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Definition df-dvdsr 17795
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 17792 . 2  class  ||r
2 vw . . 3  setvar  w
3 cvv 3087 . . 3  class  _V
4 vx . . . . . . 7  setvar  x
54cv 1436 . . . . . 6  class  x
62cv 1436 . . . . . . 7  class  w
7 cbs 15075 . . . . . . 7  class  Base
86, 7cfv 5601 . . . . . 6  class  ( Base `  w )
95, 8wcel 1870 . . . . 5  wff  x  e.  ( Base `  w
)
10 vz . . . . . . . . 9  setvar  z
1110cv 1436 . . . . . . . 8  class  z
12 cmulr 15144 . . . . . . . . 9  class  .r
136, 12cfv 5601 . . . . . . . 8  class  ( .r
`  w )
1411, 5, 13co 6305 . . . . . . 7  class  ( z ( .r `  w
) x )
15 vy . . . . . . . 8  setvar  y
1615cv 1436 . . . . . . 7  class  y
1714, 16wceq 1437 . . . . . 6  wff  ( z ( .r `  w
) x )  =  y
1817, 10, 8wrex 2783 . . . . 5  wff  E. z  e.  ( Base `  w
) ( z ( .r `  w ) x )  =  y
199, 18wa 370 . . . 4  wff  ( x  e.  ( Base `  w
)  /\  E. z  e.  ( Base `  w
) ( z ( .r `  w ) x )  =  y )
2019, 4, 15copab 4483 . . 3  class  { <. x ,  y >.  |  ( x  e.  ( Base `  w )  /\  E. z  e.  ( Base `  w ) ( z ( .r `  w
) x )  =  y ) }
212, 3, 20cmpt 4484 . 2  class  ( w  e.  _V  |->  { <. x ,  y >.  |  ( x  e.  ( Base `  w )  /\  E. z  e.  ( Base `  w ) ( z ( .r `  w
) x )  =  y ) } )
221, 21wceq 1437 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 setvar class
This definition is referenced by:  reldvdsr  17798  dvdsrval  17799
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