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Theorem islln2a 34530
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 6292 . . . . . 6  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
2 islln2a.j . . . . . . . 8  |-  .\/  =  ( join `  K )
3 islln2a.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
42, 3hlatjidm 34382 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
543adant2 1015 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
61, 5sylan9eqr 2530 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  ( P  .\/  Q )  =  Q )
7 islln2a.n . . . . . . . . . . 11  |-  N  =  ( LLines `  K )
83, 7llnneat 34527 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  N )  ->  -.  Q  e.  A
)
98adantlr 714 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  Q  e.  N
)  ->  -.  Q  e.  A )
109ex 434 . . . . . . . 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 1193 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  -.  Q  e.  N
)
1312adantr 465 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  -.  Q  e.  N )
146, 13eqneltrd 2576 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =  Q
)  ->  -.  ( P  .\/  Q )  e.  N )
1514ex 434 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  -.  ( P  .\/  Q )  e.  N
) )
1615necon2ad 2680 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  .\/  Q )  e.  N  ->  P  =/=  Q ) )
172, 3, 7llni2 34525 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  N
)
1817ex 434 . 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5588  (class class class)co 6285   joincjn 15434   Atomscatm 34277   HLchlt 34364   LLinesclln 34504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511
This theorem is referenced by:  cdleme16d  35294
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