coe_genWeightSpace_of_top — Mathlib

English

In the Noetherian setting, with χ = 0, there exists k with genWeightSpace M 0 ≤ ker((toEnd x)^k).

Русский

В рамках условия Ноetherian, при χ = 0 существует k such that genWeightSpace M 0 ⊆ ker((toEnd x)^k).

LaTeX

$$∃ k, genWeightSpace M 0 ≤ ker ((toEnd R L M x)^k)$$

Lean4

theorem coe_genWeightSpace_of_top (χ : L → R) :
    (genWeightSpace M (χ ∘ (⊤ : LieSubalgebra R L).incl) : Submodule R M) = genWeightSpace M χ :=
  by
  ext m
  simp only [mem_genWeightSpace, LieSubmodule.mem_toSubmodule, Subtype.forall]
  apply forall_congr'
  simp

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