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Theorem idl0cl 26746
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 2296 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2296 . . . 4  |-  ran  G  =  ran  G
4 idl0cl.2 . . . 4  |-  Z  =  (GId `  G )
51, 2, 3, 4isidl 26742 . . 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 470 . 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 968 1  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  Z  e.  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   RingOpscrngo 21058   Idlcidl 26735
This theorem is referenced by:  divrngidl  26756  intidl  26757  unichnidl  26759  maxidln0  26773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-idl 26738
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