-- Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
Goal:
R : Type r
inst : CommRing R
W' : Projective R
⊢ (↑2 : MvPolynomial (Fin 3) R) * HPow.hPow (α := MvPolynomial (Fin 3) R) Y (HSub.hSub (α := ℕ) 2 1) * 0 * Z +
            HPow.hPow (α := MvPolynomial (Fin 3) R) (β := ℕ) Y 2 * 0 +
          (((HMul.hMul (α := MvPolynomial (Fin 3) R) 0 X +
                    HMul.hMul (α := MvPolynomial (Fin 3) R) (β := MvPolynomial (Fin 3) R)
                      ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₁ : MvPolynomial (Fin 3) R)
                      (1 : MvPolynomial (Fin 3) R)) *
                  Y +
                HMul.hMul (α := MvPolynomial (Fin 3) R) (β := MvPolynomial (Fin 3) R)
                    ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₁ : MvPolynomial (Fin 3) R) X *
                  0) *
              Z +
            HMul.hMul (α := MvPolynomial (Fin 3) R) (β := MvPolynomial (Fin 3) R)
                  ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₁ : MvPolynomial (Fin 3) R) X *
                Y *
              0) +
        ((HMul.hMul (α := MvPolynomial (Fin 3) R) 0 Y +
              HMul.hMul (α := MvPolynomial (Fin 3) R) (β := MvPolynomial (Fin 3) R)
                ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₃ : MvPolynomial (Fin 3) R) (0 : MvPolynomial (Fin 3) R)) *
            HPow.hPow (α := MvPolynomial (Fin 3) R) (β := ℕ) Z 2 +
          HMul.hMul (α := MvPolynomial (Fin 3) R) (β := MvPolynomial (Fin 3) R)
              ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₃ : MvPolynomial (Fin 3) R) Y *
            ((↑2 : MvPolynomial (Fin 3) R) * HPow.hPow (α := MvPolynomial (Fin 3) R) Z (HSub.hSub (α := ℕ) 2 1) * 0)) -
      ((↑3 : MvPolynomial (Fin 3) R) * HPow.hPow (α := MvPolynomial (Fin 3) R) X (HSub.hSub (α := ℕ) 3 1) * 1 +
            ((0 * HPow.hPow (α := MvPolynomial (Fin 3) R) (β := ℕ) X 2 +
                  HMul.hMul (α := MvPolynomial (Fin 3) R)
                    ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₂ : MvPolynomial (Fin 3) R)
                    ((↑2 : MvPolynomial (Fin 3) R) *
                        HPow.hPow (α := MvPolynomial (Fin 3) R) X (HSub.hSub (α := ℕ) 2 1) *
                      1)) *
                Z +
              HMul.hMul (α := MvPolynomial (Fin 3) R)
                  ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₂ : MvPolynomial (Fin 3) R)
                  (HPow.hPow (α := MvPolynomial (Fin 3) R) (β := ℕ) X 2) *
                0) +
          ((HMul.hMul (α := MvPolynomial (Fin 3) R) 0 X +
                HMul.hMul (α := MvPolynomial (Fin 3) R) (β := MvPolynomial (Fin 3) R)
                  ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₄ : MvPolynomial (Fin 3) R)
                  (1 : MvPolynomial (Fin 3) R)) *
              HPow.hPow (α := MvPolynomial (Fin 3) R) (β := ℕ) Z 2 +
            HMul.hMul (α := MvPolynomial (Fin 3) R) (β := MvPolynomial (Fin 3) R)
                ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₄ : MvPolynomial (Fin 3) R) X *
              ((↑2 : MvPolynomial (Fin 3) R) * HPow.hPow (α := MvPolynomial (Fin 3) R) Z (HSub.hSub (α := ℕ) 2 1) *
                0)) +
        (0 * HPow.hPow (α := MvPolynomial (Fin 3) R) (β := ℕ) Z 3 +
          HMul.hMul (α := MvPolynomial (Fin 3) R)
            ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₆ : MvPolynomial (Fin 3) R)
            ((↑3 : MvPolynomial (Fin 3) R) * HPow.hPow (α := MvPolynomial (Fin 3) R) Z (HSub.hSub (α := ℕ) 3 1) * 0))) =
    HMul.hMul (α := MvPolynomial (Fin 3) R) (β := MvPolynomial (Fin 3) R)
          ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₁ : MvPolynomial (Fin 3) R) Y *
        Z -
      (3 * HPow.hPow (α := MvPolynomial (Fin 3) R) (β := ℕ) X 2 +
          HMul.hMul (α := MvPolynomial (Fin 3) R) (β := MvPolynomial (Fin 3) R) (2 : MvPolynomial (Fin 3) R)
                ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₂ : MvPolynomial (Fin 3) R) *
              X *
            Z +
        HMul.hMul (α := MvPolynomial (Fin 3) R) ((C : (a : R) → MvPolynomial (Fin 3) R) W'.a₄ : MvPolynomial (Fin 3) R)
          (HPow.hPow (α := MvPolynomial (Fin 3) R) (β := ℕ) Z 2))

Tactic:
ring1