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Theorem islln3 34306
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 2467 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
3 islln3.a . . 3  |-  A  =  ( Atoms `  K )
4 islln3.n . . 3  |-  N  =  ( LLines `  K )
51, 2, 3, 4islln4 34303 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X  e.  N  <->  E. p  e.  A  p (  <o  `  K ) X ) )
6 simpll 753 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  K  e.  HL )
71, 3atbase 34086 . . . . . 6  |-  ( p  e.  A  ->  p  e.  B )
87adantl 466 . . . . 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 2467 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
11 islln3.j . . . . . 6  |-  .\/  =  ( join `  K )
121, 10, 11, 2, 3cvrval3 34209 . . . . 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 1228 . . . 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 34157 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
1514ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  K  e.  AtLat )
16 simpr 461 . . . . . . . 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 34109 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  p  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
1915, 16, 17, 18syl3anc 1228 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
20 necom 2736 . . . . . . 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 2476 . . . . . . 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 710 . . . . 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 2970 . . . 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 2970 . 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 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427    <o ccvr 34059   Atomscatm 34060   AtLatcal 34061   HLchlt 34147   LLinesclln 34287
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 6574
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294
This theorem is referenced by:  islln2  34307  llni2  34308  atcvrlln2  34315  atcvrlln  34316  llnexchb2  34665
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