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Theorem islln2a 35638
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 6277 . . . . . 6  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
2 islln2a.j . . . . . . . 8  |-  .\/  =  ( join `  K )
3 islln2a.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
42, 3hlatjidm 35490 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
543adant2 1013 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
61, 5sylan9eqr 2517 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  ( P  .\/  Q )  =  Q )
7 islln2a.n . . . . . . . . . . 11  |-  N  =  ( LLines `  K )
83, 7llnneat 35635 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  N )  ->  -.  Q  e.  A
)
98adantlr 712 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  Q  e.  N
)  ->  -.  Q  e.  A )
109ex 432 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( Q  e.  N  ->  -.  Q  e.  A
) )
1110con2d 115 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( Q  e.  A  ->  -.  Q  e.  N
) )
12113impia 1191 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  -.  Q  e.  N
)
1312adantr 463 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  -.  Q  e.  N )
146, 13eqneltrd 2563 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  -.  ( P  .\/  Q )  e.  N )
1514ex 432 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  -.  ( P  .\/  Q )  e.  N
) )
1615necon2ad 2667 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  .\/  Q )  e.  N  ->  P  =/=  Q ) )
172, 3, 7llni2 35633 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  N
)
1817ex 432 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  ( P  .\/  Q
)  e.  N ) )
1916, 18impbid 191 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  .\/  Q )  e.  N  <->  P  =/=  Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   ` cfv 5570  (class class class)co 6270   joincjn 15772   Atomscatm 35385   HLchlt 35472   LLinesclln 35612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619
This theorem is referenced by:  cdleme16d  36403
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