Is the prayer of one who joins the Imam during the first Tashahhud valid if he then doubts whether he pronounced the opening Takbir (Takbirat al-Ihram) or not?

The opening Takbir (Takbirat al-Ihram) is a pillar (rukn) of prayer; it is not waived by forgetfulness or ignorance. Whoever doubts having performed it during the prayer must restart the prayer. However, doubt after finishing the prayer has no effect, because doubt after an act of worship is not taken into consideration. Whoever forgets the opening Takbir or doubts it while praying must say the Takbir immediately and act upon what he has caught up with after the Takbir.