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Theorem pridlidl 26539
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 2408 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2408 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2408 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
41, 2, 3ispridl 26538 . . 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 940 . . 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 253 . 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 607 1  |-  ( ( R  e.  RingOps  /\  P  e.  ( PrIdl `  R )
)  ->  P  e.  ( Idl `  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    e. wcel 1721    =/= wne 2571   A.wral 2670    C_ wss 3284   ran crn 4842   ` cfv 5417  (class class class)co 6044   1stc1st 6310   2ndc2nd 6311   RingOpscrngo 21920   Idlcidl 26511   PrIdlcpridl 26512
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-pridl 26515
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