The lifting-the-exponent lemma replaces a large power difference by a base valuation plus the exponent valuation. The point is to make the formal expression readable enough to audit line by line.
Start locally
The valuation \(v_p(n)\) counts factors of \(p\), and the metric \(|x|_p=p^{-v_p(x)}\) reverses the usual sense of size. Hensel lifting turns approximate roots modulo \(p\) into exact \(p\)-adic roots.
I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.
Compute before generalising
The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.
The global view
The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.
Edge conditions
A congruence root lifts uniquely only in the simple-root case. Multiple roots need stronger inequalities and may split, disappear, or lift nonuniquely.
The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.