Composition of Ring Homomorphisms is Ring Homomorphism/Proof 2

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:


 * $\paren {\psi \bullet \phi}: \struct {R_1, +_1, \circ_1} \to \struct {R_3, +_3, \circ_3}$ is a homomorphism.

To prove the above is the case, we need to demonstrate that the morphism property is held by $+_1$ and $\circ_1$ under $\psi \bullet \phi$.

We take two elements $x, y \in R_1$, and put them through the following wringer with respect to $+_1$:

The same applies to $\circ_1$:

Disentangling the confusing and tortuous expressions above, we (eventually) see that this shows that the morphism property is indeed held by both $+_1$ and $\circ_1$ under $\psi \bullet \phi$.

Thus $\paren {\psi \bullet \phi}: \struct {R_1, +_1, \circ_1} \to \struct {R_3, +_3, \circ_3}$ is a homomorphism.