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Theorem pridlc 28883
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 28882 . . . 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 484 . . 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 1002 . 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 6110 . . . . . . . 8  |-  ( a  =  A  ->  (
a H b )  =  ( A H b ) )
87eleq1d 2509 . . . . . . 7  |-  ( a  =  A  ->  (
( a H b )  e.  P  <->  ( A H b )  e.  P ) )
9 eleq1 2503 . . . . . . . 8  |-  ( a  =  A  ->  (
a  e.  P  <->  A  e.  P ) )
109orbi1d 702 . . . . . . 7  |-  ( a  =  A  ->  (
( a  e.  P  \/  b  e.  P
)  <->  ( A  e.  P  \/  b  e.  P ) ) )
118, 10imbi12d 320 . . . . . 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 6111 . . . . . . . 8  |-  ( b  =  B  ->  ( A H b )  =  ( A H B ) )
1312eleq1d 2509 . . . . . . 7  |-  ( b  =  B  ->  (
( A H b )  e.  P  <->  ( A H B )  e.  P
) )
14 eleq1 2503 . . . . . . . 8  |-  ( b  =  B  ->  (
b  e.  P  <->  B  e.  P ) )
1514orbi2d 701 . . . . . . 7  |-  ( b  =  B  ->  (
( A  e.  P  \/  b  e.  P
)  <->  ( A  e.  P  \/  B  e.  P ) ) )
1613, 15imbi12d 320 . . . . . 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 3091 . . . . 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 31 . . . 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 ) ) ) )
1918expd 436 . . 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 1202 . 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 471 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 setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   ran crn 4853   ` cfv 5430  (class class class)co 6103   1stc1st 6587   2ndc2nd 6588  CRingOpsccring 28807   Idlcidl 28819   PrIdlcpridl 28820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-grpo 23690  df-gid 23691  df-ginv 23692  df-ablo 23781  df-ass 23812  df-exid 23814  df-mgm 23818  df-sgr 23830  df-mndo 23837  df-rngo 23875  df-com2 23910  df-crngo 28808  df-idl 28822  df-pridl 28823  df-igen 28872
This theorem is referenced by:  pridlc2  28884
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