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Taking the lulav on the first day is a Torah obligation even outside of the Beis Hamikdash.

 Summary 

 Question: Where do we find a Torah obligation on the first day? 

 Response: From the following beraisa: 

 “Ulekachtem” is plural, it means everyone must take the 4 minim (not just the king or kohen gadol). “Lachem” means it must be your own. This the source for the law that one cannot fulfill his obligation with his fellowman’s lulav (33.2). “Bayom” - it must be taken even on Shabbos. “Harishon” - it must be taken on the first day even outside of the Beis Hamikdash (the possuk says that it should be taken on the first day, but also that we should rejoice for seven days. It is now explained that seven day is the mitzvah in the Beis Hamikdash, while for the rest of the world, the mitzvah is only on the first day). “Ha-rishon” - only the first day supersedes Shabbos.

 Challenge: Why do we need a possuk to permit lulav on Shabbos? It’s only a Rabbinic issue of muktzeh (33.5)!

 Rava: This beraisa permits machshirim [preparations] on Shabbos, following the Tanna of this beraisa: 

 Lulav and its machshirim supersede the Shabbos according to Rabbi Eliezer [who famously permits chopping wood to kindle a fire to fashion iron into a knife for the mitzvah of milah]. 

 Question: Rabbi Eliezer derives this law from the word “Bayom”, what do the Chachomim learn from this?

 Response: They learn from it that the lulav is taken only during the day and not at night.

 Question: How does Rabbi Eliezer know this?

 Response: From the possuk “seven days” - it means day and not night.

The Chachomim counter that “seven days” can also mean night, as in the mitzvah of sukkah. 

 Question: How do we know that the mitzvah of sukkah applies to both daytime and nighttime?

 Beraisa: “Sit in the sukkah seven days”, it means even nights. Perhaps it means only days? After all, we see that by lulav it only means days. On the other hand [the word ‘klach’ is like lech lecha], the same expression is also used for the Milu’im, which included nights. 

We are inclined to learn the law of sukkah - a mitzvah which extends throughout the day - from milu’im - which extends throughout the day, rather than from lulav, whose mitzvah can be fulfilled in a moment.

On the other hand, sukkah is more similar to lulav which is a mitzvah for all generations, as opposed to Milu’im which was a one time thing.

For this reason there’s a gezeirah shaveh: The word ‘teishvu’ is used regarding both sukkah and milu’im to teach that the mitzvah of sukkah is both daytime and nighttime.