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Theorem llnexatN 30318
Description: Given an atom on a line, there is another atom whose join equals the line. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
llnexat.l  |-  .<_  =  ( le `  K )
llnexat.j  |-  .\/  =  ( join `  K )
llnexat.a  |-  A  =  ( Atoms `  K )
llnexat.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
llnexatN  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  E. q  e.  A  ( P  =/=  q  /\  X  =  ( P  .\/  q ) ) )
Distinct variable groups:    A, q    K, q    .<_ , q    N, q    P, q    X, q
Allowed substitution hint:    .\/ ( q)

Proof of Theorem llnexatN
StepHypRef Expression
1 simp1 957 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  K  e.  HL )
2 simp3 959 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  P  e.  A )
3 simp2 958 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  X  e.  N )
41, 2, 33jca 1134 . . 3  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )
)
5 llnexat.l . . . 4  |-  .<_  =  ( le `  K )
6 eqid 2436 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
7 llnexat.a . . . 4  |-  A  =  ( Atoms `  K )
8 llnexat.n . . . 4  |-  N  =  ( LLines `  K )
95, 6, 7, 8atcvrlln2 30316 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  P (  <o  `  K ) X )
104, 9sylan 458 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  P (  <o  `  K ) X )
11 simpl1 960 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  K  e.  HL )
12 simpl3 962 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  P  e.  A
)
13 eqid 2436 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1413, 7atbase 30087 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1512, 14syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  P  e.  (
Base `  K )
)
16 simpl2 961 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  X  e.  N
)
1713, 8llnbase 30306 . . . . 5  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
1816, 17syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  X  e.  (
Base `  K )
)
19 llnexat.j . . . . 5  |-  .\/  =  ( join `  K )
2013, 5, 19, 6, 7cvrval3 30210 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  X  e.  ( Base `  K
) )  ->  ( P (  <o  `  K
) X  <->  E. q  e.  A  ( -.  q  .<_  P  /\  ( P  .\/  q )  =  X ) ) )
2111, 15, 18, 20syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  ( P ( 
<o  `  K ) X  <->  E. q  e.  A  ( -.  q  .<_  P  /\  ( P  .\/  q )  =  X ) ) )
22 simpll1 996 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  K  e.  HL )
23 hlatl 30158 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
2422, 23syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  K  e.  AtLat )
25 simpr 448 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  q  e.  A )
26 simpll3 998 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  P  e.  A )
275, 7atncmp 30110 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  ( -.  q  .<_  P  <->  q  =/=  P ) )
2824, 25, 26, 27syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  ( -.  q  .<_  P  <->  q  =/=  P ) )
2928anbi1d 686 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  (
( -.  q  .<_  P  /\  ( P  .\/  q )  =  X )  <->  ( q  =/= 
P  /\  ( P  .\/  q )  =  X ) ) )
30 necom 2685 . . . . . 6  |-  ( q  =/=  P  <->  P  =/=  q )
31 eqcom 2438 . . . . . 6  |-  ( ( P  .\/  q )  =  X  <->  X  =  ( P  .\/  q ) )
3230, 31anbi12i 679 . . . . 5  |-  ( ( q  =/=  P  /\  ( P  .\/  q )  =  X )  <->  ( P  =/=  q  /\  X  =  ( P  .\/  q
) ) )
3329, 32syl6bb 253 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  (
( -.  q  .<_  P  /\  ( P  .\/  q )  =  X )  <->  ( P  =/=  q  /\  X  =  ( P  .\/  q
) ) ) )
3433rexbidva 2722 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  ( E. q  e.  A  ( -.  q  .<_  P  /\  ( P  .\/  q )  =  X )  <->  E. q  e.  A  ( P  =/=  q  /\  X  =  ( P  .\/  q
) ) ) )
3521, 34bitrd 245 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  ( P ( 
<o  `  K ) X  <->  E. q  e.  A  ( P  =/=  q  /\  X  =  ( P  .\/  q ) ) ) )
3610, 35mpbid 202 1  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  E. q  e.  A  ( P  =/=  q  /\  X  =  ( P  .\/  q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401    <o ccvr 30060   Atomscatm 30061   AtLatcal 30062   HLchlt 30148   LLinesclln 30288
This theorem is referenced by:  lplnexllnN  30361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295
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