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Theorem igenval 28832
Description: The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)
Hypotheses
Ref Expression
igenval.1  |-  G  =  ( 1st `  R
)
igenval.2  |-  X  =  ran  G
Assertion
Ref Expression
igenval  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
Distinct variable groups:    R, j    S, j    j, X
Allowed substitution hint:    G( j)

Proof of Theorem igenval
Dummy variables  r 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 igenval.1 . . . . . 6  |-  G  =  ( 1st `  R
)
2 igenval.2 . . . . . 6  |-  X  =  ran  G
31, 2rngoidl 28795 . . . . 5  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
4 sseq2 3373 . . . . . 6  |-  ( j  =  X  ->  ( S  C_  j  <->  S  C_  X
) )
54rspcev 3068 . . . . 5  |-  ( ( X  e.  ( Idl `  R )  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
63, 5sylan 471 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
7 rabn0 3652 . . . 4  |-  ( { j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/)  <->  E. j  e.  ( Idl `  R
) S  C_  j
)
86, 7sylibr 212 . . 3  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  { j  e.  ( Idl `  R
)  |  S  C_  j }  =/=  (/) )
9 intex 4443 . . 3  |-  ( { j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/)  <->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  e.  _V )
108, 9sylib 196 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  e.  _V )
11 fvex 5696 . . . . . . 7  |-  ( 1st `  R )  e.  _V
121, 11eqeltri 2508 . . . . . 6  |-  G  e. 
_V
1312rnex 6507 . . . . 5  |-  ran  G  e.  _V
142, 13eqeltri 2508 . . . 4  |-  X  e. 
_V
1514elpw2 4451 . . 3  |-  ( S  e.  ~P X  <->  S  C_  X
)
16 simpl 457 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  r  =  R )
1716fveq2d 5690 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( Idl `  r
)  =  ( Idl `  R ) )
18 sseq1 3372 . . . . . . 7  |-  ( s  =  S  ->  (
s  C_  j  <->  S  C_  j
) )
1918adantl 466 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( s  C_  j  <->  S 
C_  j ) )
2017, 19rabeqbidv 2962 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  { j  e.  ( Idl `  r )  |  s  C_  j }  =  { j  e.  ( Idl `  R
)  |  S  C_  j } )
2120inteqd 4128 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  |^| { j  e.  ( Idl `  r
)  |  s  C_  j }  =  |^| { j  e.  ( Idl `  R )  |  S  C_  j } )
22 fveq2 5686 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2322, 1syl6eqr 2488 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
2423rneqd 5062 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
2524, 2syl6eqr 2488 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
2625pweqd 3860 . . . 4  |-  ( r  =  R  ->  ~P ran  ( 1st `  r
)  =  ~P X
)
27 df-igen 28831 . . . 4  |-  IdlGen  =  ( r  e.  RingOps ,  s  e.  ~P ran  ( 1st `  r )  |->  |^|
{ j  e.  ( Idl `  r )  |  s  C_  j } )
2821, 26, 27ovmpt2x 6214 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  ~P X  /\  |^| { j  e.  ( Idl `  R )  |  S  C_  j }  e.  _V )  ->  ( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R
)  |  S  C_  j } )
2915, 28syl3an2br 1258 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X  /\  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  e.  _V )  ->  ( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R
)  |  S  C_  j } )
3010, 29mpd3an3 1315 1  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711   {crab 2714   _Vcvv 2967    C_ wss 3323   (/)c0 3632   ~Pcpw 3855   |^|cint 4123   ran crn 4836   ` cfv 5413  (class class class)co 6086   1stc1st 6570   RingOpscrngo 23830   Idlcidl 28778    IdlGen cigen 28830
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-grpo 23646  df-gid 23647  df-ablo 23737  df-rngo 23831  df-idl 28781  df-igen 28831
This theorem is referenced by:  igenss  28833  igenidl  28834  igenmin  28835  igenidl2  28836  igenval2  28837
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