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Theorem islln3 30321
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 2296 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
3 islln3.a . . 3  |-  A  =  ( Atoms `  K )
4 islln3.n . . 3  |-  N  =  ( LLines `  K )
51, 2, 3, 4islln4 30318 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X  e.  N  <->  E. p  e.  A  p (  <o  `  K ) X ) )
6 simpll 730 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  K  e.  HL )
71, 3atbase 30101 . . . . . 6  |-  ( p  e.  A  ->  p  e.  B )
87adantl 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  p  e.  B )
9 simplr 731 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  X  e.  B )
10 eqid 2296 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
11 islln3.j . . . . . 6  |-  .\/  =  ( join `  K )
121, 10, 11, 2, 3cvrval3 30224 . . . . 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 1182 . . . 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 30172 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
1514ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  K  e.  AtLat )
16 simpr 447 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  q  e.  A )
17 simplr 731 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  p  e.  A )
1810, 3atncmp 30124 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  p  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
1915, 16, 17, 18syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
20 necom 2540 . . . . . . 7  |-  ( q  =/=  p  <->  p  =/=  q )
2119, 20syl6bb 252 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  p  =/=  q ) )
22 eqcom 2298 . . . . . . 7  |-  ( ( p  .\/  q )  =  X  <->  X  =  ( p  .\/  q ) )
2322a1i 10 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  (
( p  .\/  q
)  =  X  <->  X  =  ( p  .\/  q ) ) )
2421, 23anbi12d 691 . . . . 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 2573 . . . 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 244 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( p
(  <o  `  K ) X 
<->  E. q  e.  A  ( p  =/=  q  /\  X  =  (
p  .\/  q )
) ) )
2726rexbidva 2573 . 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 244 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076   HLchlt 30162   LLinesclln 30302
This theorem is referenced by:  islln2  30322  llni2  30323  atcvrlln2  30330  atcvrlln  30331  llnexchb2  30680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309
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