Theorems · Theorem · linear algebra

Submodule.mem_iSup_of_directed

∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {ι : Sort u_9}
  [Nonempty ι] (S : ι → Submodule R M), Directed (fun x1 x2 => x1 ≤ x2) S → ∀ {x : M}, x ∈ iSup S ↔ ∃ i, x ∈ S i

Defined in

Mathlib.LinearAlgebra.Span.Defs

Cited by

7 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleNonempty