Testing if a post is in a descendant category in WordPress
You can find a solution to this problem in WordPress Codex. The only issue being that it only allows you to pass a single post ID or an array of post IDs, not allowing you to use your category slug, not to mention multiple slugs.
The documentation of this function tells you to see
get_term_by() if you want to use category name or slug, but why not extend this function to do this for you?