Restricted Dipper Relation is Congruence for Multiplication

Theorem
Let $m, n \in \N_{>0}$ be non-zero natural numbers.

Let $\RR^*_{m, n}$ be the restricted dipper relation on $\N$:


 * $\forall x, y \in \N_{>0}: x \mathrel {\RR^*_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y} \end {cases}$

Then $\RR^*_{m, n}$ is a congruence relation for multiplication.

Proof
By definition, $\RR^*_{m, n}$ is the restriction of the dipper relation $\RR_{m, n}$ to $\N_{>0}$.

We have from Dipper Relation is Congruence for Multiplication that $\RR_{m, n}$ is a congruence relation for multiplication.

The result follows from Restriction of Congruence Relation is Congruence.

Also see

 * Restricted Dipper Relation is Congruence for Addition