conditional probability of type 1 error - hypothesis testing common malpractices

#1
If you want to test Ho = 0 versus H1≠ 1,with significance level alpha, but instead of testing this directly, you have a strategy:
-> X1,...,Xn is a random sample from a N(µ,σ2) distribution with σ2 known (I shall denote the mean of X as X(m)) and we obtain Z = X(m)/(σ/√n)
  • If Z > 0, let H1 : µ > 0 and reject H0 if Z > zα
  • If Z < 0, let H1 : µ < 0 and reject H0 if Z < −zα
How would you prove that if the null hypothesis is true, the probability of rejecting H0, conditionally on observing that Z > 0, is not equal to α but to 2α? Could you maybe give some hints? And how would this probability be unconditionlly?