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 |
$\blacksquare$