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Theorem atbtwnex 32722
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 1012 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  X )
2 simpr3 1013 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  -.  Q  .<_  X )
3 nbrne2 4435 . . . 4  |-  ( ( P  .<_  X  /\  -.  Q  .<_  X )  ->  P  =/=  Q
)
41, 2, 3syl2anc 665 . . 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 32660 . . 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 472 . 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 1042 . . . . 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 1041 . . . . . 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 1029 . . . . . . 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 1006 . . . . . . 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 1101 . . . . . . 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 1102 . . . . . . 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 1103 . . . . . . 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 1043 . . . . . . 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 32720 . . . . . . 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 1281 . . . . . 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 213 . . . . 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 1098 . . . . . . . 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 1099 . . . . . . . 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 1100 . . . . . . . 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 32670 . . . . . . . 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 1277 . . . . . . 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 32642 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  r  e.  A )  ->  ( Q  .\/  r
)  =  ( r 
.\/  Q ) )
2922, 24, 13, 28syl3anc 1264 . . . . . 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 4443 . . . . 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 1185 . . . 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 1204 . . 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 2895 . 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   E.wrex 2774   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   Basecbs 15073   lecple 15149   joincjn 16133   Atomscatm 32538   HLchlt 32625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 16117  df-poset 16135  df-plt 16148  df-lub 16164  df-glb 16165  df-join 16166  df-meet 16167  df-p0 16229  df-lat 16236  df-clat 16298  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626
This theorem is referenced by:  dalem19  32956
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