I'm trying to understand the distribution function of the minimum of a random sample.

x_1,x_2,...,x_n are i.i.d

m_n=min(x_1,x_2,...,x_n)

M_n=max(x_1,x_2,...,x_n)

We know that P(M_n<=z)= F_max (z)=(F(z))^n

My attempt to understand this:

P(m_n<=z)=P(-M_n<=z)

=P(M_n>=-z)

=1-P(M_n<-z)

After this I'm really stuck. Appreciate your help.