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Theorem igenmin 30666
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 2457 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2457 . . . . 5  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
31, 2idlss 30618 . . . 4  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_ 
ran  ( 1st `  R
) )
4 sstr 3507 . . . . . . 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 30663 . . . . . 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 30407 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  S  C_  I )  -> 
( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R
)  |  S  C_  j } )
1093impa 1191 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  ( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R )  |  S  C_  j } )
11 sseq2 3521 . . . 4  |-  ( j  =  I  ->  ( S  C_  j  <->  S  C_  I
) )
1211intminss 4315 . . 3  |-  ( ( I  e.  ( Idl `  R )  /\  S  C_  I )  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  C_  I )
13123adant1 1014 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  C_  I )
1410, 13eqsstrd 3533 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 973    = wceq 1395    e. wcel 1819   {crab 2811    C_ wss 3471   |^|cint 4288   ran crn 5009   ` cfv 5594  (class class class)co 6296   1stc1st 6797   RingOpscrngo 25504   Idlcidl 30609    IdlGen cigen 30661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-grpo 25320  df-gid 25321  df-ablo 25411  df-rngo 25505  df-idl 30612  df-igen 30662
This theorem is referenced by:  igenval2  30668
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