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Theorem igenidl 26791
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 26789 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
41, 2rngoidl 26752 . . . . 5  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
5 sseq2 3213 . . . . . 6  |-  ( j  =  X  ->  ( S  C_  j  <->  S  C_  X
) )
65rspcev 2897 . . . . 5  |-  ( ( X  e.  ( Idl `  R )  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
74, 6sylan 457 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
8 rabn0 3487 . . . 4  |-  ( { j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/)  <->  E. j  e.  ( Idl `  R
) S  C_  j
)
97, 8sylibr 203 . . 3  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  { j  e.  ( Idl `  R
)  |  S  C_  j }  =/=  (/) )
10 ssrab2 3271 . . . 4  |-  { j  e.  ( Idl `  R
)  |  S  C_  j }  C_  ( Idl `  R )
11 intidl 26757 . . . 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 1266 . . 3  |-  ( ( R  e.  RingOps  /\  {
j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/) )  ->  |^| { j  e.  ( Idl `  R )  |  S  C_  j }  e.  ( Idl `  R ) )
139, 12syldan 456 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  e.  ( Idl `  R ) )
143, 13eqeltrd 2370 1  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  e.  ( Idl `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   |^|cint 3878   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   RingOpscrngo 21058   Idlcidl 26735    IdlGen cigen 26787
This theorem is referenced by:  igenval2  26794  isfldidl  26796  ispridlc  26798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ablo 20965  df-rngo 21059  df-idl 26738  df-igen 26788
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