Theorems · Theorem · order theory
Submodule.subsingleton_iff_eq_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}, Subsingleton ↥p ↔ p = ⊥- Defined in
- Mathlib.Algebra.Module.Submodule.Lattice
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
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
- 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
- Submodule.zero_memproof · cited by 58
- Submodule.eq_bot_iffproof · cited by 31
- subsingleton_iffproof · cited by 10
Cited by10
Results whose statement or proof uses this declaration.
- Algebra.FormallySmooth.comp_surjectiveproof · cited by 6
- Submodule.eq_bot_of_subsingletonproof · cited by 5
- IsSemisimpleModule.eq_bot_or_exists_simple_leproof · cited by 4
- Submodule.nontrivial_iff_ne_botproof · cited by 3
- Submodule.annihilator_eq_top_iffproof · cited by 2
- Module.exists_basis_of_basis_baseChangeproof · cited by 1
- LinearMap.IsIdempotentElem.trace_eq_zero_iffproof · cited by 1
- IsLocalRing.split_injective_iff_lTensor_residueField_injectiveproof · cited by 1
- Submodule.dualAnnihilator_eq_bot_iff'proof · cited by 1
- ContinuousLinearMap.orthogonalComplement_iSup_eigenspaces_eq_botproof · cited by 0