Generally, yes, but once broken down you end up with a large number of items that individually each is obviously true, so you know the combined statement Is true, but you don't find out if it is saying what you think it is saying.

In combining the parts you have the correct answer to a question, but is it that question you want to know?

Consider a proof that in the future all people will be happy.

You can methodically show this to be true but at the same time inadvertently include a proof that the number of people in the future will be zero.

It doesn't make the claim wrong, it stays undoubtedly true. It's just not what you assume it means.