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Definition df-idlNEW 25547
Description: Define the class of (two-sided) ideals of a ring  R. A subset of  R is an ideal if it contains  0, is closed under addition, and is closed under multiplication on either side by any element of  R. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
Assertion
Ref Expression
df-idlNEW  |- IdlNEW  =  ( r  e.  Ring  |->  { i  e.  ~P ( Base `  r )  |  ( ( 0g `  r
)  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x ( +g  `  r ) y )  e.  i  /\  A. z  e.  ( Base `  r ) ( ( z ( .r `  r ) x )  e.  i  /\  (
x ( .r `  r ) z )  e.  i ) ) ) } )
Distinct variable group:    i, r, x, y, z

Detailed syntax breakdown of Definition df-idlNEW
StepHypRef Expression
1 cidln 25546 . 2  class IdlNEW
2 vr . . 3  set  r
3 crg 15353 . . 3  class  Ring
42cv 1631 . . . . . . 7  class  r
5 c0g 13416 . . . . . . 7  class  0g
64, 5cfv 5271 . . . . . 6  class  ( 0g
`  r )
7 vi . . . . . . 7  set  i
87cv 1631 . . . . . 6  class  i
96, 8wcel 1696 . . . . 5  wff  ( 0g
`  r )  e.  i
10 vx . . . . . . . . . . 11  set  x
1110cv 1631 . . . . . . . . . 10  class  x
12 vy . . . . . . . . . . 11  set  y
1312cv 1631 . . . . . . . . . 10  class  y
14 cplusg 13224 . . . . . . . . . . 11  class  +g
154, 14cfv 5271 . . . . . . . . . 10  class  ( +g  `  r )
1611, 13, 15co 5874 . . . . . . . . 9  class  ( x ( +g  `  r
) y )
1716, 8wcel 1696 . . . . . . . 8  wff  ( x ( +g  `  r
) y )  e.  i
1817, 12, 8wral 2556 . . . . . . 7  wff  A. y  e.  i  ( x
( +g  `  r ) y )  e.  i
19 vz . . . . . . . . . . . 12  set  z
2019cv 1631 . . . . . . . . . . 11  class  z
21 cmulr 13225 . . . . . . . . . . . 12  class  .r
224, 21cfv 5271 . . . . . . . . . . 11  class  ( .r
`  r )
2320, 11, 22co 5874 . . . . . . . . . 10  class  ( z ( .r `  r
) x )
2423, 8wcel 1696 . . . . . . . . 9  wff  ( z ( .r `  r
) x )  e.  i
2511, 20, 22co 5874 . . . . . . . . . 10  class  ( x ( .r `  r
) z )
2625, 8wcel 1696 . . . . . . . . 9  wff  ( x ( .r `  r
) z )  e.  i
2724, 26wa 358 . . . . . . . 8  wff  ( ( z ( .r `  r ) x )  e.  i  /\  (
x ( .r `  r ) z )  e.  i )
28 cbs 13164 . . . . . . . . 9  class  Base
294, 28cfv 5271 . . . . . . . 8  class  ( Base `  r )
3027, 19, 29wral 2556 . . . . . . 7  wff  A. z  e.  ( Base `  r
) ( ( z ( .r `  r
) x )  e.  i  /\  ( x ( .r `  r
) z )  e.  i )
3118, 30wa 358 . . . . . 6  wff  ( A. y  e.  i  (
x ( +g  `  r
) y )  e.  i  /\  A. z  e.  ( Base `  r
) ( ( z ( .r `  r
) x )  e.  i  /\  ( x ( .r `  r
) z )  e.  i ) )
3231, 10, 8wral 2556 . . . . 5  wff  A. x  e.  i  ( A. y  e.  i  (
x ( +g  `  r
) y )  e.  i  /\  A. z  e.  ( Base `  r
) ( ( z ( .r `  r
) x )  e.  i  /\  ( x ( .r `  r
) z )  e.  i ) )
339, 32wa 358 . . . 4  wff  ( ( 0g `  r )  e.  i  /\  A. x  e.  i  ( A. y  e.  i 
( x ( +g  `  r ) y )  e.  i  /\  A. z  e.  ( Base `  r ) ( ( z ( .r `  r ) x )  e.  i  /\  (
x ( .r `  r ) z )  e.  i ) ) )
3429cpw 3638 . . . 4  class  ~P ( Base `  r )
3533, 7, 34crab 2560 . . 3  class  { i  e.  ~P ( Base `  r )  |  ( ( 0g `  r
)  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x ( +g  `  r ) y )  e.  i  /\  A. z  e.  ( Base `  r ) ( ( z ( .r `  r ) x )  e.  i  /\  (
x ( .r `  r ) z )  e.  i ) ) ) }
362, 3, 35cmpt 4093 . 2  class  ( r  e.  Ring  |->  { i  e.  ~P ( Base `  r )  |  ( ( 0g `  r
)  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x ( +g  `  r ) y )  e.  i  /\  A. z  e.  ( Base `  r ) ( ( z ( .r `  r ) x )  e.  i  /\  (
x ( .r `  r ) z )  e.  i ) ) ) } )
371, 36wceq 1632 1  wff IdlNEW  =  ( r  e.  Ring  |->  { i  e.  ~P ( Base `  r )  |  ( ( 0g `  r
)  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x ( +g  `  r ) y )  e.  i  /\  A. z  e.  ( Base `  r ) ( ( z ( .r `  r ) x )  e.  i  /\  (
x ( .r `  r ) z )  e.  i ) ) ) } )
Colors of variables: wff set class
This definition is referenced by:  idlvalNEW  25548
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