isNilpotent_iff_isNilpotent_fst — Mathlib

English

The induced dpow structure is compatible with f as an IsDPMorphism from hI to the quotient divided powers on J.

Русский

Индуцированная dpow-структура совместима с f как IsDPMorphism от hI к DP на факторе.

LaTeX

$$$DividedPowers.Quotient.OfSurjective.isDPMorphism\\ (hI)\\ (hf)\\ (hIJ)$$$

Lean4

theorem isNilpotent_iff_isNilpotent_fst {x : TrivSqZeroExt R M} : IsNilpotent x ↔ IsNilpotent x.fst :=
  by
  constructor <;> rintro ⟨n, hn⟩
  · refine ⟨n, ?_⟩
    rw [← fst_pow, hn, fst_zero]
  · refine ⟨n * 2, ?_⟩
    rw [pow_mul]
    ext
    · rw [fst_pow, fst_pow, hn, zero_pow two_ne_zero, fst_zero]
    · rw [pow_two, snd_mul, fst_pow, hn, MulOpposite.op_zero, zero_smul, zero_smul, zero_add, snd_zero]

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