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Theorem maxidln0 30045
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 30041 . . . . 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 30018 . . . . 5  |-  ( ( R  e.  RingOps  /\  M  e.  ( Idl `  R
) )  ->  Z  e.  M )
51, 4syldan 470 . . . 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 30044 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  U  e.  M )
9 nelneq 2584 . . . 4  |-  ( ( Z  e.  M  /\  -.  U  e.  M
)  ->  -.  Z  =  U )
105, 8, 9syl2anc 661 . . 3  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  Z  =  U )
1110neqned 2670 . 2  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  Z  =/=  U )
1211necomd 2738 1  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  U  =/=  Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5586   1stc1st 6779   2ndc2nd 6780  GIdcgi 24865   RingOpscrngo 25053   Idlcidl 30007   MaxIdlcmaxidl 30009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-riota 6243  df-ov 6285  df-1st 6781  df-2nd 6782  df-grpo 24869  df-gid 24870  df-ablo 24960  df-ass 24991  df-exid 24993  df-mgm 24997  df-sgr 25009  df-mndo 25016  df-rngo 25054  df-idl 30010  df-maxidl 30012
This theorem is referenced by: (None)
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