Is it permissible to say "Yarhamuk Allah" (may Allah have mercy on you) to someone who sneezes but does not audibly praise Allah, assuming they praised Allah silently, and is it permissible to say it in one's heart if they are far away or a woman?

One should not say "Yarhamukallah" (may Allah have mercy on you) to someone who sneezes and does not praise Allah, based on the Prophet's (peace be upon him) saying: "If one of you sneezes and praises Allah, then say 'Yarhamukallah' to him. But if he does not praise Allah, then do not say it to him." It is disliked to say "Yarhamukallah" if he does not praise Allah. If he praises Allah but you do not hear his praise, then do not say it to him. You are not required to say "Yarhamukallah" to someone whose praise you did not hear, because the Muslim's right upon his brother is to say "Yarhamukallah" to him if he hears his praise. What goes on in one's mind without being uttered has no legal standing in Islamic law.