Theorems · Theorem · linear algebra
LinearMap.span_singleton_eq_range
∀ (R : Type u_1) (M : Type u_4) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (x : M), R ∙ x = (LinearMap.toSpanSingleton R M x).range
- Defined in
- Mathlib.LinearAlgebra.Span.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 29 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.
Cites10
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
- RingHom.idstatement · cited by 18,349
- 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
- LinearMap.rangestatement · cited by 893
- LinearMap.toSpanSingletonstatement · cited by 78
- LinearMap.range_toSpanSingletonproof · cited by 2
Cited by6
Results whose statement or proof uses this declaration.
- Submodule.one_eq_spanproof · cited by 15
- Submodule.one_eq_rangeproof · cited by 11
- IsSimpleModule.toSpanSingleton_surjectiveproof · cited by 4
- WittVector.isocrystal_classificationproof · cited by 0
- isNoetherian_of_finite_isArtinianproof · cited by 0
- Module.exists_surjective_quotient_of_finiteproof · cited by 0