Joy Mangano, who built a better mop and parlayed it into a multimillion-dollar housewares empire, has a story well known to her fellow Long Islanders. Starting with Sunday night’s premiere of “Joy” at the Ziegfeld theater in Manhattan, that story — or a loosely adapted version of it — will become better known to the rest of the world.

“I think all women can relate to struggling, striving to make your dream come true, struggling to pay the bills,” said Virginia Madsen (“Witches of East End”) on the red carpet. She plays Terry, the housebound, emotionally damaged mother of Joy (Jennifer Lawrence) — although, as Madsen points out, “My character was fictional. It was based on different aspects of people in Joy’s life.”

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In fact, the last name Mangano and the locale of Smithtown, where the real-life Joy’s Miracle Mop and many other inventions were born, are never uttered in the film. “I was just the inspiration for the movie,” said Mangano, noting that director and co-writer David O. Russell’s film is not a biography per se — while adding with a laugh, “Do you know many women that invented a Miracle Mop?”

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Mangano, who also has had homes in St. James and Nissequogue, does credit her hometown for developing the stick-to-itiveness that helped her overcome the initial reluctance by pioneering home-shopping channel QVC to carry her Miracle Mop — and then to go on the air and pitch it herself when initial sales proved disappointing. “I think where I grew up, the people around me, we have a certain resiliency,” she said. “And I think being resilient — and knowing that if it didn’t work one way you have to try another way — is a good thing.”

Oscar-winning star Lawrence herself took Mangano’s example to heart, saying on the red carpet that she could relate it to her own career — “knowing that I could do it and knowing that I wasn’t going to stop and having that fight in me.”

“I was told by Joy that I’m playing the one person that told her she could do it,” said three-time Oscar-nominee Diane Ladd, who plays Joy’s grandmother, Mimi. “So for her, I was that one person. And then David told me that [Mimi is] kind of like Clarence in ‘It’s a Wonderful Life.’ He said, ‘Maybe Mimi is not a human. Maybe Mimi is the soul of an angel who came in to prepare the destiny.’ ”

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Also appearing at the premiere were James Lipton and fellow “Joy” stars Robert De Niro, Bradley Cooper, Edgar Ramirez, Isabella Rossellini, Elisabeth Röhm, Dascha Polanco, Melissa Rivers and nearly 11-year-old Isabella Crovetti-Cramp, who plays Joy as a child. The film opens Dec. 25.