Theorems · Theorem · linear algebra

iSupIndep_iff_forall_dfinsupp

∀ {ι : Type u_1} {R : Type u_3} {N : Type u_6} [inst : DecidableEq ι] [inst_1 : Semiring R] [inst_2 : AddCommMonoid N]
  [inst_3 : Module R N] (p : ι → Submodule R N),
  iSupIndep p ↔
    ∀ (i : ι) (x : ↥(p i)) (v : Π₀ (i : ι), ↥(p i)),
      ((DFinsupp.lsum ℕ) fun i => (p i).subtype) (DFinsupp.erase i v) = ↑x → x = 0

Independence of a family of submodules can be expressed as a quantifier over DFinsupps. This is an intermediate result used to prove iSupIndep_of_dfinsupp_lsum_injective and iSupIndep.dfinsupp_lsum_injective.

Defined in

Mathlib.LinearAlgebra.DFinsupp

Cited by

2 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqSemiringAddCommMonoidModule