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Theorem islln2a 29683
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 oveq1 6021 . . . . . 6  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
2 islln2a.j . . . . . . . 8  |-  .\/  =  ( join `  K )
3 islln2a.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
42, 3hlatjidm 29535 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
543adant2 976 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
61, 5sylan9eqr 2435 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  ( P  .\/  Q )  =  Q )
7 islln2a.n . . . . . . . . . . 11  |-  N  =  ( LLines `  K )
83, 7llnneat 29680 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  N )  ->  -.  Q  e.  A
)
98adantlr 696 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  Q  e.  N
)  ->  -.  Q  e.  A )
109ex 424 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( Q  e.  N  ->  -.  Q  e.  A
) )
1110con2d 109 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( Q  e.  A  ->  -.  Q  e.  N
) )
12113impia 1150 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  -.  Q  e.  N
)
1312adantr 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  -.  Q  e.  N )
146, 13eqneltrd 2474 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  -.  ( P  .\/  Q )  e.  N )
1514ex 424 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  -.  ( P  .\/  Q )  e.  N
) )
1615necon2ad 2592 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  .\/  Q )  e.  N  ->  P  =/=  Q ) )
172, 3, 7llni2 29678 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  N
)
1817ex 424 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  ( P  .\/  Q
)  e.  N ) )
1916, 18impbid 184 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2544   ` cfv 5388  (class class class)co 6014   joincjn 14322   Atomscatm 29430   HLchlt 29517   LLinesclln 29657
This theorem is referenced by:  cdleme16d  30447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-undef 6473  df-riota 6479  df-poset 14324  df-plt 14336  df-lub 14352  df-glb 14353  df-join 14354  df-meet 14355  df-p0 14389  df-lat 14396  df-clat 14458  df-oposet 29343  df-ol 29345  df-oml 29346  df-covers 29433  df-ats 29434  df-atl 29465  df-cvlat 29489  df-hlat 29518  df-llines 29664
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