Theorems · Theorem · ring theory

NonUnitalSubring.mem_iSup_of_directed

∀ {R : Type u} [inst : NonUnitalNonAssocRing R] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → NonUnitalSubring R},
  Directed (fun x1 x2 => x1 ≤ x2) S → ∀ {x : R}, x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i

The underlying set of a non-empty directed Sup of NonUnitalSubrings is just a union of the NonUnitalSubrings. Note that this fails without the directedness assumption (the union of two NonUnitalSubrings is typically not a NonUnitalSubring)

Defined in

Mathlib.RingTheory.NonUnitalSubring.Basic

Cited by

2 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NonUnitalNonAssocRingNonempty