Theorems · Theorem · commutative algebra

Subring.mem_iSup_of_directed

∀ {R : Type u} [inst : NonAssocRing R] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subring 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 sSup of subrings is just a union of the subrings. Note that this fails without the directedness assumption (the union of two subrings is typically not a subring)

Defined in

Mathlib.Algebra.Ring.Subring.Basic

Cited by

3 results in Mathlib

Foundations

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

Assumes

NonAssocRingNonempty