Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset

Theorem
Let $L = \struct {S, \preceq}$ be an algebric lattice.

Let $c: S \to S$ be a closure operator that preserves directed suprema.

Then:
 * $c \sqbrk {\map K L} \subseteq \map K {\struct {c \sqbrk S, \precsim} }$

where:
 * $\map K L$ denotes the compact subset of $L$
 * $c \sqbrk S$ denotes the image of $S$ under $c$
 * $\mathord \precsim = \mathord \preceq \cap \paren {c \sqbrk S \times \map c S}$

Proof
Let $x \in c \sqbrk {\map K L}$.

By definition of image of set:
 * $\exists y \in \map K L: x = \map c y$

and
 * $x \in c \sqbrk S$

By definition of compact subset:
 * $y$ is compact in $L$.

By definition of compact element:
 * $y \ll y$

where $\ll$ denotes the way below relation.

Define $P = \struct {c \sqbrk S, \precsim}$ as an ordered subset of $L$.

We will prove that:
 * for every directed subset $D$ of $c \sqbrk S$: $x \precsim \sup_P D \implies \exists d \in D: d \precsim x$

Let $D$ be a directed subset of $c \sqbrk S$.

By definition of ordered subset:
 * $D$ is directed in $L$.

By definition of algebraic ordered set:
 * $L$ is up-complete.

By definition of up-complete:
 * $D$ admits a supremum in $L$.

By definition of image of set:
 * $\sup_L D \in c \sqbrk S$

By Supremum in Ordered Subset:
 * $\sup_L D = \sup_P D$

By definition of ordered subset:
 * $x \preceq \sup_L D$

By definition of closure operator/inflationary:
 * $y \preceq x$

By definition of transitivity:
 * $y \preceq \sup_L D$

By definition of way below relation:
 * $\exists d \in D: y \preceq d$

By definition of subset:
 * $d \in c \sqbrk S$

By definition of closure operator/idempotent:
 * $d = \map c d$

By definition of closure operator/increasing:
 * $x = \map c y \preceq \map c d = d$

Thus by definition of ordered subset:
 * $\exists d \in D: x \precsim d$

By definition of way below relation:
 * $x \ll_P x$

By definition:
 * $x$ is a compact element in $P$.

Thus by definition of compact subset:
 * $x \in \map K P$