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Theorem pridlc3 26376
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 26303 . . . 4  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
2 eldifi 3414 . . . . 5  |-  ( A  e.  ( X  \  P )  ->  A  e.  X )
3 eldifi 3414 . . . . 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 21820 . . . . 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 3415 . . . 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 26375 . . . . . . 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 3276 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 1649    e. wcel 1717    \ cdif 3262   ran crn 4821   ` cfv 5396  (class class class)co 6022   1stc1st 6288   2ndc2nd 6289   RingOpscrngo 21813  CRingOpsccring 26298   PrIdlcpridl 26311
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-grpo 21629  df-gid 21630  df-ginv 21631  df-ablo 21720  df-ass 21751  df-exid 21753  df-mgm 21757  df-sgr 21769  df-mndo 21776  df-rngo 21814  df-com2 21849  df-crngo 26299  df-idl 26313  df-pridl 26314  df-igen 26363
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