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Definition df-rlreg 16040
Description: Define the set of left-regular elements in a ring as those elements which are not left zero divisors, meaning that multiplying a nonzero element on the left by a left-regular element gives a nonzero product. (Contributed by Stefan O'Rear, 22-Mar-2015.)
Assertion
Ref Expression
df-rlreg  |- RLReg  =  ( r  e.  _V  |->  { x  e.  ( Base `  r )  |  A. y  e.  ( Base `  r ) ( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r ) ) } )
Distinct variable group:    x, r, y

Detailed syntax breakdown of Definition df-rlreg
StepHypRef Expression
1 crlreg 16036 . 2  class RLReg
2 vr . . 3  set  r
3 cvv 2801 . . 3  class  _V
4 vx . . . . . . . . 9  set  x
54cv 1631 . . . . . . . 8  class  x
6 vy . . . . . . . . 9  set  y
76cv 1631 . . . . . . . 8  class  y
82cv 1631 . . . . . . . . 9  class  r
9 cmulr 13225 . . . . . . . . 9  class  .r
108, 9cfv 5271 . . . . . . . 8  class  ( .r
`  r )
115, 7, 10co 5874 . . . . . . 7  class  ( x ( .r `  r
) y )
12 c0g 13416 . . . . . . . 8  class  0g
138, 12cfv 5271 . . . . . . 7  class  ( 0g
`  r )
1411, 13wceq 1632 . . . . . 6  wff  ( x ( .r `  r
) y )  =  ( 0g `  r
)
157, 13wceq 1632 . . . . . 6  wff  y  =  ( 0g `  r
)
1614, 15wi 4 . . . . 5  wff  ( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r ) )
17 cbs 13164 . . . . . 6  class  Base
188, 17cfv 5271 . . . . 5  class  ( Base `  r )
1916, 6, 18wral 2556 . . . 4  wff  A. y  e.  ( Base `  r
) ( ( x ( .r `  r
) y )  =  ( 0g `  r
)  ->  y  =  ( 0g `  r ) )
2019, 4, 18crab 2560 . . 3  class  { x  e.  ( Base `  r
)  |  A. y  e.  ( Base `  r
) ( ( x ( .r `  r
) y )  =  ( 0g `  r
)  ->  y  =  ( 0g `  r ) ) }
212, 3, 20cmpt 4093 . 2  class  ( r  e.  _V  |->  { x  e.  ( Base `  r
)  |  A. y  e.  ( Base `  r
) ( ( x ( .r `  r
) y )  =  ( 0g `  r
)  ->  y  =  ( 0g `  r ) ) } )
221, 21wceq 1632 1  wff RLReg  =  ( r  e.  _V  |->  { x  e.  ( Base `  r )  |  A. y  e.  ( Base `  r ) ( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r ) ) } )
Colors of variables: wff set class
This definition is referenced by:  rrgval  16044
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