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Theorem maxidlmax 25816
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 25813 . . . . . 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 3234 . . . . . 6  |-  ( j  =  I  ->  ( M  C_  j  <->  M  C_  I
) )
7 eqeq1 2322 . . . . . . 7  |-  ( j  =  I  ->  (
j  =  M  <->  I  =  M ) )
8 eqeq1 2322 . . . . . . 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 2916 . . . 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 1633    e. wcel 1701    =/= wne 2479   A.wral 2577    C_ wss 3186   ran crn 4727   ` cfv 5292   1stc1st 6162   RingOpscrngo 21095   Idlcidl 25780   MaxIdlcmaxidl 25782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fv 5300  df-maxidl 25785
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