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Theorem pridlidl 26166
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 2366 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2366 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2366 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
41, 2, 3ispridl 26165 . . 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 939 . . 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 935    e. wcel 1715    =/= wne 2529   A.wral 2628    C_ wss 3238   ran crn 4793   ` cfv 5358  (class class class)co 5981   1stc1st 6247   2ndc2nd 6248   RingOpscrngo 21353   Idlcidl 26138   PrIdlcpridl 26139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-pridl 26142
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