How to make "such that" extensible?

Here is an example:

<\equation*>
  \<ell\><rsup|p><around*|(|\<bbb-N\>|)>\<assign\><around*|{|<around*|(|a<rsub|1>,a<rsub|2>,\<ldots\>|)>\<in\>\<bbb-C\><rsup|\<bbb-N\>>\<barsuchthat\><big|sum><rsub|n=1><rsup|\<infty\>><around*|\||a<rsub|n>|\|><rsup|p>\<less\>+\<infty\>|}>
</equation*>

The size of the “such that” vertical bar is too small. I could change it to

<\equation*>
  \<ell\><rsup|p><around*|(|\<bbb-N\>|)>\<assign\><around*|{|<around*|(|a<rsub|1>,a<rsub|2>,\<ldots\>|)>\<in\>\<bbb-C\><rsup|\<bbb-N\>><mid|\|><big|sum><rsub|n=1><rsup|\<infty\>><around*|\||a<rsub|n>|\|><rsup|p>\<less\>+\<infty\>|}>
</equation*>

but the spacing does not seem to be correct.