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Intangible Assets and Goodwill (Details) - USD ($)
$ in Millions
3 Months Ended 6 Months Ended
Jun. 30, 2023
Jun. 30, 2022
Jun. 30, 2023
Jun. 30, 2022
Dec. 31, 2022
Intangible assets          
Total intangible assets, gross $ 7,508.7   $ 7,508.7   $ 7,479.3
Accumulated Amortization (5,732.3)   (5,732.3)   (5,629.2)
Intangible assets, net 1,776.4   1,776.4   1,850.1
Amortization and impairment of acquired intangible assets 52.9 $ 67.5 103.1 $ 134.4  
Expected future amortization expense, 2023 (remaining three months) 110.0   110.0    
Expected future amortization expense, 2024 200.0   200.0    
Expected future amortization expense, 2025 190.0   190.0    
Expected future amortization expense, 2026 175.0   175.0    
Expected future amortization expense, 2027 170.0   170.0    
Expected future amortization expense, 2028 165.0   165.0    
Completed technology          
Intangible assets          
Cost 7,444.7   7,444.7   7,415.3
Net 1,712.4   1,712.4   1,786.1
Accumulated Amortization $ (5,732.3)   $ (5,732.3)   (5,629.2)
Completed technology | Minimum          
Intangible assets          
Useful life 4 years   4 years    
Completed technology | Maximum          
Intangible assets          
Useful life 28 years   28 years    
Trademarks and Trade Names          
Intangible assets          
Cost and Net $ 64.0   $ 64.0   64.0
Accumulated Amortization 0.0   0.0   0.0
Indefinite-lived Intangible Assets (Excluding Goodwill) $ 64.0   $ 64.0   $ 64.0