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Theorem pridlc3 26674
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 26601 . . . 4  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
2 eldifi 3461 . . . . 5  |-  ( A  e.  ( X  \  P )  ->  A  e.  X )
3 eldifi 3461 . . . . 5  |-  ( B  e.  ( X  \  P )  ->  B  e.  X )
42, 3anim12i 550 . . . 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 21962 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
983expb 1154 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
101, 4, 9syl2an 464 . . 3  |-  ( ( R  e. CRingOps  /\  ( A  e.  ( X  \  P )  /\  B  e.  ( X  \  P
) ) )  -> 
( A H B )  e.  X )
1110adantlr 696 . 2  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  ( A H B )  e.  X )
12 eldifn 3462 . . . 4  |-  ( B  e.  ( X  \  P )  ->  -.  B  e.  P )
1312ad2antll 710 . . 3  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  -.  B  e.  P )
145, 6, 7pridlc2 26673 . . . . . . 7  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  X  /\  ( A H B )  e.  P
) )  ->  B  e.  P )
15143exp2 1171 . . . . . 6  |-  ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  ->  ( A  e.  ( X  \  P
)  ->  ( B  e.  X  ->  ( ( A H B )  e.  P  ->  B  e.  P ) ) ) )
1615imp32 423 . . . . 5  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  X ) )  -> 
( ( A H B )  e.  P  ->  B  e.  P ) )
1716con3d 127 . . . 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 635 . . 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 3323 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3309   ran crn 4871   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   RingOpscrngo 21955  CRingOpsccring 26596   PrIdlcpridl 26609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-grpo 21771  df-gid 21772  df-ginv 21773  df-ablo 21862  df-ass 21893  df-exid 21895  df-mgm 21899  df-sgr 21911  df-mndo 21918  df-rngo 21956  df-com2 21991  df-crngo 26597  df-idl 26611  df-pridl 26612  df-igen 26661
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