Theorems · Theorem · linear algebra
Submodule.mem_span_range_iff_exists_fun
∀ {α : Type u_1} {M : Type u_2} (R : Type u_3) [inst : Fintype α] [inst_1 : Semiring R] [inst_2 : AddCommMonoid M]
[inst_3 : Module R M] {v : α → M} {x : M}, x ∈ Submodule.span R (Set.range v) ↔ ∃ c, ∑ i, c i • v i = xAn element x lies in the span of v iff it can be written as sum ∑ cᵢ • vᵢ = x.
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Fintypestatement and proof · cited by 7,736
- Submodulestatement · cited by 7,192
- Finsuppproof · cited by 5,255
- Finset.sumstatement · cited by 5,195
- Set.rangestatement and proof · cited by 4,705
- Finset.univstatement and proof · cited by 3,473
- Finset.sum_congrproof · cited by 2,323
- Submodule.spanstatement and proof · cited by 1,504
- zero_smulproof · cited by 716
Cited by12
Results whose statement or proof uses this declaration.
- LinearMap.BilinForm.dualSubmodule_span_of_basisproof · cited by 7
- Ideal.mem_span_range_iff_exists_funproof · cited by 3
- Fintype.mem_span_image_iff_exists_funproof · cited by 2
- RootPairing.finrank_range_polarization_eq_finrank_span_corootproof · cited by 2
- RootPairing.range_polarizationIn_le_span_corootproof · cited by 1
- Localization.existsUnique_algebraMap_eq_of_span_eq_topproof · cited by 1
- Submodule.mem_span_iff_of_fintypeproof · cited by 1
- Submodule.mem_span_image_finset_iff_exists_funproof · cited by 1
- RootPairing.Base.eq_one_or_neg_one_of_mem_support_of_smul_mem_auxproof · cited by 1
- mem_span_of_iInf_ker_le_kerproof · cited by 1
- RootPairing.range_polarization_domRestrict_le_span_corootproof · cited by 0
- SimpleGraph.top_le_span_range_lapMatrix_ker_basis_auxproof · cited by 0