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Theorem igenidl 26627
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 26625 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
41, 2rngoidl 26588 . . . . 5  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
5 sseq2 3362 . . . . . 6  |-  ( j  =  X  ->  ( S  C_  j  <->  S  C_  X
) )
65rspcev 3044 . . . . 5  |-  ( ( X  e.  ( Idl `  R )  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
74, 6sylan 458 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
8 rabn0 3639 . . . 4  |-  ( { j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/)  <->  E. j  e.  ( Idl `  R
) S  C_  j
)
97, 8sylibr 204 . . 3  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  { j  e.  ( Idl `  R
)  |  S  C_  j }  =/=  (/) )
10 ssrab2 3420 . . . 4  |-  { j  e.  ( Idl `  R
)  |  S  C_  j }  C_  ( Idl `  R )
11 intidl 26593 . . . 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 1268 . . 3  |-  ( ( R  e.  RingOps  /\  {
j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/) )  ->  |^| { j  e.  ( Idl `  R )  |  S  C_  j }  e.  ( Idl `  R ) )
139, 12syldan 457 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  e.  ( Idl `  R ) )
143, 13eqeltrd 2509 1  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  e.  ( Idl `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   {crab 2701    C_ wss 3312   (/)c0 3620   |^|cint 4042   ran crn 4871   ` cfv 5446  (class class class)co 6073   1stc1st 6339   RingOpscrngo 21953   Idlcidl 26571    IdlGen cigen 26623
This theorem is referenced by:  igenval2  26630  isfldidl  26632  ispridlc  26634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-grpo 21769  df-gid 21770  df-ablo 21860  df-rngo 21954  df-idl 26574  df-igen 26624
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