iInf_lowerCentralSeries_eq_bot_of_isNilpotent

English

From a bot-lcs equality, one can construct a nilpotent object IsNilpotent L M.

Русский

Из равенства N.lcsk = ⊥ следует существование объекта nilpotent.

LaTeX

$$⊥ = lowerCentralSeries(R,L,M,k) ⇒ IsNilpotent L M$$

Lean4

@[simp]
theorem iInf_lowerCentralSeries_eq_bot_of_isNilpotent [IsNilpotent L M] : ⨅ k, lowerCentralSeries R L M k = ⊥ :=
  by
  obtain ⟨k, hk⟩ := IsNilpotent.nilpotent R L M
  rw [eq_bot_iff, ← hk]
  exact iInf_le _ _

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