Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  igenval Unicode version

Theorem igenval 26789
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 26752 . . . . 5  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
4 sseq2 3213 . . . . . 6  |-  ( j  =  X  ->  ( S  C_  j  <->  S  C_  X
) )
54rspcev 2897 . . . . 5  |-  ( ( X  e.  ( Idl `  R )  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
63, 5sylan 457 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  E. j  e.  ( Idl `  R
) S  C_  j
)
7 rabn0 3487 . . . 4  |-  ( { j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/)  <->  E. j  e.  ( Idl `  R
) S  C_  j
)
86, 7sylibr 203 . . 3  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  { j  e.  ( Idl `  R
)  |  S  C_  j }  =/=  (/) )
9 intex 4183 . . 3  |-  ( { j  e.  ( Idl `  R )  |  S  C_  j }  =/=  (/)  <->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  e.  _V )
108, 9sylib 188 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  |^| { j  e.  ( Idl `  R
)  |  S  C_  j }  e.  _V )
11 fvex 5555 . . . . . . 7  |-  ( 1st `  R )  e.  _V
121, 11eqeltri 2366 . . . . . 6  |-  G  e. 
_V
1312rnex 4958 . . . . 5  |-  ran  G  e.  _V
142, 13eqeltri 2366 . . . 4  |-  X  e. 
_V
1514elpw2 4191 . . 3  |-  ( S  e.  ~P X  <->  S  C_  X
)
16 simpl 443 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  r  =  R )
1716fveq2d 5545 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( Idl `  r
)  =  ( Idl `  R ) )
18 sseq1 3212 . . . . . . 7  |-  ( s  =  S  ->  (
s  C_  j  <->  S  C_  j
) )
1918adantl 452 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( s  C_  j  <->  S 
C_  j ) )
2017, 19rabeqbidv 2796 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  { j  e.  ( Idl `  r )  |  s  C_  j }  =  { j  e.  ( Idl `  R
)  |  S  C_  j } )
2120inteqd 3883 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  |^| { j  e.  ( Idl `  r
)  |  s  C_  j }  =  |^| { j  e.  ( Idl `  R )  |  S  C_  j } )
22 fveq2 5541 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2322, 1syl6eqr 2346 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
2423rneqd 4922 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
2524, 2syl6eqr 2346 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
2625pweqd 3643 . . . 4  |-  ( r  =  R  ->  ~P ran  ( 1st `  r
)  =  ~P X
)
27 df-igen 26788 . . . 4  |-  IdlGen  =  ( r  e.  RingOps ,  s  e.  ~P ran  ( 1st `  r )  |->  |^|
{ j  e.  ( Idl `  r )  |  s  C_  j } )
2821, 26, 27ovmpt2x 5992 . . 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 1222 . 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 1278 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   |^|cint 3878   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   RingOpscrngo 21058   Idlcidl 26735    IdlGen cigen 26787
This theorem is referenced by:  igenss  26790  igenidl  26791  igenmin  26792  igenidl2  26793  igenval2  26794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ablo 20965  df-rngo 21059  df-idl 26738  df-igen 26788
  Copyright terms: Public domain W3C validator