Characterization of Locale/Statement 3 Implies Statement 4

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Theorem

Let $L = \struct{S, \preceq}$ be an complete lattice satisfying the infinite join distributive law.

Then:

$L$ is a Heyting algebra

Proof

Let $a, b \in S$.

Let $a \to b = \sup \set{c \in S : a \wedge c \preceq b}$

We have:

\(\ds a \wedge \paren{a \to b}\)

\(=\)

\(\ds a \wedge \sup \set{c : a \wedge c \preceq b}\)

Definition of $a \to b$

\(\ds \)

\(=\)

\(\ds \sup \set{a \wedge c : a \wedge c \preceq b}\)

Infinite join distributive law

\(\ds \)

\(\preceq\)

\(\ds b\)

Definition of Supremum of Set

Hence:

$a \to b$ is the greatest element $c$ such that:

$c \wedge a \preceq b$

It follows that $a \to b$ is a relative psuedocomplement by definition.

Since $a, b$ were arbitrary, then:

$\forall a, b \in S : \exists a \to b \in S : a \to b$ is the greatest element $c \in L$ such that $a \wedge c \preceq b$

Hence $L$ is a Heyting algebra by definition.

$\blacksquare$