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Theorem idlss 26147
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 2366 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 idlss.2 . . . 4  |-  X  =  ran  G
4 eqid 2366 . . . 4  |-  (GId `  G )  =  (GId
`  G )
51, 2, 3, 4isidl 26145 . . 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 470 . 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 968 1  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628    C_ wss 3238   ran crn 4793   ` cfv 5358  (class class class)co 5981   1stc1st 6247   2ndc2nd 6248  GIdcgi 21165   RingOpscrngo 21353   Idlcidl 26138
This theorem is referenced by:  idlcl  26148  idlnegcl  26153  1idl  26157  divrngidl  26159  intidl  26160  unichnidl  26162  ispridl2  26169  igenmin  26195  igenidl2  26196
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-idl 26141
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