How Does White Win This Ending

The start is a bit artificial in design, but the play after that is fairly interesting and instructive.

5k2/4p1pp/4P1P1/8/8/8/1r6/4K2R w K - 0 1

1.0-0 Kg8
2.Rf8+ Kxf8
3.gxh7 Rb1+
4.Kg2 Rb2+
5.Kg3 Rb3+
6.Kg4 Rb4+
7.Kg5 Rh4
8.Kxh4 g5+
9.Kxg5 Kg7
10.h8=Q+ Kxh8
11.Kh6 Kg8
12.Kg6 Kf8
13.Kh7 Ke8
14.Kg8 Kd8
15.Kf7 Kc7
It's a winning position for white. It's was very easy for me. I saw the solution at once.