Hensel's Lemma/P-adic Integers

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 $\map {F'} X$ be the (formal) derivative of $F$.

Let $p\Z_p$ denote the principal ideal of $\Z_p$ generated by $p$.

For all $x,y \in \Z_p$, let:
 * $x \equiv y \pmod {p\Z_p}$

denote congruence modulo the principal ideal $p\Z_p$.

Let $\alpha_0 \in \Z_p$ be a $p$-adic integer:
 * $\map F {\alpha_0} \equiv 0 \pmod {p\Z_p}$
 * $\map {F'} {\alpha_0} \not\equiv 0 \pmod {p\Z_p}$

Then there exists a unique $\alpha \in \Z_p$:
 * $\alpha \equiv \alpha_0 \pmod {p\Z_p}$
 * $\map F {\alpha} = 0$

Lemma 1
Let:
 * $\alpha = \ds \sum_{n = 0}^\infty d_n p^n$

Lemma 2
From Characterization of Polynomial has Root in P-adic Integers:
 * $\map F \alpha = 0$