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For an orthomorphism $A$ in a (real or complex) Banach lattice $X$ the formula $\lvert A\rvert x=\lvert Ax\rvert$ for $x\ge0$ can be obtained relatively quickly if one uses the fact that $X$ is order isomorphic to a space of continuous functions (observing that $A$ corresponds to a multiplication operator in that space). However, the proof of the latter requires the maximal ideal theorem which cannot be shown in ZF or ZF+DC. In the lack of an order isomorphism, the only proof known to me makes use of (a real or complex form of) Freudenthal's spectral theorem.