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Theorem igenval 26355
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 26318 . . . . 5  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
4 sseq2 3306 . . . . . 6  |-  ( j  =  X  ->  ( S  C_  j  <->  S  C_  X
) )
54rspcev 2988 . . . . 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 3583 . . . 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 4290 . . 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 5675 . . . . . . 7  |-  ( 1st `  R )  e.  _V
121, 11eqeltri 2450 . . . . . 6  |-  G  e. 
_V
1312rnex 5066 . . . . 5  |-  ran  G  e.  _V
142, 13eqeltri 2450 . . . 4  |-  X  e. 
_V
1514elpw2 4298 . . 3  |-  ( S  e.  ~P X  <->  S  C_  X
)
16 simpl 444 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  r  =  R )
1716fveq2d 5665 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( Idl `  r
)  =  ( Idl `  R ) )
18 sseq1 3305 . . . . . . 7  |-  ( s  =  S  ->  (
s  C_  j  <->  S  C_  j
) )
1918adantl 453 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( s  C_  j  <->  S 
C_  j ) )
2017, 19rabeqbidv 2887 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  { j  e.  ( Idl `  r )  |  s  C_  j }  =  { j  e.  ( Idl `  R
)  |  S  C_  j } )
2120inteqd 3990 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  |^| { j  e.  ( Idl `  r
)  |  s  C_  j }  =  |^| { j  e.  ( Idl `  R )  |  S  C_  j } )
22 fveq2 5661 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2322, 1syl6eqr 2430 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
2423rneqd 5030 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
2524, 2syl6eqr 2430 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
2625pweqd 3740 . . . 4  |-  ( r  =  R  ->  ~P ran  ( 1st `  r
)  =  ~P X
)
27 df-igen 26354 . . . 4  |-  IdlGen  =  ( r  e.  RingOps ,  s  e.  ~P ran  ( 1st `  r )  |->  |^|
{ j  e.  ( Idl `  r )  |  s  C_  j } )
2821, 26, 27ovmpt2x 6134 . . 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 1649    e. wcel 1717    =/= wne 2543   E.wrex 2643   {crab 2646   _Vcvv 2892    C_ wss 3256   (/)c0 3564   ~Pcpw 3735   |^|cint 3985   ran crn 4812   ` cfv 5387  (class class class)co 6013   1stc1st 6279   RingOpscrngo 21804   Idlcidl 26301    IdlGen cigen 26353
This theorem is referenced by:  igenss  26356  igenidl  26357  igenmin  26358  igenidl2  26359  igenval2  26360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fo 5393  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-grpo 21620  df-gid 21621  df-ablo 21711  df-rngo 21805  df-idl 26304  df-igen 26354
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