Exponential on Real Numbers is Group Isomorphism/Proof 2

Proof
From Real Numbers under Addition form Abelian Group, $\struct {\R, +}$ is a group.

From Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group, $\struct {\R_{> 0}, \times}$ is a group.

We have that for all $y \in R_{> 0}$ there exists $x = \map \ln y \in R$ which satisfies $\map \exp x = y$.

Thus $\exp$ is a surjection.

Then from Exponential on Real Numbers is Injection:
 * $\exp$ is an injection.

Therefore, $\exp$ is a bijection.

Let $x, y \in R$.

From Exponential of Sum:
 * $\map \exp {x + y} = \map \exp x \, \map \exp y$

So $\exp$ is a homomorphism and a bijection.

It follows by definition that $\exp: \struct {\R, +} \to \struct {\R_{> 0}, \times}$ is an isomorphism.