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Theorem maxidln1 26656
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 2438 . . 3  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2438 . . 3  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
31, 2maxidlnr 26654 . 2  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  M  =/=  ran  ( 1st `  R
) )
4 maxidlidl 26653 . . 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 26638 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( Idl `  R
) )  ->  ( U  e.  M  <->  M  =  ran  ( 1st `  R
) ) )
87necon3bbid 2637 . . 3  |-  ( ( R  e.  RingOps  /\  M  e.  ( Idl `  R
) )  ->  ( -.  U  e.  M  <->  M  =/=  ran  ( 1st `  R ) ) )
94, 8syldan 458 . 2  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  ( -.  U  e.  M  <->  M  =/=  ran  ( 1st `  R
) ) )
103, 9mpbird 225 1  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  U  e.  M )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   ran crn 4881   ` cfv 5456   1stc1st 6349   2ndc2nd 6350  GIdcgi 21777   RingOpscrngo 21965   Idlcidl 26619   MaxIdlcmaxidl 26621
This theorem is referenced by:  maxidln0  26657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-ov 6086  df-1st 6351  df-2nd 6352  df-riota 6551  df-grpo 21781  df-gid 21782  df-ablo 21872  df-ass 21903  df-exid 21905  df-mgm 21909  df-sgr 21921  df-mndo 21928  df-rngo 21966  df-idl 26622  df-maxidl 26624
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