Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice/Lemma 1

Lemma for Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice
There do not exist $m$ and $n$ in $M$ such that $m \vee a \in F$ and $n \vee b \in F$.

Proof
such exist.

Since:
 * $m \vee \paren {n \vee b} \ge n \vee b$
 * $n \vee b \in F$
 * $F$ is a filter

it follows that:
 * $m \vee \paren {n \vee b} \in F$

Applying associativity yields:


 * $\paren {m \vee n} \vee b \in F$

By the same argument:


 * $\paren {m \vee n} \vee a \in F$

By the definition of a filter:


 * $\paren {\paren {m \vee n} \vee b} \wedge \paren {\paren {n \vee m} \vee a} \in F$

Distributing:


 * $\paren {m \vee n} \wedge \paren {b \vee a} \in F$

But by assumption:
 * $b \vee a \in M$

and by the definition of an ideal:
 * $m \vee n \in M$

so again by the definition of an ideal:


 * $\paren {m \vee n} \wedge \paren {b \vee a} \in M$

contradicting the supposition that $M$ is disjoint from $F$.