Are subgraphs of a minimum spanning tree, also minimum spanning trees? Clarification: MST = minimum spanning tree
Say I take some complete, weighted graph $G$ - I create a MST of the graph then split $G$ into 2 trees $T_1$ and $T_2$ by removing some edge connecting them called $uv$. My question is, that if I were to take the subgraph of all nodes constuting $T_1$ and $T_2$ and the edges originally connecting them in $G$, would the MSTs of these subgraphs be equivalent to the original MSTs $T_1$ and $T_2$? That is, are the subgraphs of a MST equal to the MSTs of the subgraphs?

The answer is yes; if the subgraphs created by taking the nodes and edges from $T_1$ and $T_2$ had a smaller weight than the subgraphs of the minimum spanning tree, and were combined with the original graph by adding the edge $uv$ back, the resulting graph would be a minimum spanning tree of $G$ but with a smaller weight than the original minimum spanning tree created for $G$.
This leads to a contradiction as the definition of a minimum spanning tree is such that the sum of the weights of the edges used to construct it are a minimum, hence, a smaller minimum spanning tree for $G$ cannot exist - therefore, $T_1$ and $T_2$ have to have already been minimum spanning trees as the sum of the weights of the edges in them cannot be decreased and they are already connected trees - hence fitting the definition of a minimum spanning tree.
