isNilpotent_iff_forall — Mathlib

English

Equality of left homology data implies equality of φK component again; the congruence persists.

Русский

Равенство левых гомологических данных продолжает равенство компоненты φK.

LaTeX

$$$ γ_1 = γ_2 \Rightarrow γ_1.φK = γ_2.φK $$$

Lean4

/-- Engel's theorem.

See also `LieModule.isNilpotent_iff_forall'` which assumes that `M` is Noetherian instead of `L`. -/
theorem isNilpotent_iff_forall [IsNoetherian R L] :
    LieModule.IsNilpotent L M ↔ ∀ x, _root_.IsNilpotent <| toEnd R L M x :=
  ⟨fun _ ↦ isNilpotent_toEnd_of_isNilpotent R L M, fun h => LieAlgebra.isEngelian_of_isNoetherian M h⟩

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