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Theorem idlval 26638
Description: The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idlval.1  |-  G  =  ( 1st `  R
)
idlval.2  |-  H  =  ( 2nd `  R
)
idlval.3  |-  X  =  ran  G
idlval.4  |-  Z  =  (GId `  G )
Assertion
Ref Expression
idlval  |-  ( R  e.  RingOps  ->  ( Idl `  R
)  =  { i  e.  ~P X  | 
( Z  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x G y )  e.  i  /\  A. z  e.  X  ( (
z H x )  e.  i  /\  (
x H z )  e.  i ) ) ) } )
Distinct variable groups:    x, R, y, z, i    z, X, i    i, Z    i, G    i, H
Allowed substitution hints:    G( x, y, z)    H( x, y, z)    X( x, y)    Z( x, y, z)

Proof of Theorem idlval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2 idlval.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
31, 2syl6eqr 2333 . . . . . 6  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
43rneqd 4906 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
5 idlval.3 . . . . 5  |-  X  =  ran  G
64, 5syl6eqr 2333 . . . 4  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
76pweqd 3630 . . 3  |-  ( r  =  R  ->  ~P ran  ( 1st `  r
)  =  ~P X
)
83fveq2d 5529 . . . . . 6  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  (GId
`  G ) )
9 idlval.4 . . . . . 6  |-  Z  =  (GId `  G )
108, 9syl6eqr 2333 . . . . 5  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  Z )
1110eleq1d 2349 . . . 4  |-  ( r  =  R  ->  (
(GId `  ( 1st `  r ) )  e.  i  <->  Z  e.  i
) )
123oveqd 5875 . . . . . . . 8  |-  ( r  =  R  ->  (
x ( 1st `  r
) y )  =  ( x G y ) )
1312eleq1d 2349 . . . . . . 7  |-  ( r  =  R  ->  (
( x ( 1st `  r ) y )  e.  i  <->  ( x G y )  e.  i ) )
1413ralbidv 2563 . . . . . 6  |-  ( r  =  R  ->  ( A. y  e.  i 
( x ( 1st `  r ) y )  e.  i  <->  A. y  e.  i  ( x G y )  e.  i ) )
15 fveq2 5525 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 2nd `  r )  =  ( 2nd `  R
) )
16 idlval.2 . . . . . . . . . . 11  |-  H  =  ( 2nd `  R
)
1715, 16syl6eqr 2333 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 2nd `  r )  =  H )
1817oveqd 5875 . . . . . . . . 9  |-  ( r  =  R  ->  (
z ( 2nd `  r
) x )  =  ( z H x ) )
1918eleq1d 2349 . . . . . . . 8  |-  ( r  =  R  ->  (
( z ( 2nd `  r ) x )  e.  i  <->  ( z H x )  e.  i ) )
2017oveqd 5875 . . . . . . . . 9  |-  ( r  =  R  ->  (
x ( 2nd `  r
) z )  =  ( x H z ) )
2120eleq1d 2349 . . . . . . . 8  |-  ( r  =  R  ->  (
( x ( 2nd `  r ) z )  e.  i  <->  ( x H z )  e.  i ) )
2219, 21anbi12d 691 . . . . . . 7  |-  ( r  =  R  ->  (
( ( z ( 2nd `  r ) x )  e.  i  /\  ( x ( 2nd `  r ) z )  e.  i )  <->  ( ( z H x )  e.  i  /\  ( x H z )  e.  i ) ) )
236, 22raleqbidv 2748 . . . . . 6  |-  ( r  =  R  ->  ( A. z  e.  ran  ( 1st `  r ) ( ( z ( 2nd `  r ) x )  e.  i  /\  ( x ( 2nd `  r ) z )  e.  i )  <->  A. z  e.  X  ( ( z H x )  e.  i  /\  ( x H z )  e.  i ) ) )
2414, 23anbi12d 691 . . . . 5  |-  ( r  =  R  ->  (
( A. y  e.  i  ( x ( 1st `  r ) y )  e.  i  /\  A. z  e. 
