Not necessarily. Because programs can be used for many activities, including inbound and outbound campaigns, as well as data management activities, program membership does not occur by default. In other words, depending on the program, different criteria may constitute a "member", and those aren't always related to an email. You must use the flow action in a smart campaign "Change progression status" to actually add a name as a member of a program.
So for your email program to assign a recipient as a member of the program, you should follow the "Send email" action of your smart campaign with a "change status in progression" flowstep that adds them with a status of "sent".
Hope this helps!
-Jeff