There will always be people who doubt you, who wouldn't want to work with you, who thought they could do better, who, let's face it, probably *could* do better. But they're not here now--you are. You have the seat, and you have the power to make things happen.
So shut out the noise, focus in on what you know you can achieve, focus in on what you know you need to achieve, and do the work to get it done.
Not to prove anyone wrong, not to show anyone they made a mistake in doubting you or not believing in you, but because you know you have it in you to achieve the goal and you want to prove it to yourself.
Shut out the noise and get it done.