Translation of Intersection of Subsets of Vector Space

Theorem
Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $\family {E_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of $X$.

Let $x \in X$.

Then:
 * $\ds \paren {\bigcap_{\alpha \mathop \in I} E_\alpha} + x = \bigcap_{\alpha \mathop \in I} \paren {E_\alpha + x}$

where $E_\alpha + x$ denotes the translation of $E_\alpha$ by $x$.

Proof
Let $v \in X$.

We have:
 * $\ds v \in \paren {\bigcap_{\alpha \mathop \in I} E_\alpha} + x$


 * $\ds v - x \in \bigcap_{\alpha \mathop \in I} E_\alpha$
 * $\ds v - x \in \bigcap_{\alpha \mathop \in I} E_\alpha$


 * $v - x \in E_\alpha$ for each $\alpha \in I$
 * $v - x \in E_\alpha$ for each $\alpha \in I$


 * $v \in E_\alpha + x$ for each $\alpha \in I$
 * $v \in E_\alpha + x$ for each $\alpha \in I$


 * $\ds v \in \bigcap_{\alpha \mathop \in I} \paren {E_\alpha + x}$
 * $\ds v \in \bigcap_{\alpha \mathop \in I} \paren {E_\alpha + x}$

That is:
 * $\ds \paren {\bigcap_{\alpha \mathop \in I} E_\alpha} + x = \bigcap_{\alpha \mathop \in I} \paren {E_\alpha + x}$