Infima Inheriting Ordered Subset of Complete Lattice is Complete Lattice

Theorem

Let $L = \struct {X, \preceq}$ be a complete lattice.

Let $S = \struct {T, \precsim}$ be an infima inheriting ordered subset of $L$.

Then $S$ is a complete lattice.

Proof

Let $A$ be subset of $T$.

By definition of complete lattice:

$A$ admits an infimum in $L$.

Thus by definition of infima inheriting:

$A$ admits an infimum in $S$.

Hence by dual of Lattice is Complete iff it Admits All Suprema:

$S$ is a complete lattice.

$\blacksquare$

Sources

• Mizar article YELLOW_2:30