Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice/Lemma 2

Lemma for Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice
Let $N = \set {x \in L: \exists m \in M: x \le m \vee a}$.

$N$ is an ideal in $L$.

Proof
Let:
 * $x \in N$
 * $y \in L$
 * $y \le x$

Then by the definition of $N$ there exists an $m \in M$ such that:
 * $x \le m \vee a$

Since $y \le x$, it follows that:
 * $y \le m \vee a$

so $y \in N$.

Let:
 * $x \in N$
 * $y \in N$

Then there exist $m_x$ and $m_y$ in $M$ such that:


 * $x \le m_x \vee a$
 * $y \le m_y \vee a$

Then:
 * $x \vee y \le \paren {m_x \vee a} \vee \paren {m_y \vee a} = \paren {m_x \vee m_y} \vee a$

But $m_x \vee m_y \in M$, so:


 * $x \vee y \in N$