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Theorem igenidl 30387
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 30385 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
41, 2rngoidl 30348 . . . . 5  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
5 sseq2 3531 . . . . . 6  |-  ( j  =  X  ->  ( S  C_  j  <->  S  C_  X
) )
65rspcev 3219 . . . . 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 3810 . . . 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 3590 . . . 4  |-  { j  e.  ( Idl `  R
)  |  S  C_  j }  C_  ( Idl `  R )
11 intidl 30353 . . . 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 1313 . . 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 2555 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 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   {crab 2821    C_ wss 3481   (/)c0 3790   |^|cint 4288   ran crn 5006   ` cfv 5594  (class class class)co 6295   1stc1st 6793   RingOpscrngo 25200   Idlcidl 30331    IdlGen cigen 30383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-grpo 25016  df-gid 25017  df-ablo 25107  df-rngo 25201  df-idl 30334  df-igen 30384
This theorem is referenced by:  igenval2  30390  isfldidl  30392  ispridlc  30394
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