Clarifying Confusion with Hecke Operators and Double Cosets

I am confused about the construction of the Hecke operators. I am defining them on $\Gamma_{1}(N)$ as $$(T_{m}f)(z) = \sum_{\substack{a,d \ge 1 \\ ad = m}}\langle a\rangle\left[\Gamma_{1}(N)\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}\Gamma_{1}(N)\right]_{k}f(z),$$ where $\langle a \rangle$ is the diamond operator and the bracket is the usual double coset operator. Sometimes I see the restriction $a \mid d$ in the literature. Is this a typo? Also, I want to show that this matches the formula $$(T_{m}f)(z) = m^{k-1}\sum_{\substack{ad = m \\ a,d \ge 1}}\frac{1}{d^{k}}\sum_{b \mod{d}}\langle a\rangle f\left(\frac{az+b}{d}\right),$$ and to do this I am trying to prove a disjoint decomposition of the form $$\Gamma_{1}(N)\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}\Gamma_{1}(N) = \bigcup_{b \mod{d}}\Gamma_{1}(N)\begin{pmatrix} a & b \\ 0 & d \end{pmatrix}.$$ Does this decomposition hold? It seems in the literate people have proved this when $a = 1$ and $d = p$, but does it hold more generally (I can't find a reference)? How would I go about proving it? I can't even see why the right-hand side is a subset of the left-hand side.

Any help would be appreciated. Thanks!