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Theorem igenmin 26689
Description: The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
igenmin  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  ( R  IdlGen  S )  C_  I
)

Proof of Theorem igenmin
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2283 . . . . 5  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
31, 2idlss 26641 . . . 4  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_ 
ran  ( 1st `  R
) )
4 sstr 3187 . . . . . . 7  |-  ( ( S  C_  I  /\  I  C_  ran  ( 1st `  R ) )  ->  S  C_  ran  ( 1st `  R ) )
54ancoms 439 . . . . . 6  |-  ( ( I  C_  ran  ( 1st `  R )  /\  S  C_  I )  ->  S  C_ 
ran  ( 1st `  R
) )
61, 2igenval 26686 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  C_ 
ran  ( 1st `  R
) )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
75, 6sylan2 460 . . . . 5  |-  ( ( R  e.  RingOps  /\  (
I  C_  ran  ( 1st `  R )  /\  S  C_  I ) )  -> 
( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R
)  |  S  C_  j } )
87anassrs 629 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  C_  ran  ( 1st `  R ) )  /\  S  C_  I )  -> 
( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R
)  |  S  C_  j } )
93, 8syldanl 26334 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  S  C_  I )  -> 
( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R
)  |  S  C_  j } )
1093impa 1146 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  ( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R )  |  S  C_  j } )
11 sseq2 3200 . . . 4  |-  ( j  =  I  ->  ( S  C_  j  <->  S  C_  I
) )
1211intminss 3888 . . 3  |-  ( ( I  e.  ( Idl `  R )  /\  S  C_  I )  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  C_  I )
13123adant1 973 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  C_  I )
1410, 13eqsstrd 3212 1  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  ( R  IdlGen  S )  C_  I
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   |^|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
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
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