Restriction of Homomorphism is Homomorphism

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Theorem

Let $\struct {S, \circ}$ and $\struct {T, \odot}$ be algebraic structures.

Let $\phi: S \to T$ be a homomorphism.

Let $A \subseteq S$ be a subset of $S$.


Then the restriction of $\phi$ to $A \times \Img A$ is also a homomorphism.


Proof

\(\ds \forall x, y \in A: \, \) \(\ds \map \phi {x \circ_A y}\) \(=\) \(\ds \map \phi {x \circ y}\) Definition of Restriction of Operation
\(\ds \) \(=\) \(\ds \map \phi x \odot \map \phi y\) as $x, y \in S$
\(\ds \) \(=\) \(\ds \map \phi x \odot_{\Img A} \map \phi y\) Definition of Restriction of Operation

$\blacksquare$