Is my brother's fast valid, given that he ate after the Fajr adhan from one of the channels, while no nearby mosque had called the adhan at the same time?

A fasting person must abstain from food and drink at the break of dawn, as Allah Almighty says: (And eat and drink until the white thread of dawn becomes distinct to you from the black thread [of night]). So, whoever is certain of the rising of the true dawn must abstain. He must also spit out any food in his mouth, otherwise his fast will be invalidated. As for one who is not certain of the dawn's rise, he may eat until he is certain. Similarly, if he knows that the muezzin calls the adhan before the time, or doubts it, he may eat until he is certain. However, it is more preferable to abstain as soon as one hears the adhan. The adhan on television or radio may be earlier or later than the adhan of the locality. So, there is no blame on your brother because he was not certain of the dawn's rise and did not hear an adhan close to him indicating its rise.