Theorems · Theorem · linear algebra
Submodule.span_singleton_eq_bot
∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {x : M},
R ∙ x = ⊥ ↔ x = 0- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 27 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 · 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 · cited by 7,192
- Bot.botstatement · cited by 4,720
- Submodule.spanstatement · cited by 1,504
Cited by7
Results whose statement or proof uses this declaration.
- Submodule.IsPrincipal.eq_bot_iff_generator_eq_zeroproof · cited by 7
- Submodule.span_zeroproof · cited by 6
- LocalizedModule.subsingleton_iff_support_subsetproof · cited by 5
- IsDedekindDomain.HeightOneSpectrum.intValuation_eq_exp_neg_multiplicityproof · cited by 2
- IsDiscreteValuationRing.of_ufd_of_unique_irreducibleproof · cited by 1
- NumberField.FinitePlace.apply_mul_absNorm_pow_eq_oneproof · cited by 0
- atom_iff_nonzero_spanproof · cited by 0