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Theorem maxidlmax 26668
Description: A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
maxidlnr.1  |-  G  =  ( 1st `  R
)
maxidlnr.2  |-  X  =  ran  G
Assertion
Ref Expression
maxidlmax  |-  ( ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  /\  ( I  e.  ( Idl `  R )  /\  M  C_  I ) )  ->  ( I  =  M  \/  I  =  X ) )

Proof of Theorem maxidlmax
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 maxidlnr.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
2 maxidlnr.2 . . . . . . 7  |-  X  =  ran  G
31, 2ismaxidl 26665 . . . . . 6  |-  ( 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 ) ) ) ) )
43biimpa 470 . . . . 5  |-  ( ( 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 ) ) ) )
54simp3d 969 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) )
6 sseq2 3200 . . . . . 6  |-  ( j  =  I  ->  ( M  C_  j  <->  M  C_  I
) )
7 eqeq1 2289 . . . . . . 7  |-  ( j  =  I  ->  (
j  =  M  <->  I  =  M ) )
8 eqeq1 2289 . . . . . . 7  |-  ( j  =  I  ->  (
j  =  X  <->  I  =  X ) )
97, 8orbi12d 690 . . . . . 6  |-  ( j  =  I  ->  (
( j  =  M  \/  j  =  X )  <->  ( I  =  M  \/  I  =  X ) ) )
106, 9imbi12d 311 . . . . 5  |-  ( j  =  I  ->  (
( M  C_  j  ->  ( j  =  M  \/  j  =  X ) )  <->  ( M  C_  I  ->  ( I  =  M  \/  I  =  X ) ) ) )
1110rspcva 2882 . . . 4  |-  ( ( I  e.  ( Idl `  R )  /\  A. j  e.  ( Idl `  R ) ( M 
C_  j  ->  (
j  =  M  \/  j  =  X )
) )  ->  ( M  C_  I  ->  (
I  =  M  \/  I  =  X )
) )
125, 11sylan2 460 . . 3  |-  ( ( I  e.  ( Idl `  R )  /\  ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
) )  ->  ( M  C_  I  ->  (
I  =  M  \/  I  =  X )
) )
1312ancoms 439 . 2  |-  ( ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  /\  I  e.  ( Idl `  R ) )  -> 
( M  C_  I  ->  ( I  =  M  \/  I  =  X ) ) )
1413impr 602 1  |-  ( ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  /\  ( I  e.  ( Idl `  R )  /\  M  C_  I ) )  ->  ( I  =  M  \/  I  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   ran crn 4690   ` cfv 5255   1stc1st 6120   RingOpscrngo 21042   Idlcidl 26632   MaxIdlcmaxidl 26634
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 26637
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