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Theorem idlval 30572
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 5872 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2 idlval.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
31, 2syl6eqr 2516 . . . . . 6  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
43rneqd 5240 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
5 idlval.3 . . . . 5  |-  X  =  ran  G
64, 5syl6eqr 2516 . . . 4  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
76pweqd 4020 . . 3  |-  ( r  =  R  ->  ~P ran  ( 1st `  r
)  =  ~P X
)
83fveq2d 5876 . . . . . 6  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  (GId
`  G ) )
9 idlval.4 . . . . . 6  |-  Z  =  (GId `  G )
108, 9syl6eqr 2516 . . . . 5  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  Z )
1110eleq1d 2526 . . . 4  |-  ( r  =  R  ->  (
(GId `  ( 1st `  r ) )  e.  i  <->  Z  e.  i
) )
123oveqd 6313 . . . . . . . 8  |-  ( r  =  R  ->  (
x ( 1st `  r
) y )  =  ( x G y ) )
1312eleq1d 2526 . . . . . . 7  |-  ( r  =  R  ->  (
( x ( 1st `  r ) y )  e.  i  <->  ( x G y )  e.  i ) )
1413ralbidv 2896 . . . . . 6  |-  ( r  =  R  ->  ( A. y  e.  i 
( x ( 1st `  r ) y )  e.  i  <->  A. y  e.  i  ( x G y )  e.  i ) )
15 fveq2 5872 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 2nd `  r )  =  ( 2nd `  R
) )
16 idlval.2 . . . . . . . . . . 11  |-  H  =  ( 2nd `  R
)
1715, 16syl6eqr 2516 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 2nd `  r )  =  H )
1817oveqd 6313 . . . . . . . . 9  |-  ( r  =  R  ->  (
z ( 2nd `  r
) x )  =  ( z H x ) )
1918eleq1d 2526 . . . . . . . 8  |-  ( r  =  R  ->  (
( z ( 2nd `  r ) x )  e.  i  <->  ( z H x )  e.  i ) )
2017oveqd 6313 . . . . . . . . 9  |-  ( r  =  R  ->  (
x ( 2nd `  r
) z )  =  ( x H z ) )
2120eleq1d 2526 . . . . . . . 8  |-  ( r  =  R  ->  (
( x ( 2nd `  r ) z )  e.  i  <->  ( x H z )  e.  i ) )
2219, 21anbi12d 710 . . . . . . 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 3068 . . . . . 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 710 . . . . 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 2896 . . . 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 710 . . 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 3104 . 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 30569 . 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 5882 . . . . . . 7  |-  ( 1st `  R )  e.  _V
302, 29eqeltri 2541 . . . . . 6  |-  G  e. 
_V
3130rnex 6733 . . . . 5  |-  ran  G  e.  _V
325, 31eqeltri 2541 . . . 4  |-  X  e. 
_V
3332pwex 4639 . . 3  |-  ~P X  e.  _V
3433rabex 4607 . 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 5956 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109   ~Pcpw 4015   ran crn 5009   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798  GIdcgi 25315   RingOpscrngo 25503   Idlcidl 30566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-idl 30569
This theorem is referenced by:  isidl  30573
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