Congruence Modulo a Principal Ideal of P-adic Integers

Theorem
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Let $\Z_p$ be the $p$-adic integers.

For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map B a$ denote the open ball of center $a$ of radius $\epsilon$.

For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map { {B_\epsilon}^-} a$ denote the closed ball of center $a$ of radius $\epsilon$.

For any $a \in \Z_p$ and $n \in \N$ let $a + p^n\Z_p$ denote the coset of $a$ modulo $p^n\Z_p$.

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 $p^n\Z_p$.

Let $x, y \in \Z_p$.

Let $k \in \N$.


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


 * $(2)\quad x \in y + p^{k+1} \Z_p$


 * $(3)\quad x + p^{k+1} \Z_p = y + p^{k+1} \Z_p$


 * $(4)\quad x \in \map {B_{p^{-k}}} y$


 * $(5)\quad \norm{x - y}_p < p^{-k}$


 * $(6)\quad \map {B_{p^{-k}}} x = \map {B_{p^{-k}}} y$


 * $(7)\quad x \in \map {B^{\,-}_{p^{-k-1}}} y$


 * $(8)\quad \norm{x - y}_p \le p^{-k-1}$


 * $(9)\quad \map {B^{\,-}_{p^{-k-1}}} x = \map {B^{\,-}_{p^{-k-1}}} y$

Condition (1) is Equivalent to Condition (3)
We have:

Conditions (2), (3), (4), (5) and (6) are equivalent
Follows directly from Characterization of Open Ball in P-adic Numbers.

Conditions (2), (7), (8) and (9) are equivalent
Follows directly from Characterization of Closed Ball in P-adic Numbers.