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Theorem llnexatN 34534
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 996 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  K  e.  HL )
2 simp3 998 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  P  e.  A )
3 simp2 997 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  X  e.  N )
41, 2, 33jca 1176 . . 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 2467 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
7 llnexat.a . . . 4  |-  A  =  ( Atoms `  K )
8 llnexat.n . . . 4  |-  N  =  ( LLines `  K )
95, 6, 7, 8atcvrlln2 34532 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  P (  <o  `  K ) X )
104, 9sylan 471 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  P (  <o  `  K ) X )
11 simpl1 999 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  K  e.  HL )
12 simpl3 1001 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  P  e.  A
)
13 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1413, 7atbase 34303 . . . . 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 1000 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  X  e.  N
)
1713, 8llnbase 34522 . . . . 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 34426 . . . 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 1228 . . 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 1035 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  K  e.  HL )
23 hlatl 34374 . . . . . . . 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 461 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  q  e.  A )
26 simpll3 1037 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  P  e.  A )
275, 7atncmp 34326 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  ( -.  q  .<_  P  <->  q  =/=  P ) )
2824, 25, 26, 27syl3anc 1228 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  ( -.  q  .<_  P  <->  q  =/=  P ) )
2928anbi1d 704 . . . . 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 2736 . . . . . 6  |-  ( q  =/=  P  <->  P  =/=  q )
31 eqcom 2476 . . . . . 6  |-  ( ( P  .\/  q )  =  X  <->  X  =  ( P  .\/  q ) )
3230, 31anbi12i 697 . . . . 5  |-  ( ( q  =/=  P  /\  ( P  .\/  q )  =  X )  <->  ( P  =/=  q  /\  X  =  ( P  .\/  q
) ) )
3329, 32syl6bb 261 . . . 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 2970 . . 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 253 . 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 210 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434    <o ccvr 34276   Atomscatm 34277   AtLatcal 34278   HLchlt 34364   LLinesclln 34504
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 6577
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511
This theorem is referenced by:  lplnexllnN  34577
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