Why is this King and Pawn ending lost for White? [FEN "8/8/8/3pk3/1K6/8/8/8 w - - 0 1"]


Even with White to move, the game is lost. OK, but according to what rule/principle?

With 1. Kc3 


I'm able to get first to one of the key squares (c3,d3,e3)
I get the diagonal opposition with the black King
I can occupy the black's promotion square before the black's King


In spite of all that, the ending is lost. Is it because I can't keep the oposition?

p.s.
The point of all this is to be able to evaluate this basic position only by applying rules, not having to calculate the concrete variations.
If you want to evaluate this ending with general rules, the rule of key squares is a good way to deal with it. However, this rule doesn't say "whoever reaches a key square first will get the result he is aiming for", it says "if the offensive king (here, Black) reaches one of the key squares, he will win."

Here after 1.Kc3 Ke4 2.Kd2 Kd4 you cannot prevent Black's king to reach one of the three key squares, so you lose.

[FEN "8/8/8/3pk3/1K6/8/8/8 w - - 0 1"]

1.Kc3 Ke4 2.Kd2 Kd4 3. Kc2 (3. Ke2 Kc3) Ke3