Mapping is Increasing implies Mapping at Infimum for Sequence Precedes Infimum for Composition of Mapping and Sequence

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Theorem

Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be complete lattices.

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

Let $\left({D, \precsim}\right)$ be a directed set.

Let $N: D \to S$ be a Moore-Smith sequence in $S$.

Let $j \in D$.


Then $f\left({\inf\left({N\left[{\precsim \left({j}\right)}\right]}\right)}\right) \preceq_2 \inf \left({\left({f \circ N}\right)\left[{\precsim \left({j}\right)}\right]}\right)$


Proof

By definitions of image of set and composition of mappings:

$f\left[{N\left[{\precsim \left({j}\right)}\right]}\right] = \left({f \circ N}\right)\left[{\precsim \left({j}\right)}\right]$

By definition of complete lattice:

$f\left[{N\left[{\precsim \left({j}\right)}\right]}\right]$ and $N\left[{\precsim \left({j}\right)}\right]$ admit infima.

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

$f\left({\inf\left({N\left[{\precsim \left({j}\right)}\right]}\right)}\right) \preceq_2 \inf \left({\left({f \circ N}\right)\left[{\precsim \left({j}\right)}\right]}\right)$

$\blacksquare$


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