Etymology of German "Dienst" and "Dienstag"

This may be far-fetched, but I am now curious and must know. I looked into the background of the English and German names for the months and days of the week, which is really quite interesting for those who don't know yet. In any case, I got curious about the English "Tuesday" and the German "Dienstag". According to my sources, both come from the Norse god Týr/Tïw, the god of combat, war, and victory, much like the Roman god Mars (after whom Tuesday is named in many Romance languages, e.g. mardi, martedi, martes).

"Dienstag", however, also looks a lot like "Dienst Tag" to me, so I would like to know if the word "Dienst" also came from Týr/Tïw, which would mean the German word "Dienst" was originally not as general as it is today, but earlier associated with military service. Searches online for its etymology have so far been fruitless. So, folks with an etymology dictionary, can you tell me how much truth there is to this?

Thanks in advance!