Unique Identities: Proving O₁ = O₂ in Theorem for Proof of Identity

[ver_mathstats]

Homework Statement

Prove the identity of plus.

Relevant Equations

Declaration: O₁, O₂ : ℤ

Axiom “Left-identity of +”: x = O₁ + x
Axiom “Right-identity of +”: x = x + O₂

Now, prove the following.
Theorem “Identities of + are unique”: O₁ = O₂

Theorem “Identities of + are unique”: O₁ = O₂
Proof:
O₁
= Left Identity of +
O₁ + x

I'm a little confused where to begin this proof, I don't know if that is the first step either I think it is. Proofs are not a strength of mine so I struggle to see how to show that O₁ = O₂. Any guidance would be appreciated, thank you.

 

[fresh_42]

Take your two defining equations and set $x=0_1$ in one and $x=0_2$ in the other.

 

[Mark44]

Homework Statement:: Prove the identity of plus.
Relevant Equations:: Declaration: O₁, O₂ : ℤ

Axiom “Left-identity of +”: x = O₁ + x
Axiom “Right-identity of +”: x = x + O₂

Now, prove the following.
Theorem “Identities of + are unique”: O₁ = O₂

Left Identity of +

At first glance, I didn't know what you were trying to do, but maybe it's a language translation thing.

A more usual phrasing would be "O is the additive identity" or "O₁ is the left-additive identity".
My point is that O₁ and O₂ are the left/right addition identities, the things that you can add to a number without changing it.

In a similar vein there is the concept of a multiplicative identity.

 

[fresh_42]

At first glance, I didn't know what you were trying to do, but maybe it's a language translation thing.

One can define only one sided neutrals in group theory and show that they have to be the same, even in non commutative groups. I just don't recall whether one one sided neutral is already sufficient, or whether both are needed.