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Theorem islln3 30369
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 2438 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
3 islln3.a . . 3  |-  A  =  ( Atoms `  K )
4 islln3.n . . 3  |-  N  =  ( LLines `  K )
51, 2, 3, 4islln4 30366 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X  e.  N  <->  E. p  e.  A  p (  <o  `  K ) X ) )
6 simpll 732 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  K  e.  HL )
71, 3atbase 30149 . . . . . 6  |-  ( p  e.  A  ->  p  e.  B )
87adantl 454 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  p  e.  B )
9 simplr 733 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  X  e.  B )
10 eqid 2438 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
11 islln3.j . . . . . 6  |-  .\/  =  ( join `  K )
121, 10, 11, 2, 3cvrval3 30272 . . . . 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 1185 . . . 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 30220 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
1514ad3antrrr 712 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  K  e.  AtLat )
16 simpr 449 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  q  e.  A )
17 simplr 733 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  p  e.  A )
1810, 3atncmp 30172 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  p  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
1915, 16, 17, 18syl3anc 1185 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
20 necom 2687 . . . . . . 7  |-  ( q  =/=  p  <->  p  =/=  q )
2119, 20syl6bb 254 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  p  =/=  q ) )
22 eqcom 2440 . . . . . . 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 693 . . . . 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 2724 . . . 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 246 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( p
(  <o  `  K ) X 
<->  E. q  e.  A  ( p  =/=  q  /\  X  =  (
p  .\/  q )
) ) )
2726rexbidva 2724 . 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 246 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403    <o ccvr 30122   Atomscatm 30123   AtLatcal 30124   HLchlt 30210   LLinesclln 30350
This theorem is referenced by:  islln2  30370  llni2  30371  atcvrlln2  30378  atcvrlln  30379  llnexchb2  30728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357
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