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Theorem maxidlval 26767
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 5541 . . 3  |-  ( r  =  R  ->  ( Idl `  r )  =  ( Idl `  R
) )
2 fveq2 5541 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 maxidlval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2346 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 4922 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 maxidlval.2 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2346 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
87neeq2d 2473 . . . 4  |-  ( r  =  R  ->  (
i  =/=  ran  ( 1st `  r )  <->  i  =/=  X ) )
97eqeq2d 2307 . . . . . . 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 2761 . . . 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 2796 . 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 26740 . 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 5555 . . 3  |-  ( Idl `  R )  e.  _V
1716rabex 4181 . 2  |-  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  X ) ) ) }  e.  _V
1814, 15, 17fvmpt 5618 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560    C_ wss 3165   ran crn 4706   ` cfv 5271   1stc1st 6136   RingOpscrngo 21058   Idlcidl 26735   MaxIdlcmaxidl 26737
This theorem is referenced by:  ismaxidl  26768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-maxidl 26740
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