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Theorem pridlval 30592
Description: The class of prime ideals of a ring  R. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
pridlval.1  |-  G  =  ( 1st `  R
)
pridlval.2  |-  H  =  ( 2nd `  R
)
pridlval.3  |-  X  =  ran  G
Assertion
Ref Expression
pridlval  |-  ( R  e.  RingOps  ->  ( PrIdl `  R
)  =  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. a  e.  ( Idl `  R ) A. b  e.  ( Idl `  R
) ( A. x  e.  a  A. y  e.  b  ( x H y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) } )
Distinct variable groups:    R, i, x, y, a, b    i, X    i, H
Allowed substitution hints:    G( x, y, i, a, b)    H( x, y, a, b)    X( x, y, a, b)

Proof of Theorem pridlval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . 3  |-  ( r  =  R  ->  ( Idl `  r )  =  ( Idl `  R
) )
2 fveq2 5872 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 pridlval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2516 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 5240 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 pridlval.3 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2516 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
87neeq2d 2735 . . . 4  |-  ( r  =  R  ->  (
i  =/=  ran  ( 1st `  r )  <->  i  =/=  X ) )
9 fveq2 5872 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 2nd `  r )  =  ( 2nd `  R
) )
10 pridlval.2 . . . . . . . . . . 11  |-  H  =  ( 2nd `  R
)
119, 10syl6eqr 2516 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 2nd `  r )  =  H )
1211oveqd 6313 . . . . . . . . 9  |-  ( r  =  R  ->  (
x ( 2nd `  r
) y )  =  ( x H y ) )
1312eleq1d 2526 . . . . . . . 8  |-  ( r  =  R  ->  (
( x ( 2nd `  r ) y )  e.  i  <->  ( x H y )  e.  i ) )
14132ralbidv 2901 . . . . . . 7  |-  ( r  =  R  ->  ( A. x  e.  a  A. y  e.  b 
( x ( 2nd `  r ) y )  e.  i  <->  A. x  e.  a  A. y  e.  b  ( x H y )  e.  i ) )
1514imbi1d 317 . . . . . 6  |-  ( r  =  R  ->  (
( A. x  e.  a  A. y  e.  b  ( x ( 2nd `  r ) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) )  <->  ( A. x  e.  a  A. y  e.  b  (
x H y )  e.  i  ->  (
a  C_  i  \/  b  C_  i ) ) ) )
161, 15raleqbidv 3068 . . . . 5  |-  ( r  =  R  ->  ( A. b  e.  ( Idl `  r ) ( A. x  e.  a 
A. y  e.  b  ( x ( 2nd `  r ) y )  e.  i  ->  (
a  C_  i  \/  b  C_  i ) )  <->  A. b  e.  ( Idl `  R ) ( A. x  e.  a 
A. y  e.  b  ( x H y )  e.  i  -> 
( a  C_  i  \/  b  C_  i ) ) ) )
171, 16raleqbidv 3068 . . . 4  |-  ( r  =  R  ->  ( A. a  e.  ( Idl `  r ) A. b  e.  ( Idl `  r ) ( A. x  e.  a  A. y  e.  b  (
x ( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) )  <->  A. a  e.  ( Idl `  R
) A. b  e.  ( Idl `  R
) ( A. x  e.  a  A. y  e.  b  ( x H y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) )
188, 17anbi12d 710 . . 3  |-  ( r  =  R  ->  (
( i  =/=  ran  ( 1st `  r )  /\  A. a  e.  ( Idl `  r
) A. b  e.  ( Idl `  r
) ( A. x  e.  a  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) )  <-> 
( i  =/=  X  /\  A. a  e.  ( Idl `  R ) A. b  e.  ( Idl `  R ) ( A. x  e.  a  A. y  e.  b  ( x H y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) ) )
191, 18rabeqbidv 3104 . 2  |-  ( r  =  R  ->  { i  e.  ( Idl `  r
)  |  ( i  =/=  ran  ( 1st `  r )  /\  A. a  e.  ( Idl `  r ) A. b  e.  ( Idl `  r
) ( A. x  e.  a  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) }  =  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. a  e.  ( Idl `  R ) A. b  e.  ( Idl `  R
) ( A. x  e.  a  A. y  e.  b  ( x H y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) } )
20 df-pridl 30570 . 2  |-  PrIdl  =  ( r  e.  RingOps  |->  { i  e.  ( Idl `  r
)  |  ( i  =/=  ran  ( 1st `  r )  /\  A. a  e.  ( Idl `  r ) A. b  e.  ( Idl `  r
) ( A. x  e.  a  A. y  e.  b  ( x
( 2nd `  r
) y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) } )
21 fvex 5882 . . 3  |-  ( Idl `  R )  e.  _V
2221rabex 4607 . 2  |-  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. a  e.  ( Idl `  R ) A. b  e.  ( Idl `  R
) ( A. x  e.  a  A. y  e.  b  ( x H y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) }  e.  _V
2319, 20, 22fvmpt 5956 1  |-  ( R  e.  RingOps  ->  ( PrIdl `  R
)  =  { i  e.  ( Idl `  R
)  |  ( i  =/=  X  /\  A. a  e.  ( Idl `  R ) A. b  e.  ( Idl `  R
) ( A. x  e.  a  A. y  e.  b  ( x H y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811    C_ wss 3471   ran crn 5009   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   RingOpscrngo 25503   Idlcidl 30566   PrIdlcpridl 30567
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-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
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-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-pridl 30570
This theorem is referenced by:  ispridl  30593
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