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Theorem idlss 26524
Description: An ideal of  R is a subset of  R. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idlss.1  |-  G  =  ( 1st `  R
)
idlss.2  |-  X  =  ran  G
Assertion
Ref Expression
idlss  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_  X )

Proof of Theorem idlss
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idlss.1 . . . 4  |-  G  =  ( 1st `  R
)
2 eqid 2412 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 idlss.2 . . . 4  |-  X  =  ran  G
4 eqid 2412 . . . 4  |-  (GId `  G )  =  (GId
`  G )
51, 2, 3, 4isidl 26522 . . 3  |-  ( R  e.  RingOps  ->  ( I  e.  ( Idl `  R
)  <->  ( I  C_  X  /\  (GId `  G
)  e.  I  /\  A. x  e.  I  ( A. y  e.  I 
( x G y )  e.  I  /\  A. z  e.  X  ( ( z ( 2nd `  R ) x )  e.  I  /\  (
x ( 2nd `  R
) z )  e.  I ) ) ) ) )
65biimpa 471 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  (
I  C_  X  /\  (GId `  G )  e.  I  /\  A. x  e.  I  ( A. y  e.  I  (
x G y )  e.  I  /\  A. z  e.  X  (
( z ( 2nd `  R ) x )  e.  I  /\  (
x ( 2nd `  R
) z )  e.  I ) ) ) )
76simp1d 969 1  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674    C_ wss 3288   ran crn 4846   ` cfv 5421  (class class class)co 6048   1stc1st 6314   2ndc2nd 6315  GIdcgi 21736   RingOpscrngo 21924   Idlcidl 26515
This theorem is referenced by:  idlcl  26525  idlnegcl  26530  1idl  26534  divrngidl  26536  intidl  26537  unichnidl  26539  ispridl2  26546  igenmin  26572  igenidl2  26573
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-idl 26518
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