Theorems · Theorem · linear algebra
Submodule.mem_span_singleton_self
∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (x : M), x ∈ R ∙ x- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 59 results in Mathlib
- Foundations
- Depth 20 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
- Submodule.spanstatement · cited by 1,504
- Submodule.subset_spanproof · cited by 234
Cited by59
Results whose statement or proof uses this declaration.
- finrank_span_singletonproof · cited by 16
- Submodule.mem_orthogonal_singleton_iff_inner_rightproof · cited by 10
- Submodule.mem_span_finite_of_mem_spanproof · cited by 6
- Submodule.singleton_span_isCompactElementproof · cited by 4
- exists_dual_vectorproof · cited by 3
- Submodule.span_singleton_eq_one_iffproof · cited by 3
- LinearIndependent.notMem_span_imageproof · cited by 3
- TensorProduct.exists_finite_submodule_of_setFiniteproof · cited by 3
- AffineSubspace.direction_supproof · cited by 2
- LinearPMap.supSpanSingleton_apply_selfstatement · cited by 2
- IsLocalization.coeSubmodule_isPrincipalproof · cited by 2