Proving join and meet equality

Question

Prove that in any lattice $L$, we have \begin{equation} ( (x\wedge y) \vee (x \wedge z)) \wedge ( (x \wedge y) \vee (y \wedge z)) = x \wedge y\end{equation} for all $x, y, z \in L$.

I showed $(x\wedge y)\le (x\wedge y) \vee (x \wedge z)$ and $(x\wedge y)\le (x \wedge y) \vee (y \wedge z)$ to imply that $(x\wedge y)$ is a lower bound of the pair $(x\wedge y) \vee (x \wedge z)$ and $(x \wedge y) \vee (y \wedge z)$. But I cannot show that $(x\wedge y)$ is the glb of $(x\wedge y) \vee (x \wedge z)$ and $(x \wedge y) \vee (y \wedge z)$.

Answer 1

We have that $x \wedge y \leq x$ and $x \wedge z \leq x$, so $(x \wedge y) \vee (x \wedge z) \leq x$. Similarly we get $(x \wedge y) \vee (y \wedge z) \leq y$. Putting this together we get $$ ((x \wedge y) \vee (x \wedge z)) \wedge ((x \wedge y) \vee (y \wedge z)) \leq x \wedge y, $$ which together with the direction you proved completes the argument.