## Exact meaning of $2^{\mathcal{O}(f(n))}$

In Sipser’s Introduction to the Theory of Computation he uses the notation $$2^{\mathcal{O}(f(n))}$$ to denote some asymptotic running time.

For example he says that the running time of a single-tape deterministic turing machine simulating a multi-tape non-deterministic turing machine is

$$\mathcal{O}(t(n)b^{t(n)})=2^{\mathcal{O}(t(n))}$$ where $$b$$ is the maximal number of options in the transtition function.

I was wondering if someone can clarify the exact definition of this for me:

My assumption is that if $$g(n)=2^{\mathcal{O}(f(n))}$$ then there exists $$N \in \mathbb{Z}$$ and $$c \in \mathbb{R}$$ s.t.

$$g(n) \leq 2^{cf(n)}=2^{f(n)^c}$$ for all $$n>N$$.

Thank you