ran  ( 1st `  r
) ( ( z ( 2nd `  r
) x )  e.  i  /\  ( x ( 2nd `  r
) z )  e.  i ) )  <->  ( A. y  e.  i  (
x G y )  e.  i  /\  A. z  e.  X  (
( z H x )  e.  i  /\  ( x H z )  e.  i ) ) ) )
2524ralbidv 2563 . . . 4  |-  ( r  =  R  ->  ( A. x  e.  i 
( A. y  e.  i  ( x ( 1st `  r ) y )  e.  i  /\  A. z  e. 
ran  ( 1st `  r
) ( ( z ( 2nd `  r
) x )  e.  i  /\  ( x ( 2nd `  r
) z )  e.  i ) )  <->  A. x  e.  i  ( A. y  e.  i  (
x G y )  e.  i  /\  A. z  e.  X  (
( z H x )  e.  i  /\  ( x H z )  e.  i ) ) ) )
2611, 25anbi12d 691 . . 3  |-  ( r  =  R  ->  (
( (GId `  ( 1st `  r ) )  e.  i  /\  A. x  e.  i  ( A. y  e.  i 
( x ( 1st `  r ) y )  e.  i  /\  A. z  e.  ran  ( 1st `  r ) ( ( z ( 2nd `  r
) x )  e.  i  /\  ( x ( 2nd `  r
) z )  e.  i ) ) )  <-> 
( Z  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x G y )  e.  i  /\  A. z  e.  X  ( (
z H x )  e.  i  /\  (
x H z )  e.  i ) ) ) ) )
277, 26rabeqbidv 2783 . 2  |-  ( r  =  R  ->  { i  e.  ~P ran  ( 1st `  r )  |  ( (GId `  ( 1st `  r ) )  e.  i  /\  A. x  e.  i  ( A. y  e.  i 
( x ( 1st `  r ) y )  e.  i  /\  A. z  e.  ran  ( 1st `  r ) ( ( z ( 2nd `  r
) x )  e.  i  /\  ( x ( 2nd `  r
) z )  e.  i ) ) ) }  =  { i  e.  ~P X  | 
( Z  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x G y )  e.  i  /\  A. z  e.  X  ( (
z H x )  e.  i  /\  (
x H z )  e.  i ) ) ) } )
28 df-idl 26635 . 2  |-  Idl  =  ( r  e.  RingOps  |->  { i  e.  ~P ran  ( 1st `  r )  |  ( (GId `  ( 1st `  r ) )  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x ( 1st `  r ) y )  e.  i  /\  A. z  e.  ran  ( 1st `  r ) ( ( z ( 2nd `  r
) x )  e.  i  /\  ( x ( 2nd `  r
) z )  e.  i ) ) ) } )
29 fvex 5539 . . . . . . 7  |-  ( 1st `  R )  e.  _V
302, 29eqeltri 2353 . . . . . 6  |-  G  e. 
_V
3130rnex 4942 . . . . 5  |-  ran  G  e.  _V
325, 31eqeltri 2353 . . . 4  |-  X  e. 
_V
3332pwex 4193 . . 3  |-  ~P X  e.  _V
3433rabex 4165 . 2  |-  { i  e.  ~P X  | 
( Z  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x G y )  e.  i  /\  A. z  e.  X  ( (
z H x )  e.  i  /\  (
x H z )  e.  i ) ) ) }  e.  _V
3527, 28, 34fvmpt 5602 1  |-  ( R  e.  RingOps  ->  ( Idl `  R
)  =  { i  e.  ~P X  | 
( Z  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x G y )  e.  i  /\  A. z  e.  X  ( (
z H x )  e.  i  /\  (
x H z )  e.  i ) ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   ~Pcpw 3625   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042   Idlcidl 26632
This theorem is referenced by:  isidl  26639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-idl 26635
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