Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice

Assertions
Then $M$ is a prime ideal.

Proof
$M$ is not a prime ideal.

Then by Prime Ideal in Lattice, there are elements $a$ and $b$ of $L$ such that


 * $a \wedge b \in M$
 * $a \notin M$
 * $b \notin M$

Lemma 1
, we can thus suppose that:


 * $\forall m \in M: m \vee a \notin F$

Let $N = \set {x \in L: \exists m \in M: x \le m \vee a}$.

Lemma 4
By assuming that $M$ is not a prime ideal, we have constructed an ideal $N$ properly containing $M$ that is disjoint from $F$.

This contradicts the maximality of $M$.

Thus $M$ is a prime ideal.