Polynomials of Congruent Ring Elements are Congruent

Theorem
Let $R$ be a commutative ring with unity.

Let $I$ be an ideal of $R$.

Let

Let $x, y, m \in \Z$ be integers where $m \ne 0$.

Let:
 * $x \equiv y \pmod m$

where the notation indicates congruence modlo $m$.

Let $a_0, a_1, \ldots, a_r$ be integers.

Then:
 * $\ds \sum_{k \mathop = 0}^r a_k x^k \equiv \sum_{k \mathop = 0}^r a_k y^k \pmod m$

Proof
We have that:
 * $x \equiv y \pmod m$

From Congruence of Powers:
 * $x^k \equiv y^k \pmod m$

From Modulo Multiplication is Well-Defined:
 * $\forall k \in \set {0, 2, \ldots, r}: a_k x^k \equiv a_k y^k \pmod m$

The result follows from Modulo Addition is Well-Defined.