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Definition df-pridl 26739
Description: Define the class of prime ideals of a ring  R. A proper ideal  I of  R is prime if whenever  A B  C_  I for ideals  A and  B, either  A  C_  I or  B  C_  I. The more familiar definition using elements rather than ideals is equivalent provided  R is commutative; see ispridl2 26766 and ispridlc 26798. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
df-pridl  |-  PrIdl  =  ( r  e.  RingOps  |->  { i  e.  ( Idl `  r
)  |  ( i  =/=  ran  ( 1st `  r )  /\  A. a  e.  ( Idl `  r ) A. b  e.  ( Idl `  r
) ( A. x  e.  a  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) } )
Distinct variable group:    i, r, a, b, x, y

Detailed syntax breakdown of Definition df-pridl
StepHypRef Expression
1 cpridl 26736 . 2  class  PrIdl
2 vr . . 3  set  r
3 crngo 21058 . . 3  class  RingOps
4 vi . . . . . . 7  set  i
54cv 1631 . . . . . 6  class  i
62cv 1631 . . . . . . . 8  class  r
7 c1st 6136 . . . . . . . 8  class  1st
86, 7cfv 5271 . . . . . . 7  class  ( 1st `  r )
98crn 4706 . . . . . 6  class  ran  ( 1st `  r )
105, 9wne 2459 . . . . 5  wff  i  =/= 
ran  ( 1st `  r
)
11 vx . . . . . . . . . . . . 13  set  x
1211cv 1631 . . . . . . . . . . . 12  class  x
13 vy . . . . . . . . . . . . 13  set  y
1413cv 1631 . . . . . . . . . . . 12  class  y
15 c2nd 6137 . . . . . . . . . . . . 13  class  2nd
166, 15cfv 5271 . . . . . . . . . . . 12  class  ( 2nd `  r )
1712, 14, 16co 5874 . . . . . . . . . . 11  class  ( x ( 2nd `  r
) y )
1817, 5wcel 1696 . . . . . . . . . 10  wff  ( x ( 2nd `  r
) y )  e.  i
19 vb . . . . . . . . . . 11  set  b
2019cv 1631 . . . . . . . . . 10  class  b
2118, 13, 20wral 2556 . . . . . . . . 9  wff  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i
22 va . . . . . . . . . 10  set  a
2322cv 1631 . . . . . . . . 9  class  a
2421, 11, 23wral 2556 . . . . . . . 8  wff  A. x  e.  a  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i
2523, 5wss 3165 . . . . . . . . 9  wff  a  C_  i
2620, 5wss 3165 . . . . . . . . 9  wff  b  C_  i
2725, 26wo 357 . . . . . . . 8  wff  ( a 
C_  i  \/  b  C_  i )
2824, 27wi 4 . . . . . . 7  wff  ( A. x  e.  a  A. y  e.  b  (
x ( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) )
29 cidl 26735 . . . . . . . 8  class  Idl
306, 29cfv 5271 . . . . . . 7  class  ( Idl `  r )
3128, 19, 30wral 2556 . . . . . 6  wff  A. b  e.  ( Idl `  r
) ( A. x  e.  a  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) )
3231, 22, 30wral 2556 . . . . 5  wff  A. a  e.  ( Idl `  r
) A. b  e.  ( Idl `  r
) ( A. x  e.  a  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) )
3310, 32wa 358 . . . 4  wff  ( i  =/=  ran  ( 1st `  r )  /\  A. a  e.  ( Idl `  r ) A. b  e.  ( Idl `  r
) ( A. x  e.  a  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) )
3433, 4, 30crab 2560 . . 3  class  { i  e.  ( Idl `  r
)  |  ( i  =/=  ran  ( 1st `  r )  /\  A. a  e.  ( Idl `  r ) A. b  e.  ( Idl `  r
) ( A. x  e.  a  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) }
352, 3, 34cmpt 4093 . 2  class  ( r  e.  RingOps  |->  { i  e.  ( Idl `  r
)  |  ( i  =/=  ran  ( 1st `  r )  /\  A. a  e.  ( Idl `  r ) A. b  e.  ( Idl `  r
) ( A. x  e.  a  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) } )
361, 35wceq 1632 1  wff  PrIdl  =  ( r  e.  RingOps  |->  { i  e.  ( Idl `  r
)  |  ( i  =/=  ran  ( 1st `  r )  /\  A. a  e.  ( Idl `  r ) A. b  e.  ( Idl `  r
) ( A. x  e.  a  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) } )
Colors of variables: wff set class
This definition is referenced by:  pridlval  26761
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