Intersection of Division Subrings is Division Subring

Theorem
Let $\struct {D, +, \circ}$ be a division ring.

Let $\mathbb K$ be a non-empty set of division subrings of $D$.

Then the intersection $\bigcap \mathbb K$ of the members of $\mathbb K$ is itself a division subring of $D$.

Proof
Let $L = \bigcap \mathbb K$.

By Intersection of Subgroups is Subgroup, $\struct {L, +}$ is a subgroup of $\struct {D, +}$.

By the One-Step Subgroup Test:
 * $\forall x, y \in \struct {L, +}: x + \paren {-y} \in L$

By Intersection of Subgroups is Subgroup, $\struct {L, \circ}$ a subgroup of $\struct {D, \circ}$.

By the Two-Step Subgroup Test:
 * $\forall x, y \in \struct {L, \circ}: x \circ y \in L$
 * $\forall x \in \struct {L, \circ}: x^{-1} \in L$

By the Division Subring Test it follows that $\struct {L, +, \circ}$ is a division subring of $\struct {D, +, \circ}$.