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Theorem atexchltN 29630
Description: Atom exchange property. Version of hlatexch2 29585 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 2283 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
41, 2, 3atexchcvrN 29629 . 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 29624 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  <->  P (  <o  `  K
) ( Q  .\/  R ) ) )
763adant3 975 . 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 443 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  HL )
9 simpr2 962 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
10 simpr1 961 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
11 simpr3 963 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
125, 1, 2, 3atltcvr 29624 . . . 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 1184 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( Q  .<  ( P  .\/  R )  <->  Q (  <o  `  K
) ( P  .\/  R ) ) )
14133adant3 975 . 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 259 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   ltcplt 14075   joincjn 14078    <o ccvr 29452   Atomscatm 29453   HLchlt 29540
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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  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-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-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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