Let $X$ be any normed vector lattice.
How to show that $\| x \| = \| |x| \|$ holds for all $x \in X$?
I thought its proof is trival. Because from the property of a lattice norm $\| \cdot \|$, we have $\|x\| \leq \|y\|$ whenever $|x| \leq |y|$ in $X$. So, we could set $y := |x|$ and obviously get $ |x| \leq |y|$ (precisely, $|x| = |y|$). Then, using the property of a lattice norm gives us $\| x \| = \| y \| = \| |x| \|$.
Is it a correct proof of it? Thanks in advance!