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Theorem maxidlidl 26642
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 2435 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2435 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
31, 2ismaxidl 26641 . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    C_ wss 3312   ran crn 4871   ` cfv 5446   1stc1st 6339   RingOpscrngo 21955   Idlcidl 26608   MaxIdlcmaxidl 26610
This theorem is referenced by:  maxidln1  26645  maxidln0  26646
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-maxidl 26613
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