Theorems · Theorem · linear algebra
Submodule.span_eq_iSup_of_singleton_spans
∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s : Set M),
Submodule.span R s = ⨆ x ∈ s, R ∙ x- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 66 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.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · 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
- Set.iUnionproof · cited by 2,483
- iSupstatement and proof · cited by 2,415
- Submodule.spanstatement and proof · cited by 1,504
- Set.biUnion_of_singletonproof · cited by 43
Cited by4
Results whose statement or proof uses this declaration.
- Submodule.span_range_eq_iSupproof · cited by 3
- Submodule.isOrtho_spanproof · cited by 2
- Submodule.fg_iff_compactproof · cited by 1
- Submodule.finset_span_isCompactElementproof · cited by 1