Theorems · Theorem · linear algebra

Submodule.mem_sSup_of_directed

∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
  {s : Set (Submodule R M)} {z : M}, s.Nonempty → DirectedOn (fun x1 x2 => x1 ≤ x2) s → (z ∈ sSup s ↔ ∃ y ∈ s, z ∈ y)

Defined in

Mathlib.LinearAlgebra.Span.Defs

Cited by

6 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModule