Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idl0cl Unicode version

Theorem idl0cl 26319
Description: An ideal contains  0. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idl0cl.1  |-  G  =  ( 1st `  R
)
idl0cl.2  |-  Z  =  (GId `  G )
Assertion
Ref Expression
idl0cl  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  Z  e.  I )

Proof of Theorem idl0cl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idl0cl.1 . . . 4  |-  G  =  ( 1st `  R
)
2 eqid 2387 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2387 . . . 4  |-  ran  G  =  ran  G
4 idl0cl.2 . . . 4  |-  Z  =  (GId `  G )
51, 2, 3, 4isidl 26315 . . 3  |-  ( R  e.  RingOps  ->  ( I  e.  ( Idl `  R
)  <->  ( I  C_  ran  G  /\  Z  e.  I  /\  A. x  e.  I  ( A. y  e.  I  (
x G y )  e.  I  /\  A. z  e.  ran  G ( ( z ( 2nd `  R ) x )  e.  I  /\  (
x ( 2nd `  R
) z )  e.  I ) ) ) ) )
65biimpa 471 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  (
I  C_  ran  G  /\  Z  e.  I  /\  A. x  e.  I  ( A. y  e.  I 
( x G y )  e.  I  /\  A. z  e.  ran  G
( ( z ( 2nd `  R ) x )  e.  I  /\  ( x ( 2nd `  R ) z )  e.  I ) ) ) )
76simp2d 970 1  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  Z  e.  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649    C_ wss 3263   ran crn 4819   ` cfv 5394  (class class class)co 6020   1stc1st 6286   2ndc2nd 6287  GIdcgi 21623   RingOpscrngo 21811   Idlcidl 26308
This theorem is referenced by:  divrngidl  26329  intidl  26330  unichnidl  26332  maxidln0  26346
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-rab 2658  df-v 2901  df-sbc 3105  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-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-fv 5402  df-ov 6023  df-idl 26311
  Copyright terms: Public domain W3C validator