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Theorem atbtwnex 33400
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 995 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  X )
2 simpr3 996 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  -.  Q  .<_  X )
3 nbrne2 4410 . . . 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 33338 . . 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 1025 . . . . 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 1024 . . . . . 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 1012 . . . . . . 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 989 . . . . . . 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 1084 . . . . . . 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 1085 . . . . . . 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 1086 . . . . . . 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 1026 . . . . . . 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 33398 . . . . . . 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 1236 . . . . . 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 1081 . . . . . . . 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 1082 . . . . . . . 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 1083 . . . . . . . 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 33348 . . . . . . . 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 1232 . . . . . . 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 33320 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  r  e.  A )  ->  ( Q  .\/  r
)  =  ( r 
.\/  Q ) )
2922, 24, 13, 28syl3anc 1219 . . . . . 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 4418 . . . . 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 1168 . . . 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 1187 . . 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 2924 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   Basecbs 14278   lecple 14349   joincjn 15218   Atomscatm 33216   HLchlt 33303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304
This theorem is referenced by:  dalem19  33634
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