Theorems · Theorem · order theory
Submodule.nontrivial_iff_ne_bot
∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{p : Submodule R M}, Nontrivial ↥p ↔ p ≠ ⊥- Defined in
- Mathlib.Algebra.Module.Submodule.Lattice
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement and proof · cited by 7,192
- Bot.botstatement and proof · cited by 4,720
- Nontrivialstatement · cited by 2,416
- not_nontrivial_iff_subsingletonproof · cited by 23
- iff_not_commproof · cited by 16
- Submodule.subsingleton_iff_eq_botproof · cited by 10
Cited by3
Results whose statement or proof uses this declaration.
- Module.End.disjoint_genEigenspaceproof · cited by 7
- bot_lt_isotypicComponentproof · cited by 3
- IsSemisimpleModule.exists_linearEquiv_fin_dfinsuppproof · cited by 1