Theorems · Theorem · linear algebra

LinearIndependent.eq_zero_of_smul_mem_span

∀ {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [inst : Semiring R] [inst_1 : AddCommMonoid M]
  [inst_2 : Module R M],
  LinearIndependent R v → ∀ (i : ι) (a : R), a • v i ∈ Submodule.span R (v '' (Set.univ \ {i})) → a = 0

Defined in: Mathlib.LinearAlgebra.LinearIndependent.Defs
Cited by: 3 results in Mathlib
Foundations: Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring AddCommMonoid Module

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Cites23

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 · 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
SetLike.coeproof · cited by 8,199
Submodulestatement and proof · cited by 7,192
Set.imagestatement and proof · cited by 5,609
Finsuppproof · cited by 5,255
Set.univstatement and proof · cited by 3,945
Submodule.spanstatement and proof · cited by 1,504
Finsupp.singleproof · cited by 943

Cited by3

Results whose statement or proof uses this declaration.

linearIndependent_iff_eq_zero_of_smul_mem_spanproof · cited by 2
LinearIndepOn.eq_zero_of_smul_mem_spanproof · cited by 1
LinearIndependent.notMem_spanproof · cited by 0