Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  maxidln0 Structured version   Unicode version

Theorem maxidln0 32026
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 32022 . . . . 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 31999 . . . . 5  |-  ( ( R  e.  RingOps  /\  M  e.  ( Idl `  R
) )  ->  Z  e.  M )
51, 4syldan 472 . . . 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 32025 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  U  e.  M )
9 nelneq 2537 . . . 4  |-  ( ( Z  e.  M  /\  -.  U  e.  M
)  ->  -.  Z  =  U )
105, 8, 9syl2anc 665 . . 3  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  Z  =  U )
1110neqned 2625 . 2  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  Z  =/=  U )
1211necomd 2693 1  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  U  =/=  Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   ` cfv 5592   1stc1st 6796   2ndc2nd 6797  GIdcgi 25801   RingOpscrngo 25989   Idlcidl 31988   MaxIdlcmaxidl 31990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fo 5598  df-fv 5600  df-riota 6258  df-ov 6299  df-1st 6798  df-2nd 6799  df-grpo 25805  df-gid 25806  df-ablo 25896  df-ass 25927  df-exid 25929  df-mgmOLD 25933  df-sgrOLD 25945  df-mndo 25952  df-rngo 25990  df-idl 31991  df-maxidl 31993
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator