Infimum of Image of Upper Closure of Element under Increasing Mapping
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Theorem
Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered set.
Let $f: S \to T$ be an increasing mapping.
Let $x \in S$.
Then $\map \inf {f \sqbrk {x^\succeq} } = \map f x$
Proof
By Infimum of Upper Closure of Element:
- $\inf x^\succeq = x$
By definition of infimum:
- $x$ is lower bound for $x^\succeq$
Thus by Increasing Mapping Preserves Lower Bounds:
- $\map f x$ is lower bound for $f \sqbrk {x^\succeq}$
By definition of reflexivity:
- $x \preceq x$
By definition of upper closure of element:
- $x \in x^\succeq$
By definition of image of set:
- $\map f x \in f \sqbrk {x^\succeq}$
Thus by definition:
- $\forall y \in T: y$ is lower bound for $f \sqbrk {x^\succeq} \implies y \precsim \map f x$
Thus by definition of infimum:
- $\map \inf {f \sqbrk {x^\succeq} } = \map f x$
$\blacksquare$
Sources
- Mizar article WAYBEL_9:6