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Theorem maxidlval 28844
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 5696 . . 3  |-  ( r  =  R  ->  ( Idl `  r )  =  ( Idl `  R
) )
2 fveq2 5696 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 maxidlval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2493 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 5072 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 maxidlval.2 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2493 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
87neeq2d 2627 . . . 4  |-  ( r  =  R  ->  (
i  =/=  ran  ( 1st `  r )  <->  i  =/=  X ) )
97eqeq2d 2454 . . . . . . 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 2936 . . . 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 2972 . 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 28817 . 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 5706 . . 3  |-  ( Idl `  R )  e.  _V
1716rabex 4448 . 2  |-  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  X ) ) ) }  e.  _V
1814, 15, 17fvmpt 5779 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 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   {crab 2724    C_ wss 3333   ran crn 4846   ` cfv 5423   1stc1st 6580   RingOpscrngo 23867   Idlcidl 28812   MaxIdlcmaxidl 28814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fv 5431  df-maxidl 28817
This theorem is referenced by:  ismaxidl  28845
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