If the offense recovers the fumble and advances it beyond the original first down line, yes, it is a first down. If the ball is not advanced past the original first down line then the next play is the down after the one that the fumble occurred on. For example, if a running back fumbles the ball on second down and an offensive line man recovers the fumble but does not advance the ball beyond the original first down line, it is then third down.
If the defense recovers the fumble, then it is an automatic first down wherever the player that recovered the fumble is downed.
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