∀ (R : Type u_1) [inst : CommRing R] (x y : R),
  (if x * y ∈ nonZeroDivisors R then
      (WithZero.map' (AddMonoidHom.toMultiplicative (Nat.castAddMonoidHom ℤ))) (Ring.ord R (x * y))
    else 0) =
    (if x ∈ nonZeroDivisors R then
        (WithZero.map' (AddMonoidHom.toMultiplicative (Nat.castAddMonoidHom ℤ))) (Ring.ord R x)
      else 0) *
      if y ∈ nonZeroDivisors R then
        (WithZero.map' (AddMonoidHom.toMultiplicative (Nat.castAddMonoidHom ℤ))) (Ring.ord R y)
      else 0