# Mapping at Element is Supremum of Compact Elements implies Mapping at Element is Supremum that Way Below

## Theorem

Let $\struct{S, \vee_1, \wedge_1, \preceq_1}$ and $\struct{T, \vee_2, \wedge_2, \preceq_2}$ be complete lattices.

Let $f: S \to T$ be a mapping such that

- $\forall x \in S: \map f x = \sup \leftset {\map f w: w \in S \land w \preceq_1 x \land w}$ is compact$\rightset{}$

Then

- $\forall x \in S: \map f x = \sup \set{ \map f w: w \in S \land w \ll x}$

## Proof

Let $x \in S$.

Define $X = \leftset {\map f w: w \in S \land w \preceq_1 x \land w}$ is compact$\rightset{}$

Define $Y = \set { \map f w: w \in S \land w \ll x}$

We will prove that

- $X \subseteq Y$

Let $b \in X$.

By definition of $X$:

- $\exists w \in S: b = \map f w \land w \preceq_1 x \land w$ is compact.

By definition of compact element:

- $w \ll w$

By Preceding and Way Below implies Way Below:

- $w \ll x$

Thus by definition of $Y$:

- $b \in Y$

$\Box$

We will prove that

- $\map f x$ is upper bound for $Y$.

Let $b \in Y$.

By definition of $Y$:

- $\exists w \in S: b = \map f w \land w \ll x$

By Way Below implies Preceding:

- $w \preceq_1 x$

By Mapping at Element is Supremum implies Mapping is Increasing:

- $f$ is an increasing mapping.

Thus by definition of increasing mapping:

- $b \preceq_2 \map f x$

$\Box$

We will prove that

- $\forall b \in T: b$ is upper bound for $Y \implies \map f x \preceq_2 b$

Let $b \in T$ such that

- $b$ is upper bound for $Y$.

- $b$ is upper bound for $X$.

By assumption:

- $\map f x = \sup X$

Thus by definition of supremum:

- $\map f x \preceq_2 b$

$\Box$

Thus by definition of supremum:

- $\map f x = \sup Y$

$\blacksquare$

## Sources

- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott:
*A Compendium of Continuous Lattices*

- Mizar article WAYBEL17:26