Restriction of Operation Distributivity

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Theorem

Let $\left({S, *, \circ}\right)$ be an algebraic structure.

Let $T \subseteq S$.


If the operation $\circ$ is distributive over $*$ in $\left({S, *, \circ}\right)$, then it is also distributive over $*$ on a restriction $\left({T, * \restriction_T, \circ \restriction_T}\right)$.


Proof

\(\displaystyle \) \(\) \(\displaystyle T \subseteq S\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \) \(\) \(\displaystyle \forall a, b, c \in T: a, b, c \in S\) Definition of Subset
\(\displaystyle \leadsto \ \ \) \(\displaystyle \) \(\) \(\displaystyle a \circ_T \left({b *_T c}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle a \circ \left({b * c}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({a \circ b}\right) * \left({a \circ c}\right)\) as $\circ$ is distributive over $*$
\(\displaystyle \) \(=\) \(\displaystyle \left({a \circ_T b}\right) *_T \left({a \circ_T c}\right)\)


Similarly for $\left({a *_T b}\right) \circ_T c = \left({a \circ_T c}\right) *_T \left({b \circ_T c}\right)$.

$\blacksquare$