Composite of Homomorphisms is Homomorphism

Theorem
Let: be algebraic structures.
 * $\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$
 * $\left({S_2, *_1, *_2, \ldots, *_n}\right)$
 * $\left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$

Let: be homomorphisms.
 * $\phi: \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S_2, *_1, *_2, \ldots, *_n}\right)$
 * $\psi: \left({S_2, *_1, *_2, \ldots, *_n}\right) \to \left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$

Then the composite of $\phi$ and $\psi$ is also a homomorphism.

R-Algebraic Structures
Let: be $R$-algebraic structures with the same number of operations.
 * $\left({S_1, *_1}\right)_R$
 * $\left({S_2, *_2}\right)_R$
 * $\left({S_3, *_3}\right)_R$

be homomorphisms.
 * $\phi: \left({S_1, *_1}\right)_R \to \left({S_2, *_2}\right)_R$
 * $\psi: \left({S_2, *_2}\right)_R \to \left({S_3, *_3}\right)_R$

Then the composite of $\phi$ and $\psi$ is also a homomorphism.

Proof
So as to alleviate possible confusion over notation, let the composite of $\phi$ and $\psi$ be denoted $\psi \bullet \phi$ instead of the more usual $\psi \circ \phi$.

Then what we are trying to prove is denoted:


 * $\left({\psi \bullet \phi}\right): \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$ is a homomorphism.

To prove homomomorphism, we need to demonstrate that the morphism property is held by each of the operations $\circ_1, \circ_2, \ldots, \circ_n$ under $\psi \bullet \phi$.

Let $\circ_k$ be one of these operations.


 * We take two elements $x, y \in S_1$, and put them through the following wringer:

Disentangling the confusing and tortuous expressions above, we (eventually) see that this shows that the morphism property is indeed held by $\circ_k$ under $\psi \bullet \phi$.

As this holds for any arbitrary operation $\circ_k$ in $\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$, it follows that it holds for all of them.

Thus $\left({\psi \bullet \phi}\right): \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$ is indeed a homomorphism.

R-Algebraic Structure definition

 * : $\S 28$: Theorem $28.1$