See this is a technical situation as if a married man lives with an unmarried women without any marriage than it not any offence under any law in India.
So to file a complaint against your father your mother has to file it on ground that in her lifetime her husband married again and has issues (children from that) so it amounts to offence of bigamy under 494 IPC and criminal breach of trust and is punishable under law. But for that it has to be established they are married and based on your facts it is established.
Though on that women no charges can be pressed,
along with that your mother can also fie complaint under domestic violence on mental harassment.
And she can file and seek divorce on ground of cruelty and bigamy and adultery.
As per divorce law in India, the Indian Divorce Act, 1869 governs the divorce of persons professing the Christian religion. This Act governs only Christians. If even one of the parties to marriage is a Christian, that is sufficient to give jurisdiction to decide the petition under this Act.
Under the Indian Divorce Act, a husband can seek divorce on the ground that after solemnization of marriage the wife has been guilty of adultery.
A wife, however, may petition for divorce on any of the following grounds:
Husband has changed his religion and started professing some other religion than Christianity
Husband has married another woman
Husband has been guilty of incestuous adultery, meaning adultery committed by a husband with a woman with whom, if his wife were dead, he could not lawfully contract marriage by reason of her being within the prohibited degree of consanguinity.
Husband has committed bigamy with adultery, meaning adultery with the same woman bigamy was committed
Husband has married another woman with adultery
Husband has committed rape, sodomy or bestiality
Adultery coupled with cruelty
Adultery coupled with desertion, without reasonable excuse for more than two years
As per divorce law in India, The District Court or the High Court of competent jurisdiction is the appropriate court to file a petition for divorce under Indian Divorce Act.
2) Also your mother can seek right of residence , monthly maintenance and maintenance for your Sister and alimony for divorce. under section 36 to 38 Indian Divorce Act, also under section 125 CrpC and domestice violence act.
3) After filing complaint the police shall carry preliminary investigation and register a FIR and your father may get arrested for same, also then the case shall be put before the magistrate and the magistrate shall order investigation and police file charge sheet, further there will be evidences to the effect.
on FIR there is probability or father is arrested by the police if he is not granted anticipatory bail