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Theorem idlval 28810
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 5689 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2 idlval.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
31, 2syl6eqr 2491 . . . . . 6  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
43rneqd 5065 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
5 idlval.3 . . . . 5  |-  X  =  ran  G
64, 5syl6eqr 2491 . . . 4  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
76pweqd 3863 . . 3  |-  ( r  =  R  ->  ~P ran  ( 1st `  r
)  =  ~P X
)
83fveq2d 5693 . . . . . 6  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  (GId
`  G ) )
9 idlval.4 . . . . . 6  |-  Z  =  (GId `  G )
108, 9syl6eqr 2491 . . . . 5  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  Z )
1110eleq1d 2507 . . . 4  |-  ( r  =  R  ->  (
(GId `  ( 1st `  r ) )  e.  i  <->  Z  e.  i
) )
123oveqd 6106 . . . . . . . 8  |-  ( r  =  R  ->  (
x ( 1st `  r
) y )  =  ( x G y ) )
1312eleq1d 2507 . . . . . . 7  |-  ( r  =  R  ->  (
( x ( 1st `  r ) y )  e.  i  <->  ( x G y )  e.  i ) )
1413ralbidv 2733 . . . . . 6  |-  ( r  =  R  ->  ( A. y  e.  i 
( x ( 1st `  r ) y )  e.  i  <->  A. y  e.  i  ( x G y )  e.  i ) )
15 fveq2 5689 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 2nd `  r )  =  ( 2nd `  R
) )
16 idlval.2 . . . . . . . . . . 11  |-  H  =  ( 2nd `  R
)
1715, 16syl6eqr 2491 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 2nd `  r )  =  H )
1817oveqd 6106 . . . . . . . . 9  |-  ( r  =  R  ->  (
z ( 2nd `  r
) x )  =  ( z H x ) )
1918eleq1d 2507 . . . . . . . 8  |-  ( r  =  R  ->  (
( z ( 2nd `  r ) x )  e.  i  <->  ( z H x )  e.  i ) )
2017oveqd 6106 . . . . . . . . 9  |-  ( r  =  R  ->  (
x ( 2nd `  r
) z )  =  ( x H z ) )
2120eleq1d 2507 . . . . . . . 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 2929 . . . . . 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 2733 . . . 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 2965 . 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 28807 . 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 5699 . . . . . . 7  |-  ( 1st `  R )  e.  _V
302, 29eqeltri 2511 . . . . . 6  |-  G  e. 
_V
3130rnex 6510 . . . . 5  |-  ran  G  e.  _V
325, 31eqeltri 2511 . . . 4  |-  X  e. 
_V
3332pwex 4473 . . 3  |-  ~P X  e.  _V
3433rabex 4441 . 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 5772 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 1369    e. wcel 1756   A.wral 2713   {crab 2717   _Vcvv 2970   ~Pcpw 3858   ran crn 4839   ` cfv 5416  (class class class)co 6089   1stc1st 6573   2ndc2nd 6574  GIdcgi 23672   RingOpscrngo 23860   Idlcidl 28804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-idl 28807
This theorem is referenced by:  isidl  28811
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