The 10th Mountain Division ceased to be after WWII for a time. Its members went off to civilian life and started ski resorts and boosted the skiing industry around the world, and especially the US. The 10th Infantry Division was reactivated in June of 1948. It was a training division, so soldiers were all just 'passing through'. The division was based out of Ft. Riley, Kansas and had lost its 'Mountain' designation, and thus the tab. Once the Korean War began in 1950, the role of the 10th as a training division became even more important. It was in 1954, after the Korean War/"Police Action" had ended, that the 10th Infantry Division was made into a combat division again.
The division was sent to Germany and remained there for four years, on call, should the Soviet Union ever attack Western Europe. The division was returned in 1958 to Ft. Benning, Georgia and was deactivated on June 14th of the same year for the second time.
The 10th Infantry Division did not 'do' mountains or snow. It was treated like any other flatland infantry division at the time. The specialized training was not given to its members, nor were its members from the WWII 10th Mountain Division. For all intents and purposes, this is an entirely different division than the one that served in WWII.