Composite with Constant Mapping is Constant Mapping

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Theorem

Let $f_c: S \to T$ be the constant mapping defined as:

$\forall x \in S: \map {f_c} x = c$

where $c \in T$.


Then for all mappings $g: \Dom g \to S$:

$f_c \circ g$ is a constant mapping

and for all mappings $h: T \to \Cdm h$:

$h \circ f_c$ is a constant mapping

where:

$\Dom g$ denotes the domain of $g$
$\Cdm h$ denotes the codomain of $h$
$\circ$ denotes composition of mappings.


Proof

\(\ds \forall x \in \Dom g: \, \) \(\ds \map {\paren {f_c \circ g} } x\) \(=\) \(\ds \map {f_c} {\map g x}\) Definition of Composition of Mappings
\(\ds \) \(=\) \(\ds c\) Definition of Constant Mapping

$\Box$


\(\ds \forall x \in S: \, \) \(\ds \map {\paren {h \circ f_c} } x\) \(=\) \(\ds \map h {\map {f_c} x}\) Definition of Composition of Mappings
\(\ds \) \(=\) \(\ds \map h c\) Definition of Constant Mapping

As $c$ is constant, so is $\map h c$.

$\blacksquare$