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Theorem ismaxidl 26768
Description: The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)
Hypotheses
Ref Expression
ismaxidl.1  |-  G  =  ( 1st `  R
)
ismaxidl.2  |-  X  =  ran  G
Assertion
Ref Expression
ismaxidl  |-  ( R  e.  RingOps  ->  ( M  e.  ( MaxIdl `  R )  <->  ( M  e.  ( Idl `  R )  /\  M  =/=  X  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) ) ) )
Distinct variable groups:    R, j    j, M
Allowed substitution hints:    G( j)    X( j)

Proof of Theorem ismaxidl
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 ismaxidl.1 . . . 4  |-  G  =  ( 1st `  R
)
2 ismaxidl.2 . . . 4  |-  X  =  ran  G
31, 2maxidlval 26767 . . 3  |-  ( R  e.  RingOps  ->  ( MaxIdl `  R
)  =  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  X ) ) ) } )
43eleq2d 2363 . 2  |-  ( R  e.  RingOps  ->  ( M  e.  ( MaxIdl `  R )  <->  M  e.  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  X ) ) ) } ) )
5 neeq1 2467 . . . . 5  |-  ( i  =  M  ->  (
i  =/=  X  <->  M  =/=  X ) )
6 sseq1 3212 . . . . . . 7  |-  ( i  =  M  ->  (
i  C_  j  <->  M  C_  j
) )
7 eqeq2 2305 . . . . . . . 8  |-  ( i  =  M  ->  (
j  =  i  <->  j  =  M ) )
87orbi1d 683 . . . . . . 7  |-  ( i  =  M  ->  (
( j  =  i  \/  j  =  X )  <->  ( j  =  M  \/  j  =  X ) ) )
96, 8imbi12d 311 . . . . . 6  |-  ( i  =  M  ->  (
( i  C_  j  ->  ( j  =  i  \/  j  =  X ) )  <->  ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) ) )
109ralbidv 2576 . . . . 5  |-  ( i  =  M  ->  ( A. j  e.  ( Idl `  R ) ( i  C_  j  ->  ( j  =  i  \/  j  =  X ) )  <->  A. j  e.  ( Idl `  R ) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) ) )
115, 10anbi12d 691 . . . 4  |-  ( i  =  M  ->  (
( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i  C_  j  ->  ( j  =  i  \/  j  =  X ) ) )  <->  ( M  =/=  X  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) ) ) )
1211elrab 2936 . . 3  |-  ( M  e.  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  X ) ) ) }  <->  ( M  e.  ( Idl `  R
)  /\  ( M  =/=  X  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) ) ) )
13 3anass 938 . . 3  |-  ( ( M  e.  ( Idl `  R )  /\  M  =/=  X  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) )  <-> 
( M  e.  ( Idl `  R )  /\  ( M  =/= 
X  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) ) ) )
1412, 13bitr4i 243 . 2  |-  ( M  e.  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
C_  j  ->  (
j  =  i  \/  j  =  X ) ) ) }  <->  ( M  e.  ( Idl `  R
)  /\  M  =/=  X  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) ) )
154, 14syl6bb 252 1  |-  ( R  e.  RingOps  ->  ( M  e.  ( MaxIdl `  R )  <->  ( M  e.  ( Idl `  R )  /\  M  =/=  X  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = 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:  maxidlidl  26769  maxidlnr  26770  maxidlmax  26771
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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