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Theorem igenmin 28862
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 2442 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2442 . . . . 5  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
31, 2idlss 28814 . . . 4  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_ 
ran  ( 1st `  R
) )
4 sstr 3363 . . . . . . 7  |-  ( ( S  C_  I  /\  I  C_  ran  ( 1st `  R ) )  ->  S  C_  ran  ( 1st `  R ) )
54ancoms 453 . . . . . 6  |-  ( ( I  C_  ran  ( 1st `  R )  /\  S  C_  I )  ->  S  C_ 
ran  ( 1st `  R
) )
61, 2igenval 28859 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  C_ 
ran  ( 1st `  R
) )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
75, 6sylan2 474 . . . . 5  |-  ( ( R  e.  RingOps  /\  (
I  C_  ran  ( 1st `  R )  /\  S  C_  I ) )  -> 
( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R
)  |  S  C_  j } )
87anassrs 648 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  C_  ran  ( 1st `  R ) )  /\  S  C_  I )  -> 
( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R
)  |  S  C_  j } )
93, 8syldanl 28603 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  S  C_  I )  -> 
( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R
)  |  S  C_  j } )
1093impa 1182 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  ( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R )  |  S  C_  j } )
11 sseq2 3377 . . . 4  |-  ( j  =  I  ->  ( S  C_  j  <->  S  C_  I
) )
1211intminss 4153 . . 3  |-  ( ( I  e.  ( Idl `  R )  /\  S  C_  I )  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  C_  I )
13123adant1 1006 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  C_  I )
1410, 13eqsstrd 3389 1  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  ( R  IdlGen  S )  C_  I
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {crab 2718    C_ wss 3327   |^|cint 4127   ran crn 4840   ` cfv 5417  (class class class)co 6090   1stc1st 6574   RingOpscrngo 23861   Idlcidl 28805    IdlGen cigen 28857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fo 5423  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-grpo 23677  df-gid 23678  df-ablo 23768  df-rngo 23862  df-idl 28808  df-igen 28858
This theorem is referenced by:  igenval2  28864
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