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Theorem pridlc3 31732
Description: Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
Hypotheses
Ref Expression
ispridlc.1  |-  G  =  ( 1st `  R
)
ispridlc.2  |-  H  =  ( 2nd `  R
)
ispridlc.3  |-  X  =  ran  G
Assertion
Ref Expression
pridlc3  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  ( A H B )  e.  ( X  \  P
) )

Proof of Theorem pridlc3
StepHypRef Expression
1 crngorngo 31659 . . . 4  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
2 eldifi 3564 . . . . 5  |-  ( A  e.  ( X  \  P )  ->  A  e.  X )
3 eldifi 3564 . . . . 5  |-  ( B  e.  ( X  \  P )  ->  B  e.  X )
42, 3anim12i 564 . . . 4  |-  ( ( A  e.  ( X 
\  P )  /\  B  e.  ( X  \  P ) )  -> 
( A  e.  X  /\  B  e.  X
) )
5 ispridlc.1 . . . . . 6  |-  G  =  ( 1st `  R
)
6 ispridlc.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
7 ispridlc.3 . . . . . 6  |-  X  =  ran  G
85, 6, 7rngocl 25784 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
983expb 1198 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
101, 4, 9syl2an 475 . . 3  |-  ( ( R  e. CRingOps  /\  ( A  e.  ( X  \  P )  /\  B  e.  ( X  \  P
) ) )  -> 
( A H B )  e.  X )
1110adantlr 713 . 2  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  ( A H B )  e.  X )
12 eldifn 3565 . . . 4  |-  ( B  e.  ( X  \  P )  ->  -.  B  e.  P )
1312ad2antll 727 . . 3  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  -.  B  e.  P )
145, 6, 7pridlc2 31731 . . . . . . 7  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  X  /\  ( A H B )  e.  P
) )  ->  B  e.  P )
15143exp2 1215 . . . . . 6  |-  ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  ->  ( A  e.  ( X  \  P
)  ->  ( B  e.  X  ->  ( ( A H B )  e.  P  ->  B  e.  P ) ) ) )
1615imp32 431 . . . . 5  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  X ) )  -> 
( ( A H B )  e.  P  ->  B  e.  P ) )
1716con3d 133 . . . 4  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  X ) )  -> 
( -.  B  e.  P  ->  -.  ( A H B )  e.  P ) )
183, 17sylanr2 651 . . 3  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  ( -.  B  e.  P  ->  -.  ( A H B )  e.  P
) )
1913, 18mpd 15 . 2  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  -.  ( A H B )  e.  P )
2011, 19eldifd 3424 1  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  ( A H B )  e.  ( X  \  P
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    \ cdif 3410   ran crn 4823   ` cfv 5568  (class class class)co 6277   1stc1st 6781   2ndc2nd 6782   RingOpscrngo 25777  CRingOpsccring 31654   PrIdlcpridl 31667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-grpo 25593  df-gid 25594  df-ginv 25595  df-ablo 25684  df-ass 25715  df-exid 25717  df-mgmOLD 25721  df-sgrOLD 25733  df-mndo 25740  df-rngo 25778  df-com2 25813  df-crngo 31655  df-idl 31669  df-pridl 31670  df-igen 31719
This theorem is referenced by: (None)
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