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Theorem maxidlmax 26653
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 26650 . . . . . 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 471 . . . . 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 971 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) )
6 sseq2 3370 . . . . . 6  |-  ( j  =  I  ->  ( M  C_  j  <->  M  C_  I
) )
7 eqeq1 2442 . . . . . . 7  |-  ( j  =  I  ->  (
j  =  M  <->  I  =  M ) )
8 eqeq1 2442 . . . . . . 7  |-  ( j  =  I  ->  (
j  =  X  <->  I  =  X ) )
97, 8orbi12d 691 . . . . . 6  |-  ( j  =  I  ->  (
( j  =  M  \/  j  =  X )  <->  ( I  =  M  \/  I  =  X ) ) )
106, 9imbi12d 312 . . . . 5  |-  ( j  =  I  ->  (
( M  C_  j  ->  ( j  =  M  \/  j  =  X ) )  <->  ( M  C_  I  ->  ( I  =  M  \/  I  =  X ) ) ) )
1110rspcva 3050 . . . 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 461 . . 3  |-  ( ( I  e.  ( Idl `  R )  /\  ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
) )  ->  ( M  C_  I  ->  (
I  =  M  \/  I  =  X )
) )
1312ancoms 440 . 2  |-  ( ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  /\  I  e.  ( Idl `  R ) )  -> 
( M  C_  I  ->  ( I  =  M  \/  I  =  X ) ) )
1413impr 603 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 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    C_ wss 3320   ran crn 4879   ` cfv 5454   1stc1st 6347   RingOpscrngo 21963   Idlcidl 26617   MaxIdlcmaxidl 26619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-maxidl 26622
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