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Theorem ismaxidl 28852
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 28851 . . 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 2510 . 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 2628 . . . . 5  |-  ( i  =  M  ->  (
i  =/=  X  <->  M  =/=  X ) )
6 sseq1 3389 . . . . . . 7  |-  ( i  =  M  ->  (
i  C_  j  <->  M  C_  j
) )
7 eqeq2 2452 . . . . . . . 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 2747 . . . . 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 3129 . . 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 969 . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   {crab 2731    C_ wss 3340   ran crn 4853   ` cfv 5430   1stc1st 6587   RingOpscrngo 23874   Idlcidl 28819   MaxIdlcmaxidl 28821
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 4425  ax-nul 4433  ax-pr 4543
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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5393  df-fun 5432  df-fv 5438  df-maxidl 28824
This theorem is referenced by:  maxidlidl  28853  maxidlnr  28854  maxidlmax  28855
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