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Theorem maxidlmax 30602
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 30599 . . . . . 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 1010 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) )
6 sseq2 3521 . . . . . 6  |-  ( j  =  I  ->  ( M  C_  j  <->  M  C_  I
) )
7 eqeq1 2461 . . . . . . 7  |-  ( j  =  I  ->  (
j  =  M  <->  I  =  M ) )
8 eqeq1 2461 . . . . . . 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 3208 . . . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807    C_ wss 3471   ran crn 5009   ` cfv 5594   1stc1st 6797   RingOpscrngo 25503   Idlcidl 30566   MaxIdlcmaxidl 30568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-maxidl 30571
This theorem is referenced by: (None)
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