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

Theorem maxidln0 30414
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 30410 . . . . 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 30387 . . . . 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 30413 . . . 4  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  U  e.  M )
9 nelneq 2558 . . . 4  |-  ( ( Z  e.  M  /\  -.  U  e.  M
)  ->  -.  Z  =  U )
105, 8, 9syl2anc 661 . . 3  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  -.  Z  =  U )
1110neqned 2644 . 2  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  Z  =/=  U )
1211necomd 2712 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 1381    e. wcel 1802    =/= wne 2636   ` cfv 5575   1stc1st 6780   2ndc2nd 6781  GIdcgi 25058   RingOpscrngo 25246   Idlcidl 30376   MaxIdlcmaxidl 30378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-fo 5581  df-fv 5583  df-riota 6239  df-ov 6281  df-1st 6782  df-2nd 6783  df-grpo 25062  df-gid 25063  df-ablo 25153  df-ass 25184  df-exid 25186  df-mgmOLD 25190  df-sgrOLD 25202  df-mndo 25209  df-rngo 25247  df-idl 30379  df-maxidl 30381
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator