isNilpotent_iff_eps_dvd — Mathlib

English

The epsilon in DualNumber is nilpotent; equivalently the associated dpow vanishes after squaring.

Русский

Эпсилон в DualNumber нилпотентен; эквивалентно dpow обнуляется после квадрата.

LaTeX

$$$IsNilpotent(\\varepsilon)\\,\\text{ on } R[\\varepsilon]$$$

Lean4

theorem isNilpotent_iff_eps_dvd [DivisionSemiring R] {x : R[ε]} : IsNilpotent x ↔ ε ∣ x := by
  simp only [isNilpotent_iff_isNilpotent_fst, isNilpotent_iff_eq_zero, fst_eq_zero_iff_eps_dvd]

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