Theorems · Theorem · order theory
Submodule.mem_sInf
∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{S : Set (Submodule R M)} {x : M}, x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p- Defined in
- Mathlib.Algebra.Module.Submodule.Lattice
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses propext, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- 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
- InfSet.sInfstatement · cited by 935
- Set.mem_iInter₂proof · cited by 54
Cited by11
Results whose statement or proof uses this declaration.
- Submodule.mem_set_smul_of_mem_memproof · cited by 9
- PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radicalproof · cited by 7
- Module.le_comap_jacobsonproof · cited by 6
- Submodule.mem_torsionBySet_iffproof · cited by 5
- Submodule.singleton_set_smulproof · cited by 4
- Ideal.map_sInfproof · cited by 3
- Submodule.set_smul_eq_mapproof · cited by 1
- Submodule.mem_set_smulproof · cited by 1
- nilpotent_iff_mem_primeproof · cited by 1
- IsArtinian.isSemisimpleModule_iff_jacobsonproof · cited by 1
- Submodule.empty_set_smulproof · cited by 0