$SL(n,\Bbb{C})$ is a regular submanifold of $GL(n,\Bbb{C})$ 
Let $SL(n,\Bbb{C})$ be the group of matrices of complex entries and determinant $1$.
I want to prove that $SL(n,\Bbb{C})$ is a regular submanifold of $GL(n,\Bbb{C})$.

An idea is to use the Regular Level Set Theorem because $$SL(n,\Bbb{C})=f^{-1}(1)$$ where $f(A)=\det(A)$
(I know how to prove the same for $SL(n,\Bbb{R})$ but here i believe we need to use theory of multivariable complex analysis which i do not know at all.)
Can someone help me prove this statement?
Thank you in advance.

You can avoid the calculations by using Lie group theory, because $GL(n,\mathbb{C})$ is a Lie group and $SL(n,\mathbb{C})$ a closed subgroup. Cartan's theorem then says $SL(n,\mathbb{C})$ is a regular Lie subgroup, in particular a regular submanifold. Note that this also works for $\mathbb{R}$ instead of $\mathbb{C}$.
