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

Theorem
Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be complete lattices.

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

Let $\struct {D, \precsim}$ be a directed set.

Let $N: D \to S$ be a net in $S$.

Let $j \in D$.

Then $\map f {\map \inf {N \sqbrk {\map \precsim j} } } \preceq_2 \map \inf {\paren {f \circ N} \sqbrk {\map \precsim j} }$

Proof
By definitions of image of set and composition of mappings:
 * $f \sqbrk {N \sqbrk {\map \precsim j} } = \paren {f \circ N} \sqbrk {\map \precsim j}$

By definition of complete lattice:
 * $f \sqbrk {N \sqbrk {\map \precsim j} }$ and $N \sqbrk {\map \precsim j}$ admit infima.

Thus by Subset and Image Admit Infima and Mapping is Increasing implies Infimum of Image Succeeds Mapping at Infimum:
 * $\map f{\map \inf {N \sqbrk {\map \precsim j} } } \preceq_2 \map \inf {\paren {f \circ N} \sqbrk {\map \precsim j} }$