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Theorem maxidlval 26664
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 5525 . . 3  |-  ( r  =  R  ->  ( Idl `  r )  =  ( Idl `  R
) )
2 fveq2 5525 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 maxidlval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2333 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 4906 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 maxidlval.2 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2333 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
87neeq2d 2460 . . . 4  |-  ( r  =  R  ->  (
i  =/=  ran  ( 1st `  r )  <->  i  =/=  X ) )
97eqeq2d 2294 . . . . . . 7  |-  ( r  =  R  ->  (
j  =  ran  ( 1st `  r )  <->  j  =  X ) )
109orbi2d 682 . . . . . 6  |-  ( r  =  R  ->  (
( j  =  i  \/  j  =  ran  ( 1st `  r ) )  <->  ( j  =  i  \/  j  =  X ) ) )
1110imbi2d 307 . . . . 5  |-  ( r  =  R  ->  (
( i  C_  j  ->  ( j  =  i  \/  j  =  ran  ( 1st `  r ) ) )  <->  ( i  C_  j  ->  ( j  =  i  \/  j  =  X ) ) ) )
121, 11raleqbidv 2748 . . . 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 691 . . 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 2783 . 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 26637 . 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 5539 . . 3  |-  ( Idl `  R )  e.  _V
1716rabex 4165 . 2  |-  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  X ) ) ) }  e.  _V
1814, 15, 17fvmpt 5602 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 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547    C_ wss 3152   ran crn 4690   ` cfv 5255   1stc1st 6120   RingOpscrngo 21042   Idlcidl 26632   MaxIdlcmaxidl 26634
This theorem is referenced by:  ismaxidl  26665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-maxidl 26637
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