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Theorem pridlidl 26660
Description: A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
Assertion
Ref Expression
pridlidl  |-  ( ( R  e.  RingOps  /\  P  e.  ( PrIdl `  R )
)  ->  P  e.  ( Idl `  R ) )

Proof of Theorem pridlidl
Dummy variables  x  y  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2283 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2283 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
41, 2, 3ispridl 26659 . . 3  |-  ( R  e.  RingOps  ->  ( P  e.  ( PrIdl `  R )  <->  ( P  e.  ( Idl `  R )  /\  P  =/=  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.  P  ->  ( a  C_  P  \/  b  C_  P ) ) ) ) )
5 3anass 938 . . 3  |-  ( ( P  e.  ( Idl `  R )  /\  P  =/=  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.  P  ->  ( a  C_  P  \/  b  C_  P ) ) )  <-> 
( P  e.  ( Idl `  R )  /\  ( P  =/= 
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.  P  ->  ( a  C_  P  \/  b  C_  P ) ) ) ) )
64, 5syl6bb 252 . 2  |-  ( R  e.  RingOps  ->  ( P  e.  ( PrIdl `  R )  <->  ( P  e.  ( Idl `  R )  /\  ( P  =/=  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.  P  ->  ( a  C_  P  \/  b  C_  P ) ) ) ) ) )
76simprbda 606 1  |-  ( ( R  e.  RingOps  /\  P  e.  ( PrIdl `  R )
)  ->  P  e.  ( Idl `  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   RingOpscrngo 21042   Idlcidl 26632   PrIdlcpridl 26633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-pridl 26636
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