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Definition df-div 9420
Description: Define division. Theorem divmuli 9510 relates it to multiplication, and divcli 9498 and redivcli 9523 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.)
Assertion
Ref Expression
df-div  |-  /  =  ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC ( y  x.  z )  =  x ) )
Distinct variable group:    x, y, z

Detailed syntax breakdown of Definition df-div
StepHypRef Expression
1 cdiv 9419 . 2  class  /
2 vx . . 3  set  x
3 vy . . 3  set  y
4 cc 8731 . . 3  class  CC
5 cc0 8733 . . . . 5  class  0
65csn 3642 . . . 4  class  { 0 }
74, 6cdif 3151 . . 3  class  ( CC 
\  { 0 } )
83cv 1623 . . . . . 6  class  y
9 vz . . . . . . 7  set  z
109cv 1623 . . . . . 6  class  z
11 cmul 8738 . . . . . 6  class  x.
128, 10, 11co 5820 . . . . 5  class  ( y  x.  z )
132cv 1623 . . . . 5  class  x
1412, 13wceq 1624 . . . 4  wff  ( y  x.  z )  =  x
1514, 9, 4crio 6291 . . 3  class  ( iota_ z  e.  CC ( y  x.  z )  =  x )
162, 3, 4, 7, 15cmpt2 5822 . 2  class  ( x  e.  CC ,  y  e.  ( CC  \  { 0 } ) 
|->  ( iota_ z  e.  CC ( y  x.  z
)  =  x ) )
171, 16wceq 1624 1  wff  /  =  ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC ( y  x.  z )  =  x ) )
Colors of variables: wff set class
This definition is referenced by:  1div0  9421  divval  9422  elq  10314  cnflddiv  16399  divcn  18367  1div0apr  20834
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