Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pridlc Structured version   Unicode version

Theorem pridlc 30673
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 30672 . . . 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 1010 . 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 6303 . . . . . . . 8  |-  ( a  =  A  ->  (
a H b )  =  ( A H b ) )
87eleq1d 2526 . . . . . . 7  |-  ( a  =  A  ->  (
( a H b )  e.  P  <->  ( A H b )  e.  P ) )
9 eleq1 2529 . . . . . . . 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 6304 . . . . . . . 8  |-  ( b  =  B  ->  ( A H b )  =  ( A H B ) )
1312eleq1d 2526 . . . . . . 7  |-  ( b  =  B  ->  (
( A H b )  e.  P  <->  ( A H B )  e.  P
) )
14 eleq1 2529 . . . . . . . 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 3219 . . . . 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 1211 . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   ran crn 5009   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798  CRingOpsccring 30597   Idlcidl 30609   PrIdlcpridl 30610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-grpo 25320  df-gid 25321  df-ginv 25322  df-ablo 25411  df-ass 25442  df-exid 25444  df-mgmOLD 25448  df-sgrOLD 25460  df-mndo 25467  df-rngo 25505  df-com2 25540  df-crngo 30598  df-idl 30612  df-pridl 30613  df-igen 30662
This theorem is referenced by:  pridlc2  30674
  Copyright terms: Public domain W3C validator