Ordering on Mappings Implies Galois Connection

Theorem
Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets.

Let $g: S \to T$ and $d: T \to S$ be mappings such that:
 * $g$ and $d$ are increasing mappings

and
 * $d \circ g \preceq I_S$ and $I_T \precsim g \circ d$

Then
 * $\struct {g, d}$ is Galois connection.

where
 * $\preceq, \precsim$ denote the orderings on mappings
 * $I_S$ denotes the identity mapping of $S$
 * $\circ$ denotes the composition of mappings.

Proof
We will prove that:
 * $\forall s \in S, t \in T: t \precsim \map g s \iff \map d t \preceq s$

Let $s \in S, t \in T$.

First implication:

Let
 * $t \precsim \map g s$

By definition of increasing mapping:
 * $\map d t \preceq \map d {\map g s}$

By definition of ordering on mappings:
 * $\map {\paren {d \circ g} } s \preceq \map {I_S} s$

By definition of composition:
 * $\map d {\map g s} \preceq \map {I_S} s$

By definition of identity mapping:
 * $\map d {\map g s} \preceq s$

Thus by definition of transitivity:
 * $\map d t \preceq s$

Second implication:

Let
 * $\map d t \preceq s$

By definition of increasing mapping:
 * $\map g {\map d t} \precsim \map g s$

By definition of ordering on mappings:
 * $\map {I_T} t \precsim \map {\paren {g \circ d} } t$

By definition of composition:
 * $\map {I_T} t \precsim \map g {\map d t}$

By definition of identity mapping:
 * $t \precsim \map g {\map d t}$

Thus by definition of transitivity:
 * $t \precsim \map g s$

Thus by definition:
 * $\struct {g, d}$ is Galois connection.