Split lines and sequentially number each fragment in QGIS

I have a line layer representing watermains that I need to split long lines into sections. Each line has a unique ID number, but upon splitting a line this attribute is no longer unique as the fragments each have the same attribute. I would like to be able to number each of the unique fragments sequentially so that we can identify them as originally belonging to a larger parent, but still be unique. Upon splitting, each line breaks into different number of smaller sections.

For example

• Line 1 becomes lines 1.1, 1.2, 1.3
• Line 2 becomes lines 2.1, 2.2, 2.3…..2.47
• etc

I can see how to number the whole of the data set, but not number individual subsets. Having the number as a new attribute would be acceptable.