Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  igenidl Unicode version

Theorem igenidl 26688
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 26686 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
41, 2rngoidl 26649 . . . . 5  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
5 sseq2 3200 . . . . . 6  |-  ( j  =  X  ->  ( S  C_  j  <->  S  C_  X
) )
65rspcev 2884 . . . . 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 3474 . . . 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 3258 . . . 4  |-  { j  e.  ( Idl `  R
)  |  S  C_  j }  C_  ( Idl `  R )
11 intidl 26654 . . . 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 2357 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   |^|cint 3862   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   RingOpscrngo 21042   Idlcidl 26632    IdlGen cigen 26684
This theorem is referenced by:  igenval2  26691  isfldidl  26693  ispridlc  26695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ablo 20949  df-rngo 21043  df-idl 26635  df-igen 26685
  Copyright terms: Public domain W3C validator