Subset and Image Admit Infima and Mapping is Increasing implies Infimum of Image Succeeds Mapping at Infimum

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Theorem

Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets.

Let $f: S \to T$ be a increasing mapping.

Let $D \subseteq S$ such that

$D$ admits a infimum in $S$ and $f\left[{D}\right]$ admits a infimum in $T$.


Then $f\left({\inf D}\right) \precsim \inf \left({f\left[{D}\right]}\right)$


Proof

By definition of infimum:

$\inf D$ is lower bound for $D$.

By Increasing Mapping Preserves Lower Bounds:

$f\left({\inf D}\right)$ is lower bound for $f\left[{D}\right]$.

Thus by definition of infimum:

$f\left({\inf D}\right) \precsim \inf \left({f\left[{D}\right]}\right)$

$\blacksquare$


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