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Theorem igenidl 28886
Description: The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
igenval.1  |-  G  =  ( 1st `  R
)
igenval.2  |-  X  =  ran  G
Assertion
Ref Expression
igenidl  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  e.  ( Idl `  R
) )

Proof of Theorem igenidl
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 igenval.1 . . 3  |-  G  =  ( 1st `  R
)
2 igenval.2 . . 3  |-  X  =  ran  G
31, 2igenval 28884 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
41, 2rngoidl 28847 . . . . 5  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
5 sseq2 3397 . . . . . 6  |-  ( j  =  X  ->  ( S  C_  j  <->  S  C_  X
) )
65rspcev 3092 . . . . 5  |-  ( ( X  e.  ( Idl `  R )  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
74, 6sylan 471 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
8 rabn0 3676 . . . 4  |-  ( { j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/)  <->  E. j  e.  ( Idl `  R
) S  C_  j
)
97, 8sylibr 212 . . 3  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  { j  e.  ( Idl `  R
)  |  S  C_  j }  =/=  (/) )
10 ssrab2 3456 . . . 4  |-  { j  e.  ( Idl `  R
)  |  S  C_  j }  C_  ( Idl `  R )
11 intidl 28852 . . . 4  |-  ( ( R  e.  RingOps  /\  {
j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/)  /\  {
j  e.  ( Idl `  R )  |  S  C_  j }  C_  ( Idl `  R ) )  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  e.  ( Idl `  R ) )
1210, 11mp3an3 1303 . . 3  |-  ( ( R  e.  RingOps  /\  {
j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/) )  ->  |^| { j  e.  ( Idl `  R )  |  S  C_  j }  e.  ( Idl `  R ) )
139, 12syldan 470 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  e.  ( Idl `  R ) )
143, 13eqeltrd 2517 1  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  e.  ( Idl `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   E.wrex 2735   {crab 2738    C_ wss 3347   (/)c0 3656   |^|cint 4147   ran crn 4860   ` cfv 5437  (class class class)co 6110   1stc1st 6594   RingOpscrngo 23881   Idlcidl 28830    IdlGen cigen 28882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-fo 5443  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-grpo 23697  df-gid 23698  df-ablo 23788  df-rngo 23882  df-idl 28833  df-igen 28883
This theorem is referenced by:  igenval2  28889  isfldidl  28891  ispridlc  28893
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