If the complex number system is defined to be the set $\mathbb C = \mathbb R \times \mathbb R$ (called the complex numbers) together with the two functions from $\mathbb C \times \mathbb C $ into $\mathbb C$, denoted by $+$ and $\times$ (multiplication), that are given by $(a,b) + (c,d) = (a+c, b+d)$ and $(a,b)\times(c,d) = (ac-bd, ad+bc)$ for all $a,b,c,d\in\mathbb R$.
If I am trying to prove the existence of the multiplicative inverses for $\mathbb C$, do I need to begin by laying out that $$(ac-bd, ad+bc)^{-1} \times (ac-bd, ad+bc) = 1? $$