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Theorem igenval 29008
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 28971 . . . . 5  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
4 sseq2 3485 . . . . . 6  |-  ( j  =  X  ->  ( S  C_  j  <->  S  C_  X
) )
54rspcev 3177 . . . . 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 3764 . . . 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 4555 . . 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 5808 . . . . . . 7  |-  ( 1st `  R )  e.  _V
121, 11eqeltri 2538 . . . . . 6  |-  G  e. 
_V
1312rnex 6621 . . . . 5  |-  ran  G  e.  _V
142, 13eqeltri 2538 . . . 4  |-  X  e. 
_V
1514elpw2 4563 . . 3  |-  ( S  e.  ~P X  <->  S  C_  X
)
16 simpl 457 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  r  =  R )
1716fveq2d 5802 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( Idl `  r
)  =  ( Idl `  R ) )
18 sseq1 3484 . . . . . . 7  |-  ( s  =  S  ->  (
s  C_  j  <->  S  C_  j
) )
1918adantl 466 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( s  C_  j  <->  S 
C_  j ) )
2017, 19rabeqbidv 3071 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  { j  e.  ( Idl `  r )  |  s  C_  j }  =  { j  e.  ( Idl `  R
)  |  S  C_  j } )
2120inteqd 4240 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  |^| { j  e.  ( Idl `  r
)  |  s  C_  j }  =  |^| { j  e.  ( Idl `  R )  |  S  C_  j } )
22 fveq2 5798 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2322, 1syl6eqr 2513 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
2423rneqd 5174 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
2524, 2syl6eqr 2513 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
2625pweqd 3972 . . . 4  |-  ( r  =  R  ->  ~P ran  ( 1st `  r
)  =  ~P X
)
27 df-igen 29007 . . . 4  |-  IdlGen  =  ( r  e.  RingOps ,  s  e.  ~P ran  ( 1st `  r )  |->  |^|
{ j  e.  ( Idl `  r )  |  s  C_  j } )
2821, 26, 27ovmpt2x 6328 . . 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 1259 . 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 1316 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 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799   {crab 2802   _Vcvv 3076    C_ wss 3435   (/)c0 3744   ~Pcpw 3967   |^|cint 4235   ran crn 4948   ` cfv 5525  (class class class)co 6199   1stc1st 6684   RingOpscrngo 24013   Idlcidl 28954    IdlGen cigen 29006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fo 5531  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-grpo 23829  df-gid 23830  df-ablo 23920  df-rngo 24014  df-idl 28957  df-igen 29007
This theorem is referenced by:  igenss  29009  igenidl  29010  igenmin  29011  igenidl2  29012  igenval2  29013
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