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Theorem maxidln0 26670
Description: A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
Hypotheses
Ref Expression
maxidln0.1  |-  G  =  ( 1st `  R
)
maxidln0.2  |-  H  =  ( 2nd `  R
)
maxidln0.3  |-  Z  =  (GId `  G )
maxidln0.4  |-  U  =  (GId `  H )
Assertion
Ref Expression
maxidln0  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  U  =/=  Z )

Proof of Theorem maxidln0
StepHypRef Expression
1 maxidlidl 26666 . . . . 5  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  M  e.  ( Idl `  R ) )
2 maxidln0.1 . . . . . 6  |-  G  =  ( 1st `  R
)
3 maxidln0.3 . . . . . 6  |-  Z  =  (GId `  G )
42, 3idl0cl 26643 . . . . 5  |-  ( ( R  e.  RingOps  /\  M  e.  ( Idl `  R
) )  ->  Z  e.  M )
51, 4syldan 456 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  Z  e.  M )
6 maxidln0.2 . . . . 5  |-  H  =  ( 2nd `  R
)
7 maxidln0.4 . . . . 5  |-  U  =  (GId `  H )
86, 7maxidln1 26669 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  U  e.  M )
9 nelneq 2381 . . . 4  |-  ( ( Z  e.  M  /\  -.  U  e.  M
)  ->  -.  Z  =  U )
105, 8, 9syl2anc 642 . . 3  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  Z  =  U )
11 df-ne 2448 . . 3  |-  ( Z  =/=  U  <->  -.  Z  =  U )
1210, 11sylibr 203 . 2  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  Z  =/=  U )
1312necomd 2529 1  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  U  =/=  Z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042   Idlcidl 26632   MaxIdlcmaxidl 26634
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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