Composition of Ring Homomorphisms is Ring Homomorphism/Proof 2

Proof
Let $\psi \circ \phi$ denote the composite of $\phi$ and $\psi$.

Then what we are trying to prove is denoted:


 * $\paren {\psi \circ \phi}: \struct {R_1, +_1, \odot_1} \to \struct {R_3, +_3, \odot_3}$ is a homomorphism.

To prove the above is the case, we need to demonstrate that the morphism property is held by $+_1$ and $\odot_1$ under $\psi \circ \phi$.

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

The same applies to $\odot_1$:

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 $\psi \circ \phi$.

Thus $\paren {\psi \circ \phi}: \struct {R_1, +_1, \odot_1} \to \struct {R_3, +_3, \odot_3}$ is a homomorphism.