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Definition df-div 8629
Description: Define division. Theorem divmulap 8631 relates it to multiplication, and divclap 8634 and redivclap 8687 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divvalap 8630 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
Assertion
Ref Expression
df-div / = (๐‘ฅ โˆˆ โ„‚, ๐‘ฆ โˆˆ (โ„‚ โˆ– {0}) โ†ฆ (โ„ฉ๐‘ง โˆˆ โ„‚ (๐‘ฆ ยท ๐‘ง) = ๐‘ฅ))
Distinct variable group:   ๐‘ฅ,๐‘ฆ,๐‘ง

Detailed syntax breakdown of Definition df-div
StepHypRef Expression
1 cdiv 8628 . 2 class /
2 vx . . 3 setvar ๐‘ฅ
3 vy . . 3 setvar ๐‘ฆ
4 cc 7808 . . 3 class โ„‚
5 cc0 7810 . . . . 5 class 0
65csn 3592 . . . 4 class {0}
74, 6cdif 3126 . . 3 class (โ„‚ โˆ– {0})
83cv 1352 . . . . . 6 class ๐‘ฆ
9 vz . . . . . . 7 setvar ๐‘ง
109cv 1352 . . . . . 6 class ๐‘ง
11 cmul 7815 . . . . . 6 class ยท
128, 10, 11co 5874 . . . . 5 class (๐‘ฆ ยท ๐‘ง)
132cv 1352 . . . . 5 class ๐‘ฅ
1412, 13wceq 1353 . . . 4 wff (๐‘ฆ ยท ๐‘ง) = ๐‘ฅ
1514, 9, 4crio 5829 . . 3 class (โ„ฉ๐‘ง โˆˆ โ„‚ (๐‘ฆ ยท ๐‘ง) = ๐‘ฅ)
162, 3, 4, 7, 15cmpo 5876 . 2 class (๐‘ฅ โˆˆ โ„‚, ๐‘ฆ โˆˆ (โ„‚ โˆ– {0}) โ†ฆ (โ„ฉ๐‘ง โˆˆ โ„‚ (๐‘ฆ ยท ๐‘ง) = ๐‘ฅ))
171, 16wceq 1353 1 wff / = (๐‘ฅ โˆˆ โ„‚, ๐‘ฆ โˆˆ (โ„‚ โˆ– {0}) โ†ฆ (โ„ฉ๐‘ง โˆˆ โ„‚ (๐‘ฆ ยท ๐‘ง) = ๐‘ฅ))
Colors of variables: wff set class
This definition is referenced by:  divvalap  8630  divfnzn  9620
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