This is a very good question. And let me say from the onset that I will not "dazzle" you with examples of the Trinity that deals with water, ice, eggs, four leaf clovers sun rays or anything else in like manner.
The definition of the doctrine of the Trinity is not an "assumption." It is the normative systematic theology of God in Christianity and is based on the fact that the Bible is EXPLICIT in telling us that there is, was and forever will be only ONE God AND the fact that the Bible identifies three (and only three) persons as God.
So, how is this "substantiated?" Now, please bear with me on this because it may appear complicated but it is not when you think about it. Also, the Trinity cannot be understood, argued or refuted by appealing to any single passage in the Bible. It is drawn from the whole Bible.
The Bible identifies God by: (1) His names. (2) His titles. (3) His unique attributes. (4) His unique actions. (5) His worship. If you will examine the Bible thoroughly you should be able to quickly discover that there are three and ONLY three "persons" who are identified as God by the 'COMBINATION" of the literary means I listed above.
These persons are each variously...CALLED by the NAMES of God. (YHWH and its variants) either directly or indirectly, usually both.
RECOGNIZED with the TITLES of God. (Lord, king, savior, redeemer etc.) For example God is identified at Isaiah 44:6, "Thus says the Lord, the King of Israel And his Redeemer, the Lord of host; I am the first and I am the last. And there is no God besides Me. At Isaiah 43:11, "I, even I; am the Lord; And there is no SAVIOR besides Me.
Jesus Christ is also described as the "first and the last in the book of Revelation. Jesus Christ is also identified as the "SAVIOR" at Luke 2:11, "for today in the city of David there has been born for you a "SAVIOR, who is Christ the Lord. Are you beginning to see the connection?
As a point of order let me say that when you see the word "God" in the Old Testament you "CANNOT" assume it refers to God the Father all the time. Let me pick another one! His unique actions. That is to say (Creation, origin of God's word, salvation of men etc).
Going back to Isaiah at Isaiah 44:24 the part that says, "I, the Lord am the maker of all things, Stretching out the heavens BY MYSELF, And spreading out the earth ALL ALONE."
And we have at John 1:3, "ALL things came into being by Him, and apart from Him (apart means without Him) nothing came into being that has come into being." Colossians 1:16 backs this up, "For by Him/Jesus Christ ALL things were CREATED, both in the heavens and on earth, visible and invisible, whether thrones or dominions or rulers or authorities ---all things have been created by Him and for Him." God the Father backs this up at Hebrews 1:10.
One more, God's worship. We already know God is worshiped in the Bible. Jesus also receives worship. The highest form of worship to Jesus was by Thomas at John 20:28 when he said literally to Jesus Himself, "The Lord of me and the God of me." At Hebrews 1:6 His Father says, "Let all the angels of God worship Him." The Greek word for worship in this text is "proskuneo" which is the same word used for worshiping the Father.
At this point I'm not going go over His unique attributes/characteristics which are easy to figure out on your own, i.e omnipresence, omnipotence, eternality, etc.
Keep in mind that I am saying that each person of the Trinity receives some COMBINATION of the 5 means of identifying and distinguishing God listed above. Remember, theology is not salvation. It is in Christ that we find God fully and sufficiently manifested to us human beings, AS a human being and for His own glory. And it is only through His Holy Spirit that we can know Him and be saved by His grace.
Finally, at Matthew 16:13 Jesus said, "Who do people say that the Son of Man is? At verse 16 Peter replied, "Thou art the Christ, the Son of the living God." Jesus replies to Peter, "Blessed are you Simon Barjona, because flesh and blood did not reveal this to you, but My Father who is in heaven."