The subcompleteness of diagonal Prikry forcing

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Let $$D$$ be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in $$D$$ is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in $$D$$, called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to $$D$$ is subcomplete above $$\mu $$, where $$\mu $$ is any regular cardinal below the first limit point of $$D$$.