A large bright star whose hot core has used most of its hydrogen is a?

Red giant.

The Word "most" in the question isn't quite accurate. There is still a lot of hydrogen remaining in a star when the amount of helium becomes a problem. Hydrogen in the star is fused into helium, and the helium is at least roughly analogous to the ash in a wood fire; it is what's left over from the hydrogen fusion, and too much gets in the way.

When the proportion of helium gets too high - above about 50% is enough - the helium begins to interfere with the hydrogen fusion process. Without enough new heat generated by fusion, the star begins to collapse under the influence of gravity. The compression increases the temperature, and when the pressure and temperature gets high enough, the helium "ash" in the core becomes helium "fuel" for the Red Giant phase.