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Theorem igenval 26625
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 26588 . . . . 5  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
4 sseq2 3362 . . . . . 6  |-  ( j  =  X  ->  ( S  C_  j  <->  S  C_  X
) )
54rspcev 3044 . . . . 5  |-  ( ( X  e.  ( Idl `  R )  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
63, 5sylan 458 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
7 rabn0 3639 . . . 4  |-  ( { j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/)  <->  E. j  e.  ( Idl `  R
) S  C_  j
)
86, 7sylibr 204 . . 3  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  { j  e.  ( Idl `  R
)  |  S  C_  j }  =/=  (/) )
9 intex 4348 . . 3  |-  ( { j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/)  <->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  e.  _V )
108, 9sylib 189 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  e.  _V )
11 fvex 5734 . . . . . . 7  |-  ( 1st `  R )  e.  _V
121, 11eqeltri 2505 . . . . . 6  |-  G  e. 
_V
1312rnex 5125 . . . . 5  |-  ran  G  e.  _V
142, 13eqeltri 2505 . . . 4  |-  X  e. 
_V
1514elpw2 4356 . . 3  |-  ( S  e.  ~P X  <->  S  C_  X
)
16 simpl 444 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  r  =  R )
1716fveq2d 5724 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( Idl `  r
)  =  ( Idl `  R ) )
18 sseq1 3361 . . . . . . 7  |-  ( s  =  S  ->  (
s  C_  j  <->  S  C_  j
) )
1918adantl 453 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( s  C_  j  <->  S 
C_  j ) )
2017, 19rabeqbidv 2943 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  { j  e.  ( Idl `  r )  |  s  C_  j }  =  { j  e.  ( Idl `  R
)  |  S  C_  j } )
2120inteqd 4047 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  |^| { j  e.  ( Idl `  r
)  |  s  C_  j }  =  |^| { j  e.  ( Idl `  R )  |  S  C_  j } )
22 fveq2 5720 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2322, 1syl6eqr 2485 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
2423rneqd 5089 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
2524, 2syl6eqr 2485 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
2625pweqd 3796 . . . 4  |-  ( r  =  R  ->  ~P ran  ( 1st `  r
)  =  ~P X
)
27 df-igen 26624 . . . 4  |-  IdlGen  =  ( r  e.  RingOps ,  s  e.  ~P ran  ( 1st `  r )  |->  |^|
{ j  e.  ( Idl `  r )  |  s  C_  j } )
2821, 26, 27ovmpt2x 6194 . . 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 1224 . 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 1280 1  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { j  e.  ( Idl `  R )  |  S  C_  j } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   {crab 2701   _Vcvv 2948    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   |^|cint 4042   ran crn 4871   ` cfv 5446  (class class class)co 6073   1stc1st 6339   RingOpscrngo 21953   Idlcidl 26571    IdlGen cigen 26623
This theorem is referenced by:  igenss  26626  igenidl  26627  igenmin  26628  igenidl2  26629  igenval2  26630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-grpo 21769  df-gid 21770  df-ablo 21860  df-rngo 21954  df-idl 26574  df-igen 26624
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