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Theorem igenmin 30092
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 2467 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2467 . . . . 5  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
31, 2idlss 30044 . . . 4  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_ 
ran  ( 1st `  R
) )
4 sstr 3512 . . . . . . 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 30089 . . . . . 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 29833 . . 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 3526 . . . 4  |-  ( j  =  I  ->  ( S  C_  j  <->  S  C_  I
) )
1211intminss 4308 . . 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 3538 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 1379    e. wcel 1767   {crab 2818    C_ wss 3476   |^|cint 4282   ran crn 5000   ` cfv 5588  (class class class)co 6284   1stc1st 6782   RingOpscrngo 25081   Idlcidl 30035    IdlGen cigen 30087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-grpo 24897  df-gid 24898  df-ablo 24988  df-rngo 25082  df-idl 30038  df-igen 30088
This theorem is referenced by:  igenval2  30094
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