Theorems · Theorem · order theory
Submodule.exists_mem_ne_zero_of_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}, p ≠ ⊥ → ∃ b ∈ p, b ≠ 0- Defined in
- Mathlib.Algebra.Module.Submodule.Lattice
- Cited by
- 9 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.
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 and proof · cited by 7,192
- Bot.botstatement and proof · cited by 4,720
- Submodule.ne_bot_iffproof · cited by 28
Cited by9
Results whose statement or proof uses this declaration.
- Module.End.HasEigenvalue.exists_hasEigenvectorproof · cited by 8
- Module.End.HasUnifEigenvalue.exists_hasUnifEigenvectorproof · cited by 3
- Ideal.IsPrime.exists_mem_prime_of_ne_botproof · cited by 3
- Submodule.norm_orthogonalProjectionOntoproof · cited by 2
- Valuation.ideal_isPrincipalproof · cited by 1
- DualNumber.ideal_trichotomyproof · cited by 1
- exists_integral_inj_algHom_of_quotientproof · cited by 1
- exists_mem_ne_zero_of_rank_posproof · cited by 0
- EuclideanGeometry.Sphere.inter_orthRadius_eq_empty_iffproof · cited by 0