You have gone wrong nowhere. It is a very common misconception among freshmen. An implication is false only when the antecedent is true and the consequent is false. For instance, the sentence "if it rains tomorrow, I'll carry an umbrella" is false only if it DOES rain and you DON'T carry an umbrella. If it doesn't rain, you've spoken truth, whether you carry an umbrella or not. It's also true if it does rain and, as a man or woman of your word, you do bring along an umbrella.
So, all those cases where you've gotten a false truth value for the antecedent? They straightaway mean that the full sentence is true! You need only examine whether the consequent is also true in those two other cases, and you'll have proven indeed that the above sentence is a tautology.
I'm sure you didn't doubt that anyway - after all, the antecedent implies that r is true, which is sufficient to deduce the consequent.