Theorems · Theorem · linear algebra
Submodule.isAtom_iff_finrank_eq_one
∀ {K : Type u} {V : Type v} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] {S : Submodule K V},
IsAtom S ↔ Module.finrank K ↥S = 1A submodule over a division ring is an atom of the submodule lattice iff it has finrank 1.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites25
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
- AddCommGroupstatement and proof · cited by 12,871
- Submodulestatement and proof · cited by 7,192
- Bot.botproof · cited by 4,720
- LT.lt.leproof · cited by 2,189
- FiniteDimensionalproof · cited by 1,854
- Module.finrankstatement and proof · cited by 1,770
- Submodule.spanproof · cited by 1,504
- DivisionRingstatement and proof · cited by 1,062
- LT.lt.neproof · cited by 872
- Set.singleton_subset_iffproof · cited by 206
- Submodule.span_leproof · cited by 164
Cited by1
Results whose statement or proof uses this declaration.
- LieAlgebra.IsKilling.lieIdeal_eq_inf_cartan_sup_biSup_rootSpaceproof · cited by 1