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Theorem maxidlmax 28843
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 28840 . . . . . 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 484 . . . . 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 1002 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) )
6 sseq2 3378 . . . . . 6  |-  ( j  =  I  ->  ( M  C_  j  <->  M  C_  I
) )
7 eqeq1 2449 . . . . . . 7  |-  ( j  =  I  ->  (
j  =  M  <->  I  =  M ) )
8 eqeq1 2449 . . . . . . 7  |-  ( j  =  I  ->  (
j  =  X  <->  I  =  X ) )
97, 8orbi12d 709 . . . . . 6  |-  ( j  =  I  ->  (
( j  =  M  \/  j  =  X )  <->  ( I  =  M  \/  I  =  X ) ) )
106, 9imbi12d 320 . . . . 5  |-  ( j  =  I  ->  (
( M  C_  j  ->  ( j  =  M  \/  j  =  X ) )  <->  ( M  C_  I  ->  ( I  =  M  \/  I  =  X ) ) ) )
1110rspcva 3071 . . . 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 474 . . 3  |-  ( ( I  e.  ( Idl `  R )  /\  ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
) )  ->  ( M  C_  I  ->  (
I  =  M  \/  I  =  X )
) )
1312ancoms 453 . 2  |-  ( ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  /\  I  e.  ( Idl `  R ) )  -> 
( M  C_  I  ->  ( I  =  M  \/  I  =  X ) ) )
1413impr 619 1  |-  ( ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  /\  ( I  e.  ( Idl `  R )  /\  M  C_  I ) )  ->  ( I  =  M  \/  I  =  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715    C_ wss 3328   ran crn 4841   ` cfv 5418   1stc1st 6575   RingOpscrngo 23862   Idlcidl 28807   MaxIdlcmaxidl 28809
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 4413  ax-nul 4421  ax-pr 4531
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fv 5426  df-maxidl 28812
This theorem is referenced by: (None)
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