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Theorem islln2a 29706
Description: The predicate "is a lattice line" in terms of atoms. (Contributed by NM, 15-Jul-2012.)
Hypotheses
Ref Expression
islln2a.j  |-  .\/  =  ( join `  K )
islln2a.a  |-  A  =  ( Atoms `  K )
islln2a.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
islln2a  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  .\/  Q )  e.  N  <->  P  =/=  Q ) )

Proof of Theorem islln2a
StepHypRef Expression
1 islln2a.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
2 islln2a.n . . . . . . . . . . 11  |-  N  =  ( LLines `  K )
31, 2llnneat 29703 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  N )  ->  -.  Q  e.  A
)
43adantlr 695 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  Q  e.  N
)  ->  -.  Q  e.  A )
54ex 423 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( Q  e.  N  ->  -.  Q  e.  A
) )
65con2d 107 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( Q  e.  A  ->  -.  Q  e.  N
) )
763impia 1148 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  -.  Q  e.  N
)
87adantr 451 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  -.  Q  e.  N )
9 oveq1 5865 . . . . . . 7  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
10 islln2a.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
1110, 1hlatjidm 29558 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
12113adant2 974 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
139, 12sylan9eqr 2337 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  ( P  .\/  Q )  =  Q )
1413eleq1d 2349 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  ( ( P  .\/  Q )  e.  N  <->  Q  e.  N
) )
158, 14mtbird 292 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  -.  ( P  .\/  Q )  e.  N )
1615ex 423 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  -.  ( P  .\/  Q )  e.  N
) )
1716necon2ad 2494 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  .\/  Q )  e.  N  ->  P  =/=  Q ) )
1810, 1, 2llni2 29701 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  N
)
1918ex 423 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  ( P  .\/  Q
)  e.  N ) )
2017, 19impbid 183 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  .\/  Q )  e.  N  <->  P  =/=  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   joincjn 14078   Atomscatm 29453   HLchlt 29540   LLinesclln 29680
This theorem is referenced by:  cdleme16d  30470
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  df-llines 29687
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