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Theorem pridlc 26683
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
pridlc  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  e.  P ) )  -> 
( A  e.  P  \/  B  e.  P
) )

Proof of Theorem pridlc
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ispridlc.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 ispridlc.2 . . . . 5  |-  H  =  ( 2nd `  R
)
3 ispridlc.3 . . . . 5  |-  X  =  ran  G
41, 2, 3ispridlc 26682 . . . 4  |-  ( R  e. CRingOps  ->  ( P  e.  ( PrIdl `  R )  <->  ( P  e.  ( Idl `  R )  /\  P  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( (
a H b )  e.  P  ->  (
a  e.  P  \/  b  e.  P )
) ) ) )
54biimpa 472 . . 3  |-  ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  ->  ( P  e.  ( Idl `  R
)  /\  P  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P ) ) ) )
65simp3d 972 . 2  |-  ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  ->  A. a  e.  X  A. b  e.  X  ( (
a H b )  e.  P  ->  (
a  e.  P  \/  b  e.  P )
) )
7 oveq1 6090 . . . . . . . 8  |-  ( a  =  A  ->  (
a H b )  =  ( A H b ) )
87eleq1d 2504 . . . . . . 7  |-  ( a  =  A  ->  (
( a H b )  e.  P  <->  ( A H b )  e.  P ) )
9 eleq1 2498 . . . . . . . 8  |-  ( a  =  A  ->  (
a  e.  P  <->  A  e.  P ) )
109orbi1d 685 . . . . . . 7  |-  ( a  =  A  ->  (
( a  e.  P  \/  b  e.  P
)  <->  ( A  e.  P  \/  b  e.  P ) ) )
118, 10imbi12d 313 . . . . . 6  |-  ( a  =  A  ->  (
( ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P
) )  <->  ( ( A H b )  e.  P  ->  ( A  e.  P  \/  b  e.  P ) ) ) )
12 oveq2 6091 . . . . . . . 8  |-  ( b  =  B  ->  ( A H b )  =  ( A H B ) )
1312eleq1d 2504 . . . . . . 7  |-  ( b  =  B  ->  (
( A H b )  e.  P  <->  ( A H B )  e.  P
) )
14 eleq1 2498 . . . . . . . 8  |-  ( b  =  B  ->  (
b  e.  P  <->  B  e.  P ) )
1514orbi2d 684 . . . . . . 7  |-  ( b  =  B  ->  (
( A  e.  P  \/  b  e.  P
)  <->  ( A  e.  P  \/  B  e.  P ) ) )
1613, 15imbi12d 313 . . . . . 6  |-  ( b  =  B  ->  (
( ( A H b )  e.  P  ->  ( A  e.  P  \/  b  e.  P
) )  <->  ( ( A H B )  e.  P  ->  ( A  e.  P  \/  B  e.  P ) ) ) )
1711, 16rspc2v 3060 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P ) )  -> 
( ( A H B )  e.  P  ->  ( A  e.  P  \/  B  e.  P
) ) ) )
1817com12 30 . . . 4  |-  ( A. a  e.  X  A. b  e.  X  (
( a H b )  e.  P  -> 
( a  e.  P  \/  b  e.  P
) )  ->  (
( A  e.  X  /\  B  e.  X
)  ->  ( ( A H B )  e.  P  ->  ( A  e.  P  \/  B  e.  P ) ) ) )
1918exp3a 427 . . 3  |-  ( A. a  e.  X  A. b  e.  X  (
( a H b )  e.  P  -> 
( a  e.  P  \/  b  e.  P
) )  ->  ( A  e.  X  ->  ( B  e.  X  -> 
( ( A H B )  e.  P  ->  ( A  e.  P  \/  B  e.  P
) ) ) ) )
20193imp2 1169 . 2  |-  ( ( A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P
) )  /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  e.  P ) )  ->  ( A  e.  P  \/  B  e.  P ) )
216, 20sylan 459 1  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  e.  P ) )  -> 
( A  e.  P  \/  B  e.  P
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   ran crn 4881   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350  CRingOpsccring 26607   Idlcidl 26619   PrIdlcpridl 26620
This theorem is referenced by:  pridlc2  26684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-grpo 21781  df-gid 21782  df-ginv 21783  df-ablo 21872  df-ass 21903  df-exid 21905  df-mgm 21909  df-sgr 21921  df-mndo 21928  df-rngo 21966  df-com2 22001  df-crngo 26608  df-idl 26622  df-pridl 26623  df-igen 26672
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