Theorems · Theorem · order theory
Submodule.mem_bot
∀ (R : Type u_1) {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {x : M},
x ∈ ⊥ ↔ x = 0- Defined in
- Mathlib.Algebra.Module.Submodule.Lattice
- Cited by
- 55 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses propext, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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 · cited by 7,192
- Bot.botstatement · cited by 4,720
- Set.mem_singleton_iffproof · cited by 172
Cited by55
Results whose statement or proof uses this declaration.
- LinearMap.mem_kerproof · cited by 66
- Ideal.mk_kerproof · cited by 59
- RingHom.mem_kerproof · cited by 49
- Submodule.eq_bot_iffproof · cited by 31
- Ideal.comap_map_of_surjectiveproof · cited by 30
- Ideal.mem_botproof · cited by 21
- Ideal.map_quotient_selfproof · cited by 12
- Submodule.top_orthogonal_eq_botproof · cited by 6
- Ideal.comap_bot_le_of_injectiveproof · cited by 4
- Module.End.independent_genEigenspaceproof · cited by 4
- Affine.Simplex.ExcenterExists.touchpoint_injectiveproof · cited by 4
- Ideal.mem_jacobson_botproof · cited by 4