This should be most of the hard work needed to get a multiplication map on the completion of a topological ring. This is Bourbaki GT III.6.5 Theorem I: ℤ-bilinear continuous maps from dense sub-groups into a complete Hausdorff group extend by continuity. Note that Bourbaki assumes that E and F are also complete Hausdorff, but this is a typo.
This should be most of the hard work needed to get a multiplication map on the completion of a topological ring. This is Bourbaki GT III.6.5 Theorem I: ℤ-bilinear continuous maps from dense sub-groups into a complete Hausdorff group extend by continuity. Note that Bourbaki assumes that E and F are also complete Hausdorff, but this is a typo.