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Theorem pridlval 26761
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 5541 . . 3  |-  ( r  =  R  ->  ( Idl `  r )  =  ( Idl `  R
) )
2 fveq2 5541 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 pridlval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2346 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 4922 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 pridlval.3 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2346 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
87neeq2d 2473 . . . 4  |-  ( r  =  R  ->  (
i  =/=  ran  ( 1st `  r )  <->  i  =/=  X ) )
9 fveq2 5541 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 2nd `  r )  =  ( 2nd `  R
) )
10 pridlval.2 . . . . . . . . . . 11  |-  H  =  ( 2nd `  R
)
119, 10syl6eqr 2346 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 2nd `  r )  =  H )
1211oveqd 5891 . . . . . . . . 9  |-  ( r  =  R  ->  (
x ( 2nd `  r
) y )  =  ( x H y ) )
1312eleq1d 2362 . . . . . . . 8  |-  ( r  =  R  ->  (
( x ( 2nd `  r ) y )  e.  i  <->  ( x H y )  e.  i ) )
14132ralbidv 2598 . . . . . . 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 308 . . . . . 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 2761 . . . . 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 2761 . . . 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 691 . . 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 2796 . 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 26739 . 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 5555 . . 3  |-  ( Idl `  R )  e.  _V
2221rabex 4181 . 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 5618 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 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560    C_ wss 3165   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   RingOpscrngo 21058   Idlcidl 26735   PrIdlcpridl 26736
This theorem is referenced by:  ispridl  26762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-pridl 26739
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