From Sheldon Axler's Linear Algebra Done Right (3rd Ed), the following proof was left as an exercise to the reader:
for every $\alpha\in\mathbb{C}$, there exists a unique $\beta\in\mathbb{C}$ such that $\alpha + \beta = 0$;
Thus far, the only definitions the book has given are the definitions of complex addition and multiplication. The commutative, associative, and relevant identities (i.e. $\lambda + 0 = \lambda, \lambda\cdot1 = \lambda$ for all $\lambda\in\mathbb{C}$) have already been proven using these definitions.
I went about proving the aforementioned statement statement using the following method:
Let $\alpha = a_1 + a_2i$ and $\beta = b_1 + b_2i$. It follows from the definition of complex addition that $$\alpha + \beta = (a_1 + b_1) + i(a_2 + b_2)$$ Since we want this expression to be equal to $0$, we can treat $0$ as a complex number and equate the expression's real and imaginary parts: $$(a_1 + b_1) + i(a_2 + b_2) = 0 + i0 \\ \implies a_1 + b_1 = 0, a_2 + b_2 = 0 \\ \therefore b_1 = -a_1, b_2 = -a_2 $$ From this, we deduce that $\beta = b_1 + b_2i = -a_1 - a_2i = -\alpha$ is the unique solution we were looking for, no other real numbers can be used to satisfy the above system of equations created from the complex numbers' real and imaginary parts.
However, is it valid to set these equations equal to each other given only the definitions that the books gives? Also, is it valid to assume that this proof gives a unique solution, or is there more work to be done with respect to that?