Way Below in Meet-Continuous Lattice

Theorem
Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be a meet-continuous bounded below lattice.

Let $x, y \in S$.

Then $x \ll y$
 * $\forall I \in \map {\operatorname {Ids} } {\mathscr S}: y = \sup I \implies x \in I$

where
 * $\ll$ denotes the way below relation,
 * $\map {\operatorname {Ids} } {\mathscr S}$ denotes the set of all ideals in $\mathscr S$.

Sufficient Condition
Let $x \ll y$

Let $I \in \map {\operatorname {Ids} } {\mathscr S}$ such that
 * $y = \sup I$

By definition of reflexivity:
 * $y \preceq \sup I$

By definition of meet-continuous:
 * $\mathscr S$ is up-complete.

Thus by Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal:
 * $x \in I$

Necessary Condition
Assume:
 * $\forall I \in \map {\operatorname {Ids} } {\mathscr S}: y = \sup I \implies x \in I$

We will prove that:
 * $\forall I \in \map {\operatorname {Ids} } {\mathscr S}: y \preceq \sup I \implies x \in I$

Let $I \in \map {\operatorname {Ids} } {\mathscr S}$ such that
 * $y \preceq \sup I$

By definition of ideal:
 * $I$ is directed and lower.

Define $Y := \set {y \wedge d: d \in I}$

By Set of Finite Suprema is Directed:
 * $\set {\sup A: A \in \map {\operatorname {Fin} } Y \land A \ne \O}$ is directed.

where $\map {\operatorname {Fin} } Y$ denotes the set of all finite subsets of $Y$.

By definition of up-complete:
 * $\set {\sup A: A \in \map {\operatorname {Fin} } Y \land A \ne \O}$ admits a supremum.

By definition of union:
 * $Y = \bigcup \map {\operatorname {Fin} } Y$

where $^\preceq$ denotes the lower closure of set.

By Lower Closure of Directed Subset is Ideal:
 * $\set {\sup A: A \in \map {\operatorname {Fin} } Y \land A \ne \O}^\preceq$ is an ideal in $\mathscr S$

By assumption:
 * $x \in \set {\sup A: A \in \map {\operatorname {Fin} } Y \land A \ne \O}^\preceq$

By definition of lower closure of set:
 * $\exists z \in \set {\sup A: A \in \map {\operatorname {Fin} } Y \land A \ne \O}: x \preceq z$

Then
 * $\exists A \in \map {\operatorname {Fin} } Y: z = \sup A \land A \ne \O$

We will prove as a sublemma that:
 * $A \subseteq I$

Let $a \in A$.

By definition of subset:
 * $a \in Y$

By definition of $Y$:
 * $\exists i \in I: a = y \wedge i$

By Meet Precedes Operands:
 * $a \preceq i$

Thus by definition of lower section:
 * $a \in I$

This ends the proof of sublemma.

By Directed iff Finite Subsets have Upper Bounds:
 * $\exists h \in I: \forall a \in A: a \preceq h$

By definition:
 * $h$ is upper bound for $A$

By definition of supremum:
 * $z = \sup A \preceq h$

By definition of transitivity:
 * $x \preceq h$

Thus by definition of lower section:
 * $x \in I$

Thus by Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal:
 * $x \ll y$