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Definition df-maxidl 30649
Description: Define the class of maximal ideals of a ring  R. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.)
Assertion
Ref Expression
df-maxidl  |-  MaxIdl  =  ( r  e.  RingOps  |->  { i  e.  ( Idl `  r
)  |  ( i  =/=  ran  ( 1st `  r )  /\  A. j  e.  ( Idl `  r ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  ran  ( 1st `  r ) ) ) ) } )
Distinct variable group:    i, r, j

Detailed syntax breakdown of Definition df-maxidl
StepHypRef Expression
1 cmaxidl 30646 . 2  class  MaxIdl
2 vr . . 3  setvar  r
3 crngo 25575 . . 3  class  RingOps
4 vi . . . . . . 7  setvar  i
54cv 1397 . . . . . 6  class  i
62cv 1397 . . . . . . . 8  class  r
7 c1st 6771 . . . . . . . 8  class  1st
86, 7cfv 5570 . . . . . . 7  class  ( 1st `  r )
98crn 4989 . . . . . 6  class  ran  ( 1st `  r )
105, 9wne 2649 . . . . 5  wff  i  =/= 
ran  ( 1st `  r
)
11 vj . . . . . . . . 9  setvar  j
1211cv 1397 . . . . . . . 8  class  j
135, 12wss 3461 . . . . . . 7  wff  i  C_  j
1411, 4weq 1738 . . . . . . . 8  wff  j  =  i
1512, 9wceq 1398 . . . . . . . 8  wff  j  =  ran  ( 1st `  r
)
1614, 15wo 366 . . . . . . 7  wff  ( j  =  i  \/  j  =  ran  ( 1st `  r
) )
1713, 16wi 4 . . . . . 6  wff  ( i 
C_  j  ->  (
j  =  i  \/  j  =  ran  ( 1st `  r ) ) )
18 cidl 30644 . . . . . . 7  class  Idl
196, 18cfv 5570 . . . . . 6  class  ( Idl `  r )
2017, 11, 19wral 2804 . . . . 5  wff  A. j  e.  ( Idl `  r
) ( i  C_  j  ->  ( j  =  i  \/  j  =  ran  ( 1st `  r
) ) )
2110, 20wa 367 . . . 4  wff  ( i  =/=  ran  ( 1st `  r )  /\  A. j  e.  ( Idl `  r ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  ran  ( 1st `  r ) ) ) )
2221, 4, 19crab 2808 . . 3  class  { i  e.  ( Idl `  r
)  |  ( i  =/=  ran  ( 1st `  r )  /\  A. j  e.  ( Idl `  r ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  ran  ( 1st `  r ) ) ) ) }
232, 3, 22cmpt 4497 . 2  class  ( r  e.  RingOps  |->  { i  e.  ( Idl `  r
)  |  ( i  =/=  ran  ( 1st `  r )  /\  A. j  e.  ( Idl `  r ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  ran  ( 1st `  r ) ) ) ) } )
241, 23wceq 1398 1  wff  MaxIdl  =  ( r  e.  RingOps  |->  { i  e.  ( Idl `  r
)  |  ( i  =/=  ran  ( 1st `  r )  /\  A. j  e.  ( Idl `  r ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  ran  ( 1st `  r ) ) ) ) } )
Colors of variables: wff setvar class
This definition is referenced by:  maxidlval  30676
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