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Theorem llnexatN 29528
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 955 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  K  e.  HL )
2 simp3 957 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  P  e.  A )
3 simp2 956 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  X  e.  N )
41, 2, 33jca 1132 . . 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 2316 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
7 llnexat.a . . . 4  |-  A  =  ( Atoms `  K )
8 llnexat.n . . . 4  |-  N  =  ( LLines `  K )
95, 6, 7, 8atcvrlln2 29526 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  P (  <o  `  K ) X )
104, 9sylan 457 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  P (  <o  `  K ) X )
11 simpl1 958 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  K  e.  HL )
12 simpl3 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  P  e.  A
)
13 eqid 2316 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1413, 7atbase 29297 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1512, 14syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  P  e.  (
Base `  K )
)
16 simpl2 959 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  ->  X  e.  N
)
1713, 8llnbase 29516 . . . . 5  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
1816, 17syl 15 . . . 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 29420 . . . 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 1182 . . 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 994 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  K  e.  HL )
23 hlatl 29368 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
2422, 23syl 15 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  K  e.  AtLat )
25 simpr 447 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  q  e.  A )
26 simpll3 996 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  P  e.  A )
275, 7atncmp 29320 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  ( -.  q  .<_  P  <->  q  =/=  P ) )
2824, 25, 26, 27syl3anc 1182 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  P  .<_  X )  /\  q  e.  A )  ->  ( -.  q  .<_  P  <->  q  =/=  P ) )
2928anbi1d 685 . . . . 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 2560 . . . . . 6  |-  ( q  =/=  P  <->  P  =/=  q )
31 eqcom 2318 . . . . . 6  |-  ( ( P  .\/  q )  =  X  <->  X  =  ( P  .\/  q ) )
3230, 31anbi12i 678 . . . . 5  |-  ( ( q  =/=  P  /\  ( P  .\/  q )  =  X )  <->  ( P  =/=  q  /\  X  =  ( P  .\/  q
) ) )
3329, 32syl6bb 252 . . . 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 2594 . . 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 244 . 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 201 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 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127    <o ccvr 29270   Atomscatm 29271   AtLatcal 29272   HLchlt 29358   LLinesclln 29498
This theorem is referenced by:  lplnexllnN  29571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505
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