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Theorem maxidln1 26772
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 2296 . . 3  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2296 . . 3  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
31, 2maxidlnr 26770 . 2  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  M  =/=  ran  ( 1st `  R
) )
4 maxidlidl 26769 . . 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 26754 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( Idl `  R
) )  ->  ( U  e.  M  <->  M  =  ran  ( 1st `  R
) ) )
87necon3bbid 2493 . . 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 1632    e. wcel 1696    =/= wne 2459   ran crn 4706   ` cfv 5271   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   RingOpscrngo 21058   Idlcidl 26735   MaxIdlcmaxidl 26737
This theorem is referenced by:  maxidln0  26773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059  df-idl 26738  df-maxidl 26740
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