Brouwerian Lattice iff Meet-Continuous and Distributive

Theorem
Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.

Then
 * $\mathscr S$ is a Brouwerian lattice


 * $\mathscr S$ is meet-continuous and distributive.

Sufficient Condition
Let $\mathscr S$ be a Brouwerian lattice.

By Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice:
 * for every $x \in S$, a mapping $f: S \to S$ if $\forall y \in S: f\left({y}\right) = x \wedge y$, then $f$ is lower adjoint of Galois connection

By Shift Mapping is Lower Adjoint iff Meet is Distributive with Supremum:
 * $\forall x \in S, X \subseteq S: x \wedge \sup X = \sup \left\{ {x \wedge y: y \in X}\right\}$

Thus by definition of complete lattice:
 * $\mathscr S$ is up-complete.

Thus
 * for every an element $x \in S$ and a directed subset $D$ of $S$: $x \wedge \sup D = \sup \left\{ {x \wedge d: d \in D}\right\}$

Thus by definition
 * $\mathscr S$ is meet-continuous.

Thus by Brouwerian Lattice is Distributive:
 * $\mathscr S$ is distributive.

Necessary Condition
Assume that
 * $\mathscr S$ is meet-continuous and distributive.

Let $x \in S$.

Define a mapping $f: S \to S$:
 * $\forall y \in S: f\left({y}\right) = x \wedge y$

By Meet-Continuous and Distributive implies Shift Mapping Preserves Finite Suprema:
 * $f$ preserves finite suprema.

By Meet-Continuous implies Shift Mapping Preserves Directed Suprema:
 * $f$ preserves directed suprema.

By Mapping Preserves Finite and Directed Suprema:
 * $f$ preserves all suprema.

By All Suprema Preserving Mapping is Lower Adjoint of Galois Connection:
 * $f$ is lower adjoint of Galois connection.

Thus by Brouwerian Lattice iff Shift Mapping is Lower Adjoint:
 * $\mathscr S$ is Brouwerian lattice.