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Theorem maxidln1 26669
Description: One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)
Hypotheses
Ref Expression
maxidln1.1  |-  H  =  ( 2nd `  R
)
maxidln1.2  |-  U  =  (GId `  H )
Assertion
Ref Expression
maxidln1  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  U  e.  M )

Proof of Theorem maxidln1
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2283 . . 3  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
31, 2maxidlnr 26667 . 2  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  M  =/=  ran  ( 1st `  R
) )
4 maxidlidl 26666 . . 3  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  M  e.  ( Idl `  R ) )
5 maxidln1.1 . . . . 5  |-  H  =  ( 2nd `  R
)
6 maxidln1.2 . . . . 5  |-  U  =  (GId `  H )
71, 5, 2, 61idl 26651 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( Idl `  R
) )  ->  ( U  e.  M  <->  M  =  ran  ( 1st `  R
) ) )
87necon3bbid 2480 . . 3  |-  ( ( R  e.  RingOps  /\  M  e.  ( Idl `  R
) )  ->  ( -.  U  e.  M  <->  M  =/=  ran  ( 1st `  R ) ) )
94, 8syldan 456 . 2  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  ( -.  U  e.  M  <->  M  =/=  ran  ( 1st `  R
) ) )
103, 9mpbird 223 1  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  U  e.  M )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ran crn 4690   ` cfv 5255   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042   Idlcidl 26632   MaxIdlcmaxidl 26634
This theorem is referenced by:  maxidln0  26670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043  df-idl 26635  df-maxidl 26637
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