Theorems · Theorem · nonassociative algebras
LieSubmodule.mem_bot
∀ {R : Type u} {L : Type v} {M : Type w} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : AddCommGroup M]
[inst_3 : Module R M] [inst_4 : LieRingModule L M] (x : M), x ∈ ⊥ ↔ x = 0- Defined in
- Mathlib.Algebra.Lie.Submodule
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 19 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- Bot.botstatement · cited by 4,720
- LieRingstatement and proof · cited by 1,548
- LieRingModulestatement and proof · cited by 727
- LieSubmodulestatement · cited by 489
- Set.mem_singleton_iffproof · cited by 172
Cited by11
Results whose statement or proof uses this declaration.
- LieSubmodule.nontrivial_iff_ne_botproof · cited by 6
- LieModule.exists_forall_pow_toEnd_eq_zeroproof · cited by 3
- LieModule.isNilpotent_toEnd_of_isNilpotent₂proof · cited by 1
- LieModule.isNilpotent_toEnd_of_mem_rootSpaceproof · cited by 1
- LieAlgebra.toLieSubmodule_le_rootSpace_zeroproof · cited by 1
- LieAlgebra.isSolvable_tensorProduct_iffproof · cited by 0
- LieDerivation.maxTrivSubmodule_eq_bot_of_center_eq_botproof · cited by 0
- LieSubmodule.bot_lieproof · cited by 0
- LieDerivation.IsKilling.ad_apply_eq_zero_iffproof · cited by 0
- LieAlgebra.InvariantForm.isSemisimple_of_nondegenerateproof · cited by 0
- LieAlgebra.IsSemisimple.isSimple_of_isAtomproof · cited by 0