What type of breast cancer that cause the breast to swell and cause under the armpit to have a painful lump? and does that mean the cancer has spread

May not be cancer: Breast cancer usually does not present with pain and swelling in most cases. Pain in armpit may be related to some nearby infection in hair follicle or in breast. See you doctor and get checked asap.

Possibly IBC: Inflammatory breast cancer occurs suddenly with breast pain, redness, or swelling. It is very fast-growing and spreads quickly. It looks like an infection, so doctors often try antibiotics first, delaying diagnosis. However, infection is also a possibility. See a doctor immediately for these symptoms. If it is cancer, your symptoms don't mean it has spread. But you need a biopsy of any lump!

Infiltrating duct: All carcinomas of the breast will eventually metastasize or spread to the axilla. Intraductal lesions in the process of converting are within the duct system, non palpable and wont spread until they become invasive. Swelling occurs when the tumor enlarges but not to the extent seen with inflammatory Ca where the swelling is within the subcutaneous lymphatic system only.