Decrypting RSA: Using d_K, d_p, d_q, M_p, M_q and x_p,x_q to Prove y^d=x mod n

[hope2009]

In RSA: d_K (y)=y^d mod n and n=pq. Define

d_p=d mod(p-1)

d_q=d mod(q-1)
Let

M_p=q^(-1) mod p
M_q=p^(-1) mod q
And

x_p=y^(d_p ) mod p
x_q=y^(d_q ) mod q
x=M_p qx_p+M_q px_q mod n

Show that y^d=x mod n
any help would be appraciated, thanks

 

[farful]

homework eh?

use fermat's thm to prove y^d = y^(d_p) mod p (same for q)
show x = x_p mod p (same for q)
then use CRT to solve for x