Equivalence of Definitions of Bottom of Lattice

Theorem
Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.

Let $\bot$ be a bottom of $\struct {S, \vee, \wedge, \preceq}$.

Proof
By definition, $\bot$ is the smallest element of $S$ for all $a \in S$:


 * $\bot \preceq a$

By Ordering in terms of Join, this is equivalent to:


 * $a \vee \bot = a$

If this equality holds for all $a \in S$, then by definition $\bot$ is an identity for $\vee$.

The result follows.