Exponential on Real Numbers is Group Isomorphism/Proof 1

Theorem
Let $\left({\R, +}\right)$ be the additive group of real numbers.

Let $\left({\R_{> 0}, \times}\right)$ be the multiplicative group of positive real numbers.

Let $\exp: \left({\R, +}\right) \to \left({\R_{> 0}, \times}\right)$ be the mapping:
 * $x \mapsto \exp \left({x}\right)$

where $\exp$ is the exponential function.

Then $\exp$ is a group isomorphism.

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

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

Then we have that Exponential is Strictly Increasing and Strictly Convex.

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.