Does anyone know where I can find a proof of the Iwasawa decomposition for reductive groups? I know that there are a couple of related results that are called the Iwasawa decomposition, but I am interested in the following statement:

Let G be a complex reductive group, let O be Taylor series with complex coefficients, and let K be Laurent series with complex coefficients. The G(K) = G(O) * T(K) * U(K), where T is a maximal torus and U is a maximal unipotent subgroup of G.

I am interesting in finding the proof because this is how one shows that the semi-infinite cells in the affine Grassmannian cover the entire space. I have been using this fact for quite a while now and am becoming uncomfortable about not knowing where to find the proof. A proof in the case where K is a p-adic field and O is its ring of integers would also be great since I am sure a proof would carry over to the above case.