If a denial of discharge is entered in a chapter 7 case as provided 11
usc, rule 4006 is the case considered dismissed after notices are
mailed as provided 11 usc, rule 2002 by the clerk? If so why? If not,
why not? please provide the law that applies.

I can't make sense of the question.

Sorry for the confusion.  let me restate my question.

A denial of discharge was entered in a chapter 7 case as provided under 11
usc, rule 4006.  Given the fact that this is the case, we are
wondering whether the case would be considered to be dismissed (i.e.
finished, ended) under 11 usc, rule 2002? If the case is considered to
be dismissed/ended, then could you provide a brief explanation as to
why it is considered to be, and if it is not considered
dismissed/ended then why not? Also, could you also provide the law
that applies.

You know I'm not sure.  Have you asked your attorney?  It was my
understanding that debts (that is individual debts) are either
discharged in bankruptcy or denied discharge.  And cases (that is the
bankruptcy case itself) are either dismissed or not.  Check out the
appropriate chapter in title 11.