Theorem: LieModule.isNilpotent_toEnd_sub_algebraMap
File: Mathlib/Algebra/Lie/Weights/Basic.lean

Proof steps:
state_before: K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : LieAlgebra.IsNilpotent R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M
χ : L → R
x : L
⊢ _root_.IsNilpotent ((toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x))
state_after: K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : LieAlgebra.IsNilpotent R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M
χ : L → R
x : L
this :
  (toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x) =
    LinearMap.restrict ((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ⋯
⊢ _root_.IsNilpotent ((toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x))
tactic: have : toEnd R L (weightSpace M χ) x - algebraMap R _ (χ x) =
      (toEnd R L M x - algebraMap R _ (χ x)).restrict
        (fun m hm ↦ sub_mem (LieSubmodule.lie_mem _ hm) (Submodule.smul_mem _ _ hm)) := by
    rfl
state_before: K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : LieAlgebra.IsNilpotent R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M
χ : L → R
x : L
this :
  (toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x) =
    LinearMap.restrict ((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ⋯
⊢ _root_.IsNilpotent ((toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x))
state_after: case intro
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : LieAlgebra.IsNilpotent R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M
χ : L → R
x : L
this :
  (toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x) =
    LinearMap.restrict ((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ⋯
k : ℕ
hk : ↑(weightSpace M χ) ≤ LinearMap.ker (((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ^ k)
⊢ _root_.IsNilpotent ((toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x))
tactic: obtain ⟨k, hk⟩ := exists_weightSpace_le_ker_of_isNoetherian M χ x
state_before: case intro
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : LieAlgebra.IsNilpotent R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M
χ : L → R
x : L
this :
  (toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x) =
    LinearMap.restrict ((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ⋯
k : ℕ
hk : ↑(weightSpace M χ) ≤ LinearMap.ker (((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ^ k)
⊢ _root_.IsNilpotent ((toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x))
state_after: case h
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : LieAlgebra.IsNilpotent R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M
χ : L → R
x : L
this :
  (toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x) =
    LinearMap.restrict ((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ⋯
k : ℕ
hk : ↑(weightSpace M χ) ≤ LinearMap.ker (((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ^ k)
⊢ ((toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x)) ^ k = 0
tactic: use k
state_before: case h
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : LieAlgebra.IsNilpotent R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M
χ : L → R
x : L
this :
  (toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x) =
    LinearMap.restrict ((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ⋯
k : ℕ
hk : ↑(weightSpace M χ) ≤ LinearMap.ker (((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ^ k)
⊢ ((toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x)) ^ k = 0
state_after: case h.h.mk.a
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : LieAlgebra.IsNilpotent R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M
χ : L → R
x : L
this :
  (toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x) =
    LinearMap.restrict ((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ⋯
k : ℕ
hk : ↑(weightSpace M χ) ≤ LinearMap.ker (((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ^ k)
m : M
hm : m ∈ ↑(weightSpace M χ)
⊢ ↑((((toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x)) ^ k) ⟨m, hm⟩) =
    ↑(0 ⟨m, hm⟩)
tactic: ext ⟨m, hm⟩
state_before: case h.h.mk.a
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : LieAlgebra.IsNilpotent R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : IsNoetherian R M
χ : L → R
x : L
this :
  (toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x) =
    LinearMap.restrict ((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ⋯
k : ℕ
hk : ↑(weightSpace M χ) ≤ LinearMap.ker (((toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ^ k)
m : M
hm : m ∈ ↑(weightSpace M χ)
⊢ ↑((((toEnd R L ↥↑(weightSpace M χ)) x - (algebraMap R (Module.End R ↥↑(weightSpace M χ))) (χ x)) ^ k) ⟨m, hm⟩) =
    ↑(0 ⟨m, hm⟩)
state_after: no goals
tactic: simpa [this, LinearMap.pow_restrict _, LinearMap.restrict_apply] using hk hm