Hensel's Lemma/P-adic Integers/Lemma 4

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

Let $\map F X \in \Z_p \sqbrk X$ be a polynomial.

Let $\alpha_0 \in \Z_p$ be a $p$-adic integer:
 * $\map F {\alpha_0} \equiv 0 \pmod {p\Z_p}$

Let $T$ be the set of $p$-adic digits.

Let:
 * $S_0 = \set{\tuple{b_0} \subseteq T^1 : \map F {b_0} \equiv 0 \pmod{p\Z_p} \land b_0 \equiv \alpha_0 \pmod{p\Z_p}}$

Let $d_0$ be the first $p$-adic digit of the canonical expansion of $\alpha_0$.

Then:
 * $\tuple{d_0} \in S_0$

Proof
Let the $p$-adic expansion for $\alpha_0$ be:
 * $\alpha_0 = \ds \sum_{n = 0}^\infty d_n p^n$

We have:

From User:Leigh.Samphier/Sandbox/Polynomials of Congruent Ring Elements are Congruent:
 * $\map F {d_0} \equiv \map F {\alpha_0} \equiv 0 \pmod {p\Z_p}$

Hence:
 * $\tuple {d_0} \in S_0$