Lemma. Let K be a field. A subset L of K is a subfield if and only if the following conditions are satisfied:
(a) $0 \in L$ and $1 \in L$;
(b) if $x \in L$ and $y \in L$ then $x + y \in $L and $xy \in L$;
(c) if $x \in L$ then $−x \in L$;
(d) if $x \in L$ \ {$0$} then $x^{−1} \in L$.
How do I go about proving this? I'm aware that this is a biconditional and that the proof strategy is of the form $P \rightarrow Q$ and $Q \rightarrow P$.
Why does showing L satisfy the above imply it is a field?
Thanks - greatly appreciated.