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

Theorem maxidln0 26346
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 26342 . . . . 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 26319 . . . . 5  |-  ( ( R  e.  RingOps  /\  M  e.  ( Idl `  R
) )  ->  Z  e.  M )
51, 4syldan 457 . . . 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 26345 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  U  e.  M )
9 nelneq 2485 . . . 4  |-  ( ( Z  e.  M  /\  -.  U  e.  M
)  ->  -.  Z  =  U )
105, 8, 9syl2anc 643 . . 3  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  Z  =  U )
1110neneqad 2620 . 2  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  Z  =/=  U )
1211necomd 2633 1  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  U  =/=  Z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   ` cfv 5394   1stc1st 6286   2ndc2nd 6287  GIdcgi 21623   RingOpscrngo 21811   Idlcidl 26308   MaxIdlcmaxidl 26310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-ov 6023  df-1st 6288  df-2nd 6289  df-riota 6485  df-grpo 21627  df-gid 21628  df-ablo 21718  df-ass 21749  df-exid 21751  df-mgm 21755  df-sgr 21767  df-mndo 21774  df-rngo 21812  df-idl 26311  df-maxidl 26313
  Copyright terms: Public domain W3C validator