Theorems · Theorem · nonassociative algebras

Function.Injective.lieModuleIsNilpotent

∀ {R : Type u} {L : Type v} {M : Type w} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : LieRingModule L M] [LieModule R L M] {L₂ : Type u_1}
  {M₂ : Type u_2} [inst_7 : LieRing L₂] [inst_8 : LieAlgebra R L₂] [inst_9 : AddCommGroup M₂] [inst_10 : Module R M₂]
  [inst_11 : LieRingModule L₂ M₂] {f : L →ₗ⁅R⁆ L₂} {g : M →ₗ[R] M₂},
  (∀ (x : L) (m : M), ⁅f x, g m⁆ = g ⁅x, m⁆) →
    ∀ [LieModule R L₂ M₂], Function.Injective ⇑g → ∀ [LieModule.IsNilpotent L₂ M₂], LieModule.IsNilpotent L M

Defined in

Mathlib.Algebra.Lie.Nilpotent

Cited by

1 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModuleLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModuleLieModule.IsNilpotent