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Theorem maxidlval 30331
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 5871 . . 3  |-  ( r  =  R  ->  ( Idl `  r )  =  ( Idl `  R
) )
2 fveq2 5871 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 maxidlval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 5235 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 maxidlval.2 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2526 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
87neeq2d 2745 . . . 4  |-  ( r  =  R  ->  (
i  =/=  ran  ( 1st `  r )  <->  i  =/=  X ) )
97eqeq2d 2481 . . . . . . 7  |-  ( r  =  R  ->  (
j  =  ran  ( 1st `  r )  <->  j  =  X ) )
109orbi2d 701 . . . . . 6  |-  ( r  =  R  ->  (
( j  =  i  \/  j  =  ran  ( 1st `  r ) )  <->  ( j  =  i  \/  j  =  X ) ) )
1110imbi2d 316 . . . . 5  |-  ( r  =  R  ->  (
( i  C_  j  ->  ( j  =  i  \/  j  =  ran  ( 1st `  r ) ) )  <->  ( i  C_  j  ->  ( j  =  i  \/  j  =  X ) ) ) )
121, 11raleqbidv 3077 . . . 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 710 . . 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 3113 . 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 30304 . 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 5881 . . 3  |-  ( Idl `  R )  e.  _V
1716rabex 4603 . 2  |-  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  X ) ) ) }  e.  _V
1814, 15, 17fvmpt 5956 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 setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   {crab 2821    C_ wss 3481   ran crn 5005   ` cfv 5593   1stc1st 6792   RingOpscrngo 25168   Idlcidl 30299   MaxIdlcmaxidl 30301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-maxidl 30304
This theorem is referenced by:  ismaxidl  30332
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