Can a batter carry the bat to first base?

Can a batter carry the bat to first base? Any batter-runner who carries the bat during a live ball and legally reaches or touches 1st base while still holding the bat, will be declared out. Should this be the 3rd out of the inning, no preceding runner shall score.

Any batter-runner who carries the bat during a live ball and legally reaches or touches 1st base while still holding the bat, will be declared out. Should this be the 3rd out of the inning, no preceding runner shall score.