Theorems · Theorem · field theory

Subfield.mem_iSup_of_directed

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

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

Defined in

Mathlib.Algebra.Field.Subfield.Basic

Cited by

2 results in Mathlib

Foundations

Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingNonempty