Prove that for any monotone map \(P \xrightarrow{f} Q\):
if there exist \(a \lor b \in P\) and \(f(a) \lor f(b) \in Q\):
\(f(a) \lor_Q f(b) \leq f(a \lor_P b)\)
Let’s abbreviate \(f(a\ \lor_P\ b)\) as \(JF\) (join-first) and \(f(b)\ \lor_Q\ f(a)\) as \(JL\) (join-last)
This exercise is to show that \(JL \leq JF\)
The property of joins gives us, in \(P\), that \(a\ \leq\ (a \lor b)\) and \(b\ \leq\ (a \lor b)\)
Monotonicity then gives us, in \(Q\), that \(f(a) \leq JF\) and \(f(b) \leq JF\)
We also know from the property of joins, in \(Q\), that \(f(a) \leq JL\) and \(f(b) \leq JL\)
The only way that \(JF\) could be strictly smaller than \(JL\), given that both are \(\geq f(a)\) and \(\geq f(b)\) is for \(f(a) \leq JF < JL\) and \(f(b) \leq JF < JL\)
But, \(JL \in Q\) is the smallest thing (or equal to it) that is greater than \(f(a)\) and \(f(b)\), so this situation is not possible.