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Theorem pridlnr 25809
Description: A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
pridlnr.1  |-  G  =  ( 1st `  R
)
prdilnr.2  |-  X  =  ran  G
Assertion
Ref Expression
pridlnr  |-  ( ( R  e.  RingOps  /\  P  e.  ( PrIdl `  R )
)  ->  P  =/=  X )

Proof of Theorem pridlnr
Dummy variables  x  y  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pridlnr.1 . . . 4  |-  G  =  ( 1st `  R
)
2 eqid 2316 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 prdilnr.2 . . . 4  |-  X  =  ran  G
41, 2, 3ispridl 25807 . . 3  |-  ( R  e.  RingOps  ->  ( P  e.  ( PrIdl `  R )  <->  ( P  e.  ( Idl `  R )  /\  P  =/=  X  /\  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 3anan12 947 . . 3  |-  ( ( P  e.  ( Idl `  R )  /\  P  =/=  X  /\  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  =/=  X  /\  ( P  e.  ( Idl `  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  =/=  X  /\  ( P  e.  ( Idl `  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  =/=  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   A.wral 2577    C_ wss 3186   ran crn 4727   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163   RingOpscrngo 21095   Idlcidl 25780   PrIdlcpridl 25781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-pridl 25784
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