Theorems · Theorem · linear algebra
Finsupp.range_linearCombination
∀ {α : Type u_1} {M : Type u_2} (R : Type u_5) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{v : α → M}, (Finsupp.linearCombination R v).range = Submodule.span R (Set.range v)- Cited by
- 24 results in Mathlib
- Foundations
- Depth 79 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.
Cites24
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- 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
- Finsuppstatement and proof · cited by 5,255
- Set.rangestatement and proof · cited by 4,705
- Submodule.spanstatement and proof · cited by 1,504
- one_smulproof · cited by 1,374
- Finsupp.singleproof · cited by 943
- LinearMap.rangestatement and proof · cited by 893
Cited by24
Results whose statement or proof uses this declaration.
- LinearMap.linearIndependent_iff_of_injOnproof · cited by 12
- Module.FinitePresentation.fg_kerproof · cited by 6
- Module.finitePresentation_of_surjectiveproof · cited by 6
- Module.FinitePresentation.exists_finproof · cited by 4
- Module.Basis.mem_submodule_iffproof · cited by 4
- Fintype.range_linearCombinationproof · cited by 4
- Module.projective_of_localization_maximalproof · cited by 3
- Module.FinitePresentation.exists_lift_of_isLocalizedModuleproof · cited by 3
- Module.Relations.range_mapproof · cited by 3
- Finsupp.span_eq_range_linearCombinationproof · cited by 3
- KaehlerDifferential.linearCombination_surjectiveproof · cited by 3
- Basis.le_span''proof · cited by 1