Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atexchltN Structured version   Unicode version

Theorem atexchltN 30139
Description: Atom exchange property. Version of hlatexch2 30094 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
atexchlt.s  |-  .<  =  ( lt `  K )
atexchlt.j  |-  .\/  =  ( join `  K )
atexchlt.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atexchltN  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  ->  ( P  .<  ( Q  .\/  R
)  ->  Q  .<  ( P  .\/  R ) ) )

Proof of Theorem atexchltN
StepHypRef Expression
1 atexchlt.j . . 3  |-  .\/  =  ( join `  K )
2 atexchlt.a . . 3  |-  A  =  ( Atoms `  K )
3 eqid 2435 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
41, 2, 3atexchcvrN 30138 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  ->  ( P
(  <o  `  K )
( Q  .\/  R
)  ->  Q (  <o  `  K ) ( P  .\/  R ) ) )
5 atexchlt.s . . . 4  |-  .<  =  ( lt `  K )
65, 1, 2, 3atltcvr 30133 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  <->  P (  <o  `  K
) ( Q  .\/  R ) ) )
763adant3 977 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  ->  ( P  .<  ( Q  .\/  R
)  <->  P (  <o  `  K
) ( Q  .\/  R ) ) )
8 simpl 444 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  HL )
9 simpr2 964 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
10 simpr1 963 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
11 simpr3 965 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
125, 1, 2, 3atltcvr 30133 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  P  e.  A  /\  R  e.  A
) )  ->  ( Q  .<  ( P  .\/  R )  <->  Q (  <o  `  K
) ( P  .\/  R ) ) )
138, 9, 10, 11, 12syl13anc 1186 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( Q  .<  ( P  .\/  R )  <->  Q (  <o  `  K
) ( P  .\/  R ) ) )
14133adant3 977 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  ->  ( Q  .<  ( P  .\/  R
)  <->  Q (  <o  `  K
) ( P  .\/  R ) ) )
154, 7, 143imtr4d 260 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  ->  ( P  .<  ( Q  .\/  R
)  ->  Q  .<  ( P  .\/  R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   ltcplt 14388   joincjn 14391    <o ccvr 29961   Atomscatm 29962   HLchlt 30049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-lat 14465  df-clat 14527  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050
  Copyright terms: Public domain W3C validator