This is a question from Silverman's book 'the arithmetic of elliptic curves', chapter Ⅵ.
If $\Lambda$ is a lattice in $\Bbb C$ the map $$z\mapsto (\wp(z),\wp'(z))$$ is a parametrisation of the complex points of the elliptic curve $$E:\qquad y^2=4x^3-g_2x-g_3$$ where $g_2$ and $g_3$ depend on $\Lambda$. It induces a group isomorphism $Φ$：$\Bbb C/\Lambda\to E(\Bbb C)$ : $t\mapsto (\wp(t),\wp'(t))$ .
To show this is isomorphism as Rieman surface, the books reads it is enough to prove $Φ$ is local analytic isomorphism ( because $Φ$ is bijective).
So, we only need to check
Φ*(dx/y)＝$d\wp(z)/\wp'(z)$＝$z$・・・①.
Once ① is proved, this is holomorphic and non vanishing at every point of domain, so $Φ$ is local analytic function.
My question: Why is is enough to show that $Φ$ is local analytic isomorphism ? Why bijectivity of $Φ$ ensures that $Φ$ is exactly global analytic isomorphism?