rw₆ :
  midpoint R p₁ p₂ -ᵥ p =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p₂) +
      (p₂ -ᵥ p)
rw₇ :
  midpoint R p₁ p₂ -ᵥ p =
    (HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p₂) +ᵥ
        p₂) -ᵥ
      p
rw₈ :
  midpoint R p₂ p₁ -ᵥ p =
    (HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p₂) +ᵥ
        p₂) -ᵥ
      p
rw₉ :
  (lineMap p₂ p₁ : (a : R) → P) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) -ᵥ p =
    (HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p₂) +ᵥ
        p₂) -ᵥ
      p
rw₁₀ rw₁₁ :
  (HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p₂) +ᵥ
        p₂) -ᵥ
      p =
    (HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p₂) +ᵥ
        p₂) -ᵥ
      p
⊢ midpoint R p₁ p₂ -ᵥ p =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p)

Tactic:
rw [← vsub_sub_vsub_cancel_right p₁ p p₂, smul_sub, sub_eq_add_neg, ← smul_neg,
    neg_vsub_eq_vsub_rev, add_assoc, invOf_two_smul_add_invOf_two_smul, ← vadd_vsub_assoc,
    midpoint_comm, midpoint, lineMap_apply]

Premises used: vadd_vsub_assoc, add_assoc, sub_eq_add_neg, midpoint_comm, smul_neg, invOf_two_smul_add_invOf_two_smul, AffineMap.lineMap_apply, smul_sub, vsub_sub_vsub_cancel_right, neg_vsub_eq_vsub_rev, midpoint