Goodwill and Intangible Assets (Schedule of Changes To Goodwill) (Details) - USD ($) $ in Millions	3 Months Ended	12 Months Ended
	Jan. 01, 2022	Jan. 01, 2022	Jan. 02, 2021	Dec. 28, 2019
Goodwill [Roll Forward]
Goodwill, beginning balance		$ 1,518.2	$ 1,501.3
Impairment Charge	$ (33.0)	(33.0)	(10.5)	$ 0.0
Acquisitions		2,576.8
Translation and Other		(22.8)	27.4
Goodwill, ending balance	4,039.2	4,039.2	1,518.2	1,501.3
Cumulative Goodwill Impairment Charges	328.7	328.7
Commercial Systems
Goodwill [Roll Forward]
Goodwill, beginning balance		433.3	426.6
Impairment Charge		0.0	0.0
Acquisitions		0.0
Translation and Other		(4.4)	6.7
Goodwill, ending balance	428.9	428.9	433.3	426.6
Cumulative Goodwill Impairment Charges	183.2	183.2
Industrial Systems
Goodwill [Roll Forward]
Goodwill, beginning balance		163.7	170.8
Impairment Charge		(33.0)	(10.5)
Acquisitions		0.0
Translation and Other		(1.9)	3.4
Goodwill, ending balance	128.8	128.8	163.7	170.8
Cumulative Goodwill Impairment Charges	105.1	105.1
Climate Solutions
Goodwill [Roll Forward]
Goodwill, beginning balance		330.8	331.2
Impairment Charge		0.0	0.0
Acquisitions		0.0
Translation and Other		(0.3)	(0.4)
Goodwill, ending balance	330.5	330.5	330.8	331.2
Cumulative Goodwill Impairment Charges	17.2	17.2
Motion Control Solutions
Goodwill [Roll Forward]
Goodwill, beginning balance		590.4	572.7
Impairment Charge		0.0	0.0
Acquisitions		2,576.8
Translation and Other		(16.2)	17.7
Goodwill, ending balance	3,151.0	3,151.0	$ 590.4	$ 572.7
Cumulative Goodwill Impairment Charges	$ 23.2	$ 23.2