Certainly there should not be any duplication of effort.
The product life cycle defines the general approach to the life cycle, from conceptualization to development to manufacturing to any post-market activities (maintenance, service, etc.) to decommissioning. I would suggest that the life cycle is a top-level view of how your company develops product and is not product specific.
Project management details the deliverables, resources, and schedule implementing the life cycle for that product. The life cycle, essentially, drives project management. Think of it as “customer requirements” for project management.
Project management, in and of itself, has few deliverables: schedule, status reports, etc.
Project life cycle documents (requirements specifications, design documents, etc.) are artifacts from implementing the life cycle and are tracked in project management.
