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Theorem pridlc3 30073
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 30000 . . . 4  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
2 eldifi 3626 . . . . 5  |-  ( A  e.  ( X  \  P )  ->  A  e.  X )
3 eldifi 3626 . . . . 5  |-  ( B  e.  ( X  \  P )  ->  B  e.  X )
42, 3anim12i 566 . . . 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 25060 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
983expb 1197 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
101, 4, 9syl2an 477 . . 3  |-  ( ( R  e. CRingOps  /\  ( A  e.  ( X  \  P )  /\  B  e.  ( X  \  P
) ) )  -> 
( A H B )  e.  X )
1110adantlr 714 . 2  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  ( A H B )  e.  X )
12 eldifn 3627 . . . 4  |-  ( B  e.  ( X  \  P )  ->  -.  B  e.  P )
1312ad2antll 728 . . 3  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  -.  B  e.  P )
145, 6, 7pridlc2 30072 . . . . . . 7  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  X  /\  ( A H B )  e.  P
) )  ->  B  e.  P )
15143exp2 1214 . . . . . 6  |-  ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  ->  ( A  e.  ( X  \  P
)  ->  ( B  e.  X  ->  ( ( A H B )  e.  P  ->  B  e.  P ) ) ) )
1615imp32 433 . . . . 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 653 . . 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 3487 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 369    = wceq 1379    e. wcel 1767    \ cdif 3473   ran crn 5000   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   RingOpscrngo 25053  CRingOpsccring 29995   PrIdlcpridl 30008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-grpo 24869  df-gid 24870  df-ginv 24871  df-ablo 24960  df-ass 24991  df-exid 24993  df-mgm 24997  df-sgr 25009  df-mndo 25016  df-rngo 25054  df-com2 25089  df-crngo 29996  df-idl 30010  df-pridl 30011  df-igen 30060
This theorem is referenced by: (None)
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