Theorems · Theorem · nonassociative algebras
LieSubmodule.lie_abelian_iff_lie_self_eq_bot
∀ {R : Type u} {L : Type v} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L] (I : LieIdeal R L),
IsLieAbelian ↥I ↔ ⁅I, I⁆ = ⊥- Defined in
- Mathlib.Algebra.Lie.Abelian
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingLieRingLieAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Bot.botstatement · cited by 4,720
- LieRingstatement and proof · cited by 1,548
- LieAlgebrastatement and proof · cited by 1,246
- Bracket.bracketstatement and proof · cited by 642
- LieIdealstatement and proof · cited by 282
- IsLieAbelianstatement and proof · cited by 55
- LieIdeal.toLieSubalgebraproof · cited by 48
- LieModule.IsTrivial.trivialproof · cited by 6
- LieSubalgebra.coe_bracketproof · cited by 1
- LieSubalgebra.coe_zero_iff_zeroproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- lie_eq_self_of_isAtom_of_nonabelianproof · cited by 2
- LieAlgebra.abelian_iff_derived_one_eq_botproof · cited by 1
- LieIdeal.isCompl_killingComplproof · cited by 0