Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Ideals

Theorem
Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice.

Then
 * $\mathscr S$ is meet-continuous
 * for every ideals $I, J$ in $\mathscr S$: $\paren {\sup I} \wedge \paren {\sup J} = \sup \set {i \wedge j: i \in I, j \in J}$

Sufficient Condition
Let $\mathscr S$ be meet-continuous.

Define $\II$, the set of all ideals in $\mathscr S$

Define a mapping $f: \II \to S$ such that
 * $\forall I \in \II: \map f I = \sup I$

By Meet-Continuous iff Ideal Supremum is Meet Preserving:
 * $f$ preserves meet.

Let $I, J \in \II$.

By definition of mapping preserves meet:
 * $f$ preserves the infimum of $\set {I, J}$

Thus

Necessary Condition
Assume that for every ideals $I, J$ in $\mathscr S$:
 * $\paren {\sup I} \wedge \paren {\sup J} = \sup \set {i \wedge j: i \in I, j \in J}$

By Meet of Suprema equals Supremum of Meet of Ideals implies Ideal Supremum is Meet Preserving:
 * $f$ preserves meet.

Thus by Meet-Continuous iff Ideal Supremum is Meet Preserving:
 * $\mathscr S$ is meet-continuous.