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Theorem pridlc 26696
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 26695 . . . 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 470 . . 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 969 . 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 5865 . . . . . . . 8  |-  ( a  =  A  ->  (
a H b )  =  ( A H b ) )
87eleq1d 2349 . . . . . . 7  |-  ( a  =  A  ->  (
( a H b )  e.  P  <->  ( A H b )  e.  P ) )
9 eleq1 2343 . . . . . . . 8  |-  ( a  =  A  ->  (
a  e.  P  <->  A  e.  P ) )
109orbi1d 683 . . . . . . 7  |-  ( a  =  A  ->  (
( a  e.  P  \/  b  e.  P
)  <->  ( A  e.  P  \/  b  e.  P ) ) )
118, 10imbi12d 311 . . . . . 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 5866 . . . . . . . 8  |-  ( b  =  B  ->  ( A H b )  =  ( A H B ) )
1312eleq1d 2349 . . . . . . 7  |-  ( b  =  B  ->  (
( A H b )  e.  P  <->  ( A H B )  e.  P
) )
14 eleq1 2343 . . . . . . . 8  |-  ( b  =  B  ->  (
b  e.  P  <->  B  e.  P ) )
1514orbi2d 682 . . . . . . 7  |-  ( b  =  B  ->  (
( A  e.  P  \/  b  e.  P
)  <->  ( A  e.  P  \/  B  e.  P ) ) )
1613, 15imbi12d 311 . . . . . 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 2890 . . . . 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 27 . . . 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 425 . . 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 1166 . 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 457 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 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  CRingOpsccring 26620   Idlcidl 26632   PrIdlcpridl 26633
This theorem is referenced by:  pridlc2  26697
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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