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:

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

if and only if:

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

if and only if:

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

if and only if:

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

if and only if:

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

That is:

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

$\blacksquare$