Ideals form Arithmetic Lattice

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below lattice.

Let $I = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an inclusion ordered set

where
 * $\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$,
 * $\mathord\precsim = \mathord\subseteq \cap \left({\mathit{Ids}\left({L}\right) \times \mathit{Ids}\left({L}\right)}\right)$

Then $I$ is an arithmetic lattice.

Proof
We will prove that
 * $\forall x, y \in K\left({I}\right): \exists z \in K\left({I}\right): z \precsim x \land z \precsim y \land \forall v \in K\left({I}\right): v \precsim x \land v \precsim y \implies v \precsim z$

By definition:
 * $K\left({I}\right)$ form a meet semilattice.

By Compact Subset is Join Subsemilattice:
 * $K\left({I}\right)$ form a join semilattice.

By definition:
 * $K\left({I}\right)$ form a lattice.

By Ideals form Algebraic Lattice:
 * $I$ is an algebraic lattice.

Thus by Arithmetic iff Compact Subset form Lattice in Algebraic Lattice:
 * $I$ is an arithmetic lattice.