Theorems · Theorem · linear algebra
Finsupp.span_image_eq_map_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} (s : Set α),
Submodule.span R (v '' s) = Submodule.map (Finsupp.linearCombination R v) (Finsupp.supported R R s)- Cited by
- 7 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.
Cites29
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · 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
- Set.imagestatement and proof · cited by 5,609
- Finsuppstatement and proof · cited by 5,255
- Submodule.spanstatement and proof · cited by 1,504
- one_smulproof · cited by 1,374
- Finsupp.singleproof · cited by 943
Cited by7
Results whose statement or proof uses this declaration.
- Finsupp.mem_span_image_iff_linearCombinationproof · cited by 17
- LinearIndependent.eq_zero_of_smul_mem_spanproof · cited by 3
- Module.Basis.constr_rangeproof · cited by 2
- Submodule.range_ker_disjointproof · cited by 2
- Submodule.codisjoint_span_image_of_codisjointproof · cited by 1
- LinearIndependent.linearCombination_ne_of_notMem_supportproof · cited by 1
- Finsupp.linearCombinationOn_rangeproof · cited by 0