Adding new row to dataframe with group by and summarise

I want to group my df by three columns and add a new row which will be a sum of a fourth column.

My data looks like

fc <- c("F", "F", "E", "E", "TF", "TF")
group_code <- c("Egg_x", "Egg_y", "Egg_x", "Egg_y", "Egg_x", "Egg_y")
id <- c(1, 1, 1, 1, 1, 1)
value <- c(2, 21, 4, 3, 20, 15)

df <-data.frame(cbind(fc, group_code, id, value))

> df
  fc group_code id value
1  F      Egg_x  1     2
2  F      Egg_y  1    21
3  E      Egg_x  1     4
4  E      Egg_y  1     3
5 TF      Egg_x  1    20
6 TF      Egg_y  1    15

In this example I want to create a new group which will include both Egg_x and Egg_y, I can do this using df$group <- sub('\\_.*', '', df$group_code), we have

> df
  fc group_code id value main_group
1  F      Egg_x  1     2  Egg
2  F      Egg_y  1    21  Egg
3  E      Egg_x  1     4  Egg
4  E      Egg_y  1     3  Egg
5 TF      Egg_x  1    20  Egg
6 TF      Egg_y  1    15  Egg

I want to add in a new row for each value of the fc column wehreby I group on fc, id and main_group and get the sum of the value column.

My end df should look like:

> df
  fc group_code id value main_group
1  F      Egg_x  1     2  Egg
2  F      Egg_y  1    21  Egg
3  F      Egg    1    23  Egg
4  E      Egg_x  1     4  Egg
5  E      Egg_y  1     3  Egg
6  E      Egg    1     7  Egg
7 TF      Egg_x  1    20  Egg
8 TF      Egg_y  1    15  Egg
9 TF      Egg    1    35  Egg 

In the above df every third row the value element in a sum of the previous two elements.

Thanks