Structure Induced by Distributive Operation is Distributive

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Theorem

Let $\struct {T, +, \times}$ be an algebraic structure, and let $S$ be a set.

Let $\struct {T^S, \oplus, \otimes}$ be the structure on $T^S$ induced by $+$ and $\times$.

Let $\times$ be distributive over $+$:

$\forall a, b, c \in S$:

$a \times \paren {b + c} = \paren{a \times b} + \paren{a \times c}$

and:

$\paren {b + c} \times a= \paren{b \times a} + \paren{c \times a}$

Then the pointwise operation $\otimes$ is distributive over the pointwise operation $\oplus$ on $T^S$:

$\forall f, g, h \in T^S$:

$f \otimes \paren {g \oplus h} = \paren{f \otimes g} \oplus \paren{f \otimes h}$

and:

$\paren {g \oplus h} \otimes f = \paren{g \otimes f} \oplus \paren{h \otimes f}$

Proof

By definition of distributive operation:

$\times$ is left distributive over $+$

and

$\times$ is right distributive over $+$

From Structure Induced by Left Distributive Operation is Left Distributive:

$\otimes$ is left distributive over $\oplus$

From Structure Induced by Right Distributive Operation is Right Distributive:

$\otimes$ is right distributive over $\oplus$

It follws that $\otimes$ is distributive over $\oplus$ by definition.

$\blacksquare$