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Theorem pridlval 26328
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 5662 . . 3  |-  ( r  =  R  ->  ( Idl `  r )  =  ( Idl `  R
) )
2 fveq2 5662 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 pridlval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2431 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 5031 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 pridlval.3 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2431 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
87neeq2d 2558 . . . 4  |-  ( r  =  R  ->  (
i  =/=  ran  ( 1st `  r )  <->  i  =/=  X ) )
9 fveq2 5662 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 2nd `  r )  =  ( 2nd `  R
) )
10 pridlval.2 . . . . . . . . . . 11  |-  H  =  ( 2nd `  R
)
119, 10syl6eqr 2431 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 2nd `  r )  =  H )
1211oveqd 6031 . . . . . . . . 9  |-  ( r  =  R  ->  (
x ( 2nd `  r
) y )  =  ( x H y ) )
1312eleq1d 2447 . . . . . . . 8  |-  ( r  =  R  ->  (
( x ( 2nd `  r ) y )  e.  i  <->  ( x H y )  e.  i ) )
14132ralbidv 2685 . . . . . . 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 309 . . . . . 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 2853 . . . . 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 2853 . . . 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 692 . . 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 2888 . 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 26306 . 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 5676 . . 3  |-  ( Idl `  R )  e.  _V
2221rabex 4289 . 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 5739 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 set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2544   A.wral 2643   {crab 2647    C_ wss 3257   ran crn 4813   ` cfv 5388  (class class class)co 6014   1stc1st 6280   2ndc2nd 6281   RingOpscrngo 21805   Idlcidl 26302   PrIdlcpridl 26303
This theorem is referenced by:  ispridl  26329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pr 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-sbc 3099  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-iota 5352  df-fun 5390  df-fv 5396  df-ov 6017  df-pridl 26306
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