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Theorem maxidlval 26651
Description: The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)
Hypotheses
Ref Expression
maxidlval.1  |-  G  =  ( 1st `  R
)
maxidlval.2  |-  X  =  ran  G
Assertion
Ref Expression
maxidlval  |-  ( R  e.  RingOps  ->  ( MaxIdl `  R
)  =  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  X ) ) ) } )
Distinct variable group:    R, i, j
Allowed substitution hints:    G( i, j)    X( i, j)

Proof of Theorem maxidlval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5730 . . 3  |-  ( r  =  R  ->  ( Idl `  r )  =  ( Idl `  R
) )
2 fveq2 5730 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 maxidlval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2488 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 5099 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 maxidlval.2 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2488 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
87neeq2d 2617 . . . 4  |-  ( r  =  R  ->  (
i  =/=  ran  ( 1st `  r )  <->  i  =/=  X ) )
97eqeq2d 2449 . . . . . . 7  |-  ( r  =  R  ->  (
j  =  ran  ( 1st `  r )  <->  j  =  X ) )
109orbi2d 684 . . . . . 6  |-  ( r  =  R  ->  (
( j  =  i  \/  j  =  ran  ( 1st `  r ) )  <->  ( j  =  i  \/  j  =  X ) ) )
1110imbi2d 309 . . . . 5  |-  ( r  =  R  ->  (
( i  C_  j  ->  ( j  =  i  \/  j  =  ran  ( 1st `  r ) ) )  <->  ( i  C_  j  ->  ( j  =  i  \/  j  =  X ) ) ) )
121, 11raleqbidv 2918 . . . 4  |-  ( r  =  R  ->  ( A. j  e.  ( Idl `  r ) ( i  C_  j  ->  ( j  =  i  \/  j  =  ran  ( 1st `  r ) ) )  <->  A. j  e.  ( Idl `  R ) ( i  C_  j  ->  ( j  =  i  \/  j  =  X ) ) ) )
138, 12anbi12d 693 . . 3  |-  ( r  =  R  ->  (
( i  =/=  ran  ( 1st `  r )  /\  A. j  e.  ( Idl `  r
) ( i  C_  j  ->  ( j  =  i  \/  j  =  ran  ( 1st `  r
) ) ) )  <-> 
( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i  C_  j  ->  ( j  =  i  \/  j  =  X ) ) ) ) )
141, 13rabeqbidv 2953 . 2  |-  ( r  =  R  ->  { i  e.  ( Idl `  r
)  |  ( i  =/=  ran  ( 1st `  r )  /\  A. j  e.  ( Idl `  r ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  ran  ( 1st `  r ) ) ) ) }  =  { i  e.  ( Idl `  R )  |  ( i  =/= 
X  /\  A. j  e.  ( Idl `  R
) ( i  C_  j  ->  ( j  =  i  \/  j  =  X ) ) ) } )
15 df-maxidl 26624 . 2  |-  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 ) ) ) ) } )
16 fvex 5744 . . 3  |-  ( Idl `  R )  e.  _V
1716rabex 4356 . 2  |-  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  X ) ) ) }  e.  _V
1814, 15, 17fvmpt 5808 1  |-  ( R  e.  RingOps  ->  ( MaxIdl `  R
)  =  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  X ) ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711    C_ wss 3322   ran crn 4881   ` cfv 5456   1stc1st 6349   RingOpscrngo 21965   Idlcidl 26619   MaxIdlcmaxidl 26621
This theorem is referenced by:  ismaxidl  26652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fv 5464  df-maxidl 26624
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