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Theorem atbtwnex 34912
Description: Given atoms  P in  X and  Q not in  X, there exists an atom  r not in  X such that the line  Q  .\/  r intersects  X at  P. (Contributed by NM, 1-Aug-2012.)
Hypotheses
Ref Expression
atbtwn.b  |-  B  =  ( Base `  K
)
atbtwn.l  |-  .<_  =  ( le `  K )
atbtwn.j  |-  .\/  =  ( join `  K )
atbtwn.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atbtwnex  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  P  .<_  ( Q  .\/  r ) ) )
Distinct variable groups:    A, r    B, r    K, r    .<_ , r    P, r    Q, r    X, r
Allowed substitution hint:    .\/ ( r)

Proof of Theorem atbtwnex
StepHypRef Expression
1 simpr2 1004 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  X )
2 simpr3 1005 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  -.  Q  .<_  X )
3 nbrne2 4455 . . . 4  |-  ( ( P  .<_  X  /\  -.  Q  .<_  X )  ->  P  =/=  Q
)
41, 2, 3syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  =/=  Q )
5 atbtwn.l . . . 4  |-  .<_  =  ( le `  K )
6 atbtwn.j . . . 4  |-  .\/  =  ( join `  K )
7 atbtwn.a . . . 4  |-  A  =  ( Atoms `  K )
85, 6, 7hlsupr 34850 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r  .<_  ( P  .\/  Q ) ) )
94, 8syldan 470 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r  .<_  ( P  .\/  Q ) ) )
10 simp32 1034 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  =/=  Q )
11 simp31 1033 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  =/=  P )
12 simp1l 1021 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) )
13 simp2 998 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  e.  A )
14 simp1r1 1093 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  X  e.  B )
15 simp1r2 1094 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  P  .<_  X )
16 simp1r3 1095 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  -.  Q  .<_  X )
17 simp33 1035 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  .<_  ( P  .\/  Q ) )
18 atbtwn.b . . . . . . . 8  |-  B  =  ( Base `  K
)
1918, 5, 6, 7atbtwn 34910 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( r  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) )  ->  (
r  =/=  P  <->  -.  r  .<_  X ) )
2012, 13, 14, 15, 16, 17, 19syl123anc 1246 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  ( r  =/=  P  <->  -.  r  .<_  X ) )
2111, 20mpbid 210 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  -.  r  .<_  X )
22 simp1l1 1090 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  HL )
23 simp1l2 1091 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  A )
24 simp1l3 1092 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  A )
255, 6, 7hlatexch2 34860 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  r  =/=  Q )  ->  ( r  .<_  ( P  .\/  Q
)  ->  P  .<_  ( r  .\/  Q ) ) )
2622, 13, 23, 24, 10, 25syl131anc 1242 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  ( r  .<_  ( P  .\/  Q
)  ->  P  .<_  ( r  .\/  Q ) ) )
2717, 26mpd 15 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  P  .<_  ( r  .\/  Q ) )
286, 7hlatjcom 34832 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  r  e.  A )  ->  ( Q  .\/  r
)  =  ( r 
.\/  Q ) )
2922, 24, 13, 28syl3anc 1229 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  ( Q  .\/  r )  =  ( r  .\/  Q ) )
3027, 29breqtrrd 4463 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  P  .<_  ( Q  .\/  r ) )
3110, 21, 303jca 1177 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  ( r  =/=  Q  /\  -.  r  .<_  X  /\  P  .<_  ( Q  .\/  r ) ) )
32313exp 1196 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  ( r  e.  A  ->  ( ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P  .\/  Q
) )  ->  (
r  =/=  Q  /\  -.  r  .<_  X  /\  P  .<_  ( Q  .\/  r ) ) ) ) )
3332reximdvai 2915 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  ( E. r  e.  A  (
r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P  .\/  Q
) )  ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  P  .<_  ( Q  .\/  r ) ) ) )
349, 33mpd 15 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  P  .<_  ( Q  .\/  r ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14509   lecple 14581   joincjn 15447   Atomscatm 34728   HLchlt 34815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-preset 15431  df-poset 15449  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-clat 15612  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816
This theorem is referenced by:  dalem19  35146
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