Please refer to an equation in Pattern recognition and Machine Learning by Bishop. My query is related to manipulation of an error term, equation 3.18: $$E_{D}(\boldsymbol{w}) = \frac{1}{2}\sum_{n=1}^N\{t_{n} - w_{0} - \sum_{j=1}^{M-1}w_{j}\phi_{j}(x_{n})\}^{2} \tag{3.18}$$ the author goes on to get: $$w_{0} = \overline{t} - \sum_{n=1}^{M-1}w_{j}\overline\phi_{j} \tag{3.19} $$ $$where \quad \overline{t} = \frac{1}{N}\sum_{n=1}^Nt_{n} \qquad \overline{\phi_j} = \frac{1}{N}\sum_{n=1}^{N}\phi_{j}(x_{n}) \tag{3.20}$$ If I take the derivate of 3.18, w.r.t $w_{0}$, set it to zero, and divide both sides by $N$, I get: $$w_{0} = \frac{1}{N}\sum_{n=1}^{N}t_{n} - \frac{1}{N}\sum_{n=1}^{N}\sum_{j=1}^{M-1}w_{j}\phi_{j}(x_{n})$$
Am I creating any error above? If so, please help. If not, my problem is related to the second term (R.H.S., above equation):
The right term, $\frac{1}{N}\sum_{n=1}^{N}\sum_{j=1}^{M-1}w_{j}\phi_{j}(x_{n})$, is the average of double sum. I can't understand how we can simply 'push' $\frac{1}{N}$ inside to get $\sum_{j=1}^{M-1}\frac{1}{N}\sum_{n=1}^{N}w_{j}\phi_{j}(x_{n})$ and eventually equation for $\overline\phi_{j}$ (ref. 3.20), since:
$$\frac{1}{2}[(1+2) + (3+5)] \quad != 1 + 3 + \frac{1}{2}(2 + 5)$$
LHS is average of sum of sums and RHS is average 'pushed' inside.
