Polynomials of Congruent Ring Elements are Congruent

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

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

Let $x, y \in R$.

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

where the notation indicates congruence modulo $I$.

Let $\map F X \in R \sqbrk X$ be a polynomial in one variable over $R$.

Then:
 * $\ds \map F x \equiv \map F y \pmod I$

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.