"مَدينة لوس أَنْجِلِس قَريبة مِن اَلْجَبَل."
Translation:Los Angeles city is close to the mountains.
Nope, because "Madinah" here is connected to (Los Angeles), in other words, there is a Genitive relation between the two. Unlike English where the definite article goes on "city" in such case, in Arabic the definite article would go to the second term in the Genitive relation. However, here we have a proper name of a city thus it is defined by itself and AL is not added.
Let's take another example, if I want to say (the city of angels). Close enough. Then in Arabic it would be (مدينة الملائكة) [madínat-ul-Mala'ikah], the definite article (AL) is merged onto the second term (Mala'ikah: angels). As for why AL is UL here, it's just how the vowel harmony works where the vowel (U) at the end of the first word is merged to the (AL) on the second word to make a non-stopping speech.