Exponential on Real Numbers is Group Isomorphism/Proof 1

Proof
From Exponent of Sum we have:
 * $\forall x, y \in \R: \map \exp {x + y} = \exp x \cdot \exp y$

That is, $\exp$ is a group homomorphism.

Then we have that Exponential is Strictly Increasing.

From Strictly Monotone Function is Bijective, it follows that $\exp$ is a bijection.

So $\exp$ is a bijective group homomorphism, and so a group isomorphism.