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Theorem pridlc3 26698
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 26625 . . . 4  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
2 eldifi 3298 . . . . 5  |-  ( A  e.  ( X  \  P )  ->  A  e.  X )
3 eldifi 3298 . . . . 5  |-  ( B  e.  ( X  \  P )  ->  B  e.  X )
42, 3anim12i 549 . . . 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 21049 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
983expb 1152 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
101, 4, 9syl2an 463 . . 3  |-  ( ( R  e. CRingOps  /\  ( A  e.  ( X  \  P )  /\  B  e.  ( X  \  P
) ) )  -> 
( A H B )  e.  X )
1110adantlr 695 . 2  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  ( A H B )  e.  X )
12 eldifn 3299 . . . 4  |-  ( B  e.  ( X  \  P )  ->  -.  B  e.  P )
1312ad2antll 709 . . 3  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  -.  B  e.  P )
145, 6, 7pridlc2 26697 . . . . . . 7  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  X  /\  ( A H B )  e.  P
) )  ->  B  e.  P )
15143exp2 1169 . . . . . 6  |-  ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  ->  ( A  e.  ( X  \  P
)  ->  ( B  e.  X  ->  ( ( A H B )  e.  P  ->  B  e.  P ) ) ) )
1615imp32 422 . . . . 5  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  X ) )  -> 
( ( A H B )  e.  P  ->  B  e.  P ) )
1716con3d 125 . . . 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 634 . . 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 14 . 2  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  ( X  \  P
)  /\  B  e.  ( X  \  P ) ) )  ->  -.  ( A H B )  e.  P )
20 eldif 3162 . 2  |-  ( ( A H B )  e.  ( X  \  P )  <->  ( ( A H B )  e.  X  /\  -.  ( A H B )  e.  P ) )
2111, 19, 20sylanbrc 645 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 358    = wceq 1623    e. wcel 1684    \ cdif 3149   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   RingOpscrngo 21042  CRingOpsccring 26620   PrIdlcpridl 26633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043  df-com2 21078  df-crngo 26621  df-idl 26635  df-pridl 26636  df-igen 26685
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