∀ (p : ℕ) [Fact (Nat.Prime p)] (x y : ℚ),
  (if x * y = 0 then 0 else WithZero.exp (-padicValRat p (x * y))) =
    (if x = 0 then 0 else WithZero.exp (-padicValRat p x)) * if y = 0 then 0 else WithZero.exp (-padicValRat p y)