-- Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
lemma addZ_eq' {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) :
    W'.addZ P Q * (P z * Q z) = (P x * Q z - Q x * P z) ^ 3

Proof state:
R : Type r
inst✝ : CommRing R
W' : Projective R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
⊢ W'.addZ P Q * (P z * Q z) = (P x * Q z - Q x * P z) ^ 3

Tactic:
linear_combination (norm := (rw [addZ]; ring1))
  Q z ^ 3 * (equation_iff _).mp hP - P z ^ 3 * (equation_iff _).mp hQ

Result:
no goals

Proof state:
case a.a
R : Type r
inst✝ : CommRing R
W' : Projective R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
⊢ W'.addZ P Q * (P z * Q z) +
        (Q z ^ 3 * 0 +
          P z ^ 3 *
            (Q y ^ 2 * Q z + W'.a₁ * Q x * Q y * Q z + W'.a₃ * Q y * Q z ^ 2 -
              (Q x ^ 3 + W'.a₂ * Q x ^ 2 * Q z + W'.a₄ * Q x * Q z ^ 2 + W'.a₆ * Q z ^ 3))) -
      ((P x * Q z - Q x * P z) ^ 3 +
        (Q z ^ 3 *
            (P y ^ 2 * P z + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 2 -
              (P x ^ 3 + W'.a₂ * P x ^ 2 * P z + W'.a₄ * P x * P z ^ 2 + W'.a₆ * P z ^ 3)) +
          P z ^ 3 * 0)) =
    0

Tactic:
rw [addZ]

Result:
case a.a
R : Type r
inst✝ : CommRing R
W' : Projective R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
⊢ (-3 * P x ^ 2 * Q x * Q z + 3 * P x * Q x ^ 2 * P z + P y ^ 2 * Q z ^ 2 - Q y ^ 2 * P z ^ 2 +
                          W'.a₁ * P x * P y * Q z ^ 2 -
                        W'.a₁ * Q x * Q y * P z ^ 2 -
                      W'.a₂ * P x ^ 2 * Q z ^ 2 +
                    W'.a₂ * Q x ^ 2 * P z ^ 2 +
                  W'.a₃ * P y * P z * Q z ^ 2 -
                W'.a₃ * Q y * P z ^ 2 * Q z -
              W'.a₄ * P x * P z * Q z ^ 2 +
            W'.a₄ * Q x * P z ^ 2 * Q z) *
          (P z * Q z) +
        (Q z ^ 3 * 0 +
          P z ^ 3 *
            (Q y ^ 2 * Q z + W'.a₁ * Q x * Q y * Q z + W'.a₃ * Q y * Q z ^ 2 -
              (Q x ^ 3 + W'.a₂ * Q x ^ 2 * Q z + W'.a₄ * Q x * Q z ^ 2 + W'.a₆ * Q z ^ 3))) -
      ((P x * Q z - Q x * P z) ^ 3 +
        (Q z ^ 3 *
            (P y ^ 2 * P z + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 2 -
              (P x ^ 3 + W'.a₂ * P x ^ 2 * P z + W'.a₄ * P x * P z ^ 2 + W'.a₆ * P z ^ 3)) +
          P z ^ 3 * 0)) =
    0

Proof state:
case a.a
R : Type r
inst✝ : CommRing R
W' : Projective R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
⊢ (-3 * P x ^ 2 * Q x * Q z + 3 * P x * Q x ^ 2 * P z + P y ^ 2 * Q z ^ 2 - Q y ^ 2 * P z ^ 2 +
                          W'.a₁ * P x * P y * Q z ^ 2 -
                        W'.a₁ * Q x * Q y * P z ^ 2 -
                      W'.a₂ * P x ^ 2 * Q z ^ 2 +
                    W'.a₂ * Q x ^ 2 * P z ^ 2 +
                  W'.a₃ * P y * P z * Q z ^ 2 -
                W'.a₃ * Q y * P z ^ 2 * Q z -
              W'.a₄ * P x * P z * Q z ^ 2 +
            W'.a₄ * Q x * P z ^ 2 * Q z) *
          (P z * Q z) +
        (Q z ^ 3 * 0 +
          P z ^ 3 *
            (Q y ^ 2 * Q z + W'.a₁ * Q x * Q y * Q z + W'.a₃ * Q y * Q z ^ 2 -
              (Q x ^ 3 + W'.a₂ * Q x ^ 2 * Q z + W'.a₄ * Q x * Q z ^ 2 + W'.a₆ * Q z ^ 3))) -
      ((P x * Q z - Q x * P z) ^ 3 +
        (Q z ^ 3 *
            (P y ^ 2 * P z + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 2 -
              (P x ^ 3 + W'.a₂ * P x ^ 2 * P z + W'.a₄ * P x * P z ^ 2 + W'.a₆ * P z ^ 3)) +
          P z ^ 3 * 0)) =
    0

Tactic:
ring1

Result:
no goals