Is my act valid after I doubted the completeness of Al-Fatiha, reread it, and performed the prostration of forgetfulness?

If someone doubts whether they recited Al-Fatiha or any part of it, they must recite it, because it is a pillar [of prayer], and doubting the omission of a pillar is like omitting it, thus it is obligatory to perform it. As for one who repeated Al-Fatiha or doubts having repeated it, some jurists deem prostration of forgetfulness (sujud al-sahw) permissible for them. The addition of verbal elements in prayer does not necessitate sujud al-sahw unless the verbal element is an obligatory one. Therefore, repeating Al-Fatiha out of forgetfulness warrants sujud al-sahw, whereas repeating it intentionally does not require prostration, and the more valid opinion is that the prayer is not invalidated, though sin is incurred.