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 Riemann surface, the books reads it is enough to prove $Φ$ is local analytic isomorphism (I understand this means we can prove $Φ$ is holomorphic map). So, we only need to check
$$Φ^*(dx/y)＝d\wp(z)/\wp'(z)＝z\tag1$$
My question Why (1) ensures $Φ$ is holomorphic ( or local isomorphism) ?