Theorems · Theorem · linear algebra
Finsupp.mem_span_image_iff_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} {x : M},
x ∈ Submodule.span R (v '' s) ↔ ∃ l ∈ Finsupp.supported R R s, (Finsupp.linearCombination R v) l = x- Cited by
- 17 results in Mathlib
- Foundations
- Depth 80 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.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · 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
- LinearMapstatement · cited by 10,215
- Submodulestatement and proof · cited by 7,192
- Set.imagestatement · cited by 5,609
- Finsuppstatement and proof · cited by 5,255
- Submodule.spanstatement · cited by 1,504
- Finsupp.linearCombinationstatement and proof · cited by 269
Cited by17
Results whose statement or proof uses this declaration.
- Submodule.fg_of_fg_map_of_fg_inf_kerproof · cited by 7
- Submodule.mem_span_setproof · cited by 6
- Submodule.mem_span_iff_exists_finset_subsetproof · cited by 3
- maximal_orthonormal_iff_orthogonalComplement_eq_botproof · cited by 3
- Module.Basis.mem_span_repr_supportproof · cited by 3
- linearIndependent_iff_eq_zero_of_smul_mem_spanproof · cited by 2
- weightedVSub_mem_vectorSpanproof · cited by 2
- Module.Basis.repr_support_subset_of_mem_spanproof · cited by 2
- Submodule.mem_span_image_iff_exists_funproof · cited by 1
- prod_mem_ideal_map_of_mem_conductorproof · cited by 1
- AlgebraicClosure.spanCoeffs_ne_topproof · cited by 1
- FixedPoints.linearIndependent_smul_of_linearIndependentproof · cited by 1