Pseudoprime Element is Prime in Arithmetic Lattice

Theorem

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below arithmetic lattice.

Let $p \in S$.

Then if $p$ is pseudoprime element, then $p$ is prime element.

Proof

By Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice:

$\ll$ is a multiplicative relation.

Thus by Way Below Relation is Multiplicative implies Pseudoprime Element is Prime:

the result holds.

$\blacksquare$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_8:21