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Theorem maxidlidl 26335
Description: A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
Assertion
Ref Expression
maxidlidl  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  M  e.  ( Idl `  R ) )

Proof of Theorem maxidlidl
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 eqid 2380 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2380 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
31, 2ismaxidl 26334 . . 3  |-  ( R  e.  RingOps  ->  ( M  e.  ( MaxIdl `  R )  <->  ( M  e.  ( Idl `  R )  /\  M  =/=  ran  ( 1st `  R
)  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  ran  ( 1st `  R
) ) ) ) ) )
4 3anass 940 . . 3  |-  ( ( M  e.  ( Idl `  R )  /\  M  =/=  ran  ( 1st `  R
)  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  ran  ( 1st `  R
) ) ) )  <-> 
( M  e.  ( Idl `  R )  /\  ( M  =/= 
ran  ( 1st `  R
)  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  ran  ( 1st `  R
) ) ) ) ) )
53, 4syl6bb 253 . 2  |-  ( R  e.  RingOps  ->  ( M  e.  ( MaxIdl `  R )  <->  ( M  e.  ( Idl `  R )  /\  ( M  =/=  ran  ( 1st `  R )  /\  A. j  e.  ( Idl `  R ) ( M 
C_  j  ->  (
j  =  M  \/  j  =  ran  ( 1st `  R ) ) ) ) ) ) )
65simprbda 607 1  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  M  e.  ( Idl `  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642    C_ wss 3256   ran crn 4812   ` cfv 5387   1stc1st 6279   RingOpscrngo 21804   Idlcidl 26301   MaxIdlcmaxidl 26303
This theorem is referenced by:  maxidln1  26338  maxidln0  26339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fv 5395  df-maxidl 26306
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