Inverse of Element in Semidirect Product

Theorem
Let $N$ and $H$ be groups.

Let $H$ act by automorphisms on $N$ via $\phi$.

Let $N \rtimes_\phi H$ be the corresponding (outer) semidirect product.

Let $\tuple {n, h} \in N \rtimes_\phi H$.

Then:

Proof
Follows from Semidirect Product of Groups is Group.

The alternatives follow from the fact that $H$ acts by automorphisms.