Composite of Ordered Ring Monomorphisms is Ordered Ring Monomorphism

Theorem
Let $\left({S_1, +_1, \circ_1, \preccurlyeq_1}\right), \left({S_2, +_2, \circ_2, \preccurlyeq_2}\right), \left({S_3, +_3, \circ_3, \preccurlyeq_3}\right)$ be ordered rings.

Let $\phi: S_1 \to S_2$ and $\psi: S_2 \to S_3$ be ordered ring monomorphisms.

Then the composite mapping $\psi \circ \phi: S_1 \to S_3$ is also an ordered ring monomorphism.

Proof
From Composite of Order Embeddings is Order Embedding, $\psi \circ \phi: \left({S_1, \preceq_1}\right) \to \left({S_3, \preceq_3}\right)$ is an order embedding.

From Composite of Monomorphisms is Monomorphism, $\psi \circ \phi$ is a monomorphism.

From Group Monomorphism preserves Groups, it follows that $\psi \circ \phi$ is a group monomorphism from $\left({S_1, +_1}\right)$ to $\left({S_3, +_3}\right)$.

From Semigroup Monomorphism preserves Semigroups, it follows that $\psi \circ \phi$ is a semigroup monomorphism from $\left({S_1, \circ_1}\right)$ to $\left({S_3, \circ_3}\right)$.

Hence the result.