Theorems · Theorem · linear algebra

LinearIndependent.notMem_span_image

∀ {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] [Nontrivial R],
  LinearIndependent R v → ∀ {s : Set ι} {x : ι}, x ∉ s → v x ∉ Submodule.span R (v '' s)

Defined in: Mathlib.LinearAlgebra.LinearIndependent.Basic
Cited by: 3 results in Mathlib
Foundations: Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring AddCommMonoid Module Nontrivial

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Cites14

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement and proof · 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
Submodulestatement · cited by 7,192
Set.imagestatement and proof · cited by 5,609
Nontrivialstatement and proof · cited by 2,416
Submodule.spanstatement and proof · cited by 1,504
LinearIndependentstatement and proof · cited by 560
Set.image_singletonproof · cited by 174
Submodule.mem_span_singleton_selfproof · cited by 59
Submodule.disjoint_defproof · cited by 21

Cited by3

Results whose statement or proof uses this declaration.

InnerProductSpace.gramSchmidt_ne_zero_coeproof · cited by 2
LinearIndependent.linearCombination_ne_of_notMem_supportproof · cited by 1
RootPairing.Base.induction_on_cartanMatrixproof · cited by 0