JOEL - Yes they were the very same ONE Church. You might notice as you read in the Book of Mormon that when Jesus visited the Nephites that He called 12 "disciples" to lead His church there. They were not called "Apostles", even though they were given similar authority to act in the name of Jesus for the people living there. The Twelve Apostles that He chose to lead His Church on the eastern continent were the leaders of the Church for the entire earth. Here is what you might call an "official" statement from one of the Presidents of our Church today.
"While in every instance the Nephite twelve are spoken of as disciples, the fact remains that they had been endowed with divine authority to be special witnesses for Christ among their own people. Therefore, they were acting like apostles to the Nephite race although their jurisdiction was eventually to be subject to the authority and jurisdiction of Peter and the Twelve chosen in Palestine." (Joseph Fielding Smith)
There has only been one Church of Christ on the earth at a given moment in time. However, Jesus knew that a core of leadership would be needed in the western continent in order to establish His Gospel and Church there. Otherwise, those people would never be able to have access to the leadership of the presiding Twelve Apostles on the eastern continent.
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