Let $(L,\le,\bot,\top)$ be a bounded lattice and $\neg: L \rightarrow L$ be a map that satisfies the following laws:
- $a \wedge b = \bot \Leftrightarrow a \le \neg b$
- $\neg\neg a =a$
I'd like to show distributivity, namely, $a\wedge(b\vee c) = (a \wedge b) \vee (a \wedge c)$ and $a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$.
I've already shown the following laws:
- $a \wedge \neg a = \bot$, $a \vee \neg a = \top$
- The de Morgan's laws: $\neg (a \wedge b) = \neg a \vee \neg b$ and $ \neg (a \vee b) = \neg a \wedge \neg b$
I've also shown the distributive inequalities that hold for all lattices, and that the two forms of distributivity are equivalent. How should I use these laws to show what I'd like to?