Congruence Modulo Equivalence for Integers in P-adic Integers

Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.

For any $a, b \in \Z_p$ and $n \in \N$, let $x \equiv y \pmod{p^n \Z_p}$ denote congruence modulo the principal ideal $p^n\Z_p$.

For any integers $a, b \in \Z$ and $n \in \N$, let $x \equiv y \pmod{p^n}$ denote congruence modulo integer $p^n$.

Let $x, y \in \Z$ be integers.

Let $k \in \N_{>0}$.


 * $(1)\quad x \equiv y \pmod{p^k \Z_p}$


 * $(2)\quad x \equiv y \pmod{p^k}$


 * $(3)\quad p^k \divides \paren{x - y}$

Lemma
We have: