= Solution
Interpret the <normal-operator source condition> as the displayed intersection condition: there is $v\in U$ with $w=K^*Kv\in\partial J(u^\dagger)$. Fix any $\alpha>0$ and put $\bar u=u^\dagger+\alpha v$. The <subgradient optimality condition> for the artificial-data objective is
$$
0\in K^*(Ku^\dagger-K\bar u)+\alpha\partial J(u^\dagger).
$$
It holds because its first term equals $-\alpha K^*Kv=-\alpha w$. By <convexity>, this is sufficient for global minimality, proving the forward implication. Conversely, if $u^\dagger$ minimizes that objective, the same optimality condition gives
$$
w=\frac1\alpha K^*K(\bar u-u^\dagger)\in\partial J(u^\dagger).
$$
Taking $v=(\bar u-u^\dagger)/\alpha$ proves the intersection condition. Thus \b[the two conditions are equivalent when the source vector is allowed to be zero].
The printed additional requirement $v\ne0$ is not equivalent in general. For $K=I$ and $J\equiv0$, the displayed source condition holds, and choosing $\bar u=u^\dagger$ makes $u^\dagger$ the minimizer, but $\partial J(u^\dagger)=\{0\}$ forces $v=0$. A nonzero-source version needs an additional hypothesis, such as requiring $w\ne0$; it cannot be inferred from the printed intersection alone.
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