Equivalence of Definitions of Dipper Semigroup

Theorem
Let $m \in \N$ be a natural number.

Let $n \in \N_{>0}$ be a non-zero natural number.

Proof
We have established in Dipper Semigroup is Commutative Semigroup that $\struct {\N_{< \paren {m + n} }, \oplus_{m, n} }$ is a (commutative) semigroup.

Let $\phi_{m, n}$ be the canonical surjection from $\N$ onto $\map D {m, n}$.

It will be established that the restriction of $\phi_{m, n}$ to $\N_{< \paren {m + n} }$ is an isomorphism from the semigroup $\struct {\N_{< \paren {m + n} }, +_{m, n} }$ onto $\struct {\map D {m, n}, \oplus_{m, n} }$.

From Dipper Relation is Equivalence Relation we have that:
 * $\RR_{m, n}$ is an equivalence relation

By definition of canonical surjection:
 * $\phi_{m, n}: \N_{< \paren {m + n} } \to \map D {m, n}: \map {\phi_{m, n} } x = \eqclass x {\RR_{m, n} }$

By definition $\phi_{m, n}$ is indeed a surjection.

Then we have:

Hence $\phi_{m, n}$ is an injection.

Thus $\phi_{m, n}$ is by definition a bijection.

It remains to show that $\phi_{m, n}$ has the morphism property.

Let $x, y \in \N_{< \paren {m + n} }$ be arbitrary.

We have: