Structure Induced by Right Distributive Operation is Right 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 right distributive over $+$:

$\forall a, b, c \in S: \paren {b + c} \times a= \paren{b \times a} + \paren{c \times a}$

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

$\forall f, g, h \in T^S: \paren {g \oplus h} \otimes f = \paren{g \otimes f} \oplus \paren{h \otimes f}$

Proof

Let $f, g, h \in T^S$.

Then:

\(\ds \forall x \in S: \, \)

\(\ds \map {\paren{\paren {g \oplus h} \otimes f} } x\)

\(=\)

\(\ds \map {\paren{g \otimes h} } x \times \map f x\)

Definition of Pointwise Operation

\(\ds \)

\(=\)

\(\ds \paren{ \map g x + \map h x} \times \map f x\)

Definition of Pointwise Operation

\(\ds \)

\(=\)

\(\ds \paren{\map g x \times \map f x} + \paren{\map h x \times \map f x}\)

Definition of Right Distributive Operation

\(\ds \)

\(=\)

\(\ds \map {\paren{g \otimes f } } x + \map {\paren{h \otimes f} } x\)

Definition of Pointwise Operation

\(\ds \)

\(=\)

\(\ds \map {\paren{\paren{g \otimes f} \oplus \paren{h \otimes f} } } x\)

Definition of Pointwise Operation

From Equality of Mappings:

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

Since $f, g, h$ were arbitrary:

$\forall f, g, h \in T^S: \paren {g \oplus h} \otimes f = \paren{g \otimes f} \oplus \paren{h \otimes f}$

Hence $\otimes$ is right distributive over $\oplus$ by definition.

$\blacksquare$