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Definition df-dvdsr 15439
Description: Define the (right) divisibility relation in a ring. Access to the left divisibility relation is available through  ( ||r `
 (oppr
`  R ) ). (Contributed by Mario Carneiro, 1-Dec-2014.)
Assertion
Ref Expression
df-dvdsr  |-  ||r  =  (
w  e.  _V  |->  {
<. x ,  y >.  |  ( x  e.  ( Base `  w
)  /\  E. z  e.  ( Base `  w
) ( z ( .r `  w ) x )  =  y ) } )
Distinct variable group:    x, w, y, z

Detailed syntax breakdown of Definition df-dvdsr
StepHypRef Expression
1 cdsr 15436 . 2  class  ||r
2 vw . . 3  set  w
3 cvv 2801 . . 3  class  _V
4 vx . . . . . . 7  set  x
54cv 1631 . . . . . 6  class  x
62cv 1631 . . . . . . 7  class  w
7 cbs 13164 . . . . . . 7  class  Base
86, 7cfv 5271 . . . . . 6  class  ( Base `  w )
95, 8wcel 1696 . . . . 5  wff  x  e.  ( Base `  w
)
10 vz . . . . . . . . 9  set  z
1110cv 1631 . . . . . . . 8  class  z
12 cmulr 13225 . . . . . . . . 9  class  .r
136, 12cfv 5271 . . . . . . . 8  class  ( .r
`  w )
1411, 5, 13co 5874 . . . . . . 7  class  ( z ( .r `  w
) x )
15 vy . . . . . . . 8  set  y
1615cv 1631 . . . . . . 7  class  y
1714, 16wceq 1632 . . . . . 6  wff  ( z ( .r `  w
) x )  =  y
1817, 10, 8wrex 2557 . . . . 5  wff  E. z  e.  ( Base `  w
) ( z ( .r `  w ) x )  =  y
199, 18wa 358 . . . 4  wff  ( x  e.  ( Base `  w
)  /\  E. z  e.  ( Base `  w
) ( z ( .r `  w ) x )  =  y )
2019, 4, 15copab 4092 . . 3  class  { <. x ,  y >.  |  ( x  e.  ( Base `  w )  /\  E. z  e.  ( Base `  w ) ( z ( .r `  w
) x )  =  y ) }
212, 3, 20cmpt 4093 . 2  class  ( w  e.  _V  |->  { <. x ,  y >.  |  ( x  e.  ( Base `  w )  /\  E. z  e.  ( Base `  w ) ( z ( .r `  w
) x )  =  y ) } )
221, 21wceq 1632 1  wff  ||r  =  (
w  e.  _V  |->  {
<. x ,  y >.  |  ( x  e.  ( Base `  w
)  /\  E. z  e.  ( Base `  w
) ( z ( .r `  w ) x )  =  y ) } )
Colors of variables: wff set class
This definition is referenced by:  reldvdsr  15442  dvdsrval  15443
  Copyright terms: Public domain W3C validator