Ashura and misconception
One may refer to the hadith quoted in the Bukhari, ref. no. 971 and 970 to find out the background of Ashura.
It was narrated by Ibn Abbas (#971) that, after arriving in Medina Monawara, the Prophet (pbh) observed that Jews were fasting on this day. When asked about the reason it was said that Musa (Moses) (pbh) used to fast on this day as Bani Israel and Musa (pbh) were saved from Pharaoh on this day. Than the Prophet (pbh) said that Musa belonged more to us than the Jews. It was also mentioned (not in this Hadith) that 'Allah's messenger intended to fast if he lives till next year, two days, one day after or before the 10th Muharram to distinguish the Muslim fast from the Jews.
So Ashura was established much before the Great Imams were slain in Karbala.