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Theorem islln3 32507
Description: The predicate "is a lattice line". (Contributed by NM, 17-Jun-2012.)
Hypotheses
Ref Expression
islln3.b  |-  B  =  ( Base `  K
)
islln3.j  |-  .\/  =  ( join `  K )
islln3.a  |-  A  =  ( Atoms `  K )
islln3.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
islln3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X  e.  N  <->  E. p  e.  A  E. q  e.  A  (
p  =/=  q  /\  X  =  ( p  .\/  q ) ) ) )
Distinct variable groups:    q, p, A    B, p, q    K, p, q    X, p, q
Allowed substitution hints:    .\/ ( q, p)    N( q, p)

Proof of Theorem islln3
StepHypRef Expression
1 islln3.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2402 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
3 islln3.a . . 3  |-  A  =  ( Atoms `  K )
4 islln3.n . . 3  |-  N  =  ( LLines `  K )
51, 2, 3, 4islln4 32504 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X  e.  N  <->  E. p  e.  A  p (  <o  `  K ) X ) )
6 simpll 752 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  K  e.  HL )
71, 3atbase 32287 . . . . . 6  |-  ( p  e.  A  ->  p  e.  B )
87adantl 464 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  p  e.  B )
9 simplr 754 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  X  e.  B )
10 eqid 2402 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
11 islln3.j . . . . . 6  |-  .\/  =  ( join `  K )
121, 10, 11, 2, 3cvrval3 32410 . . . . 5  |-  ( ( K  e.  HL  /\  p  e.  B  /\  X  e.  B )  ->  ( p (  <o  `  K ) X  <->  E. q  e.  A  ( -.  q ( le `  K ) p  /\  ( p  .\/  q )  =  X ) ) )
136, 8, 9, 12syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( p
(  <o  `  K ) X 
<->  E. q  e.  A  ( -.  q ( le `  K ) p  /\  ( p  .\/  q )  =  X ) ) )
14 hlatl 32358 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
1514ad3antrrr 728 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  K  e.  AtLat )
16 simpr 459 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  q  e.  A )
17 simplr 754 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  p  e.  A )
1810, 3atncmp 32310 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  p  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
1915, 16, 17, 18syl3anc 1230 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
20 necom 2672 . . . . . . 7  |-  ( q  =/=  p  <->  p  =/=  q )
2119, 20syl6bb 261 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  p  =/=  q ) )
22 eqcom 2411 . . . . . . 7  |-  ( ( p  .\/  q )  =  X  <->  X  =  ( p  .\/  q ) )
2322a1i 11 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  (
( p  .\/  q
)  =  X  <->  X  =  ( p  .\/  q ) ) )
2421, 23anbi12d 709 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  (
( -.  q ( le `  K ) p  /\  ( p 
.\/  q )  =  X )  <->  ( p  =/=  q  /\  X  =  ( p  .\/  q
) ) ) )
2524rexbidva 2914 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( E. q  e.  A  ( -.  q ( le `  K ) p  /\  ( p  .\/  q )  =  X )  <->  E. q  e.  A  ( p  =/=  q  /\  X  =  ( p  .\/  q
) ) ) )
2613, 25bitrd 253 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( p
(  <o  `  K ) X 
<->  E. q  e.  A  ( p  =/=  q  /\  X  =  (
p  .\/  q )
) ) )
2726rexbidva 2914 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( E. p  e.  A  p (  <o  `  K ) X  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  ( p  .\/  q
) ) ) )
285, 27bitrd 253 1  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X  e.  N  <->  E. p  e.  A  E. q  e.  A  (
p  =/=  q  /\  X  =  ( p  .\/  q ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   lecple 14914   joincjn 15895    <o ccvr 32260   Atomscatm 32261   AtLatcal 32262   HLchlt 32348   LLinesclln 32488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495
This theorem is referenced by:  islln2  32508  llni2  32509  atcvrlln2  32516  atcvrlln  32517  llnexchb2  32866
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