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Theorem ismaxidl 30371
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 30370 . . 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 2537 . 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 2748 . . . . 5  |-  ( i  =  M  ->  (
i  =/=  X  <->  M  =/=  X ) )
6 sseq1 3530 . . . . . . 7  |-  ( i  =  M  ->  (
i  C_  j  <->  M  C_  j
) )
7 eqeq2 2482 . . . . . . . 8  |-  ( i  =  M  ->  (
j  =  i  <->  j  =  M ) )
87orbi1d 702 . . . . . . 7  |-  ( i  =  M  ->  (
( j  =  i  \/  j  =  X )  <->  ( j  =  M  \/  j  =  X ) ) )
96, 8imbi12d 320 . . . . . 6  |-  ( i  =  M  ->  (
( i  C_  j  ->  ( j  =  i  \/  j  =  X ) )  <->  ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) ) )
109ralbidv 2906 . . . . 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 710 . . . 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 3266 . . 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 977 . . 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 252 . 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 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   {crab 2821    C_ wss 3481   ran crn 5006   ` cfv 5594   1stc1st 6793   RingOpscrngo 25208   Idlcidl 30338   MaxIdlcmaxidl 30340
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 4574  ax-nul 4582  ax-pr 4692
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 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fv 5602  df-maxidl 30343
This theorem is referenced by:  maxidlidl  30372  maxidlnr  30373  maxidlmax  30374
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