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Theorem ismaxidl 26077
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 26076 . . 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 2350 . 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 2454 . . . . 5  |-  ( i  =  M  ->  (
i  =/=  X  <->  M  =/=  X ) )
6 sseq1 3199 . . . . . . 7  |-  ( i  =  M  ->  (
i  C_  j  <->  M  C_  j
) )
7 eqeq2 2292 . . . . . . . 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 2563 . . . . 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 2923 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547    C_ wss 3152   ran crn 4690   ` cfv 5255   1stc1st 6120   RingOpscrngo 21042   Idlcidl 26044   MaxIdlcmaxidl 26046
This theorem is referenced by:  maxidlidl  26078  maxidlnr  26079  maxidlmax  26080
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 26049
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