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Theorem atbtwn 32930
Description: Property of a 3rd atom  R on a line  P  .\/  Q intersecting element  X at  P. (Contributed by NM, 30-Jul-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
atbtwn  |-  ( ( ( 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 ) )

Proof of Theorem atbtwn
StepHypRef Expression
1 simpl33 1071 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  R  .<_  ( P  .\/  Q ) )
2 simpr 461 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  R  .<_  X )
3 simpl11 1063 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  K  e.  HL )
4 hllat 32848 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  K  e.  Lat )
6 simpl2l 1041 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  R  e.  A )
7 atbtwn.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
8 atbtwn.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
97, 8atbase 32774 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  B )
106, 9syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  R  e.  B )
11 simpl1 991 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  -> 
( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )
)
12 atbtwn.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
137, 12, 8hlatjcl 32851 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
1411, 13syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  -> 
( P  .\/  Q
)  e.  B )
15 simpl2r 1042 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  X  e.  B )
16 atbtwn.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
17 eqid 2438 . . . . . . . . 9  |-  ( meet `  K )  =  (
meet `  K )
187, 16, 17latlem12 15240 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( R  e.  B  /\  ( P  .\/  Q
)  e.  B  /\  X  e.  B )
)  ->  ( ( R  .<_  ( P  .\/  Q )  /\  R  .<_  X )  <->  R  .<_  ( ( P  .\/  Q ) ( meet `  K
) X ) ) )
195, 10, 14, 15, 18syl13anc 1220 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  -> 
( ( R  .<_  ( P  .\/  Q )  /\  R  .<_  X )  <-> 
R  .<_  ( ( P 
.\/  Q ) (
meet `  K ) X ) ) )
201, 2, 19mpbi2and 912 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  R  .<_  ( ( P 
.\/  Q ) (
meet `  K ) X ) )
21 simpl12 1064 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  P  e.  A )
22 simpl13 1065 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  Q  e.  A )
23 simpl31 1069 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  P  .<_  X )
24 simpl32 1070 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  -.  Q  .<_  X )
257, 16, 12, 17, 82atjm 32929 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q ) ( meet `  K
) X )  =  P )
263, 21, 22, 15, 23, 24, 25syl132anc 1236 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  -> 
( ( P  .\/  Q ) ( meet `  K
) X )  =  P )
2720, 26breqtrd 4311 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  R  .<_  P )
28 hlatl 32845 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
293, 28syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  K  e.  AtLat )
3016, 8atcmp 32796 . . . . . 6  |-  ( ( K  e.  AtLat  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .<_  P  <->  R  =  P ) )
3129, 6, 21, 30syl3anc 1218 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  -> 
( R  .<_  P  <->  R  =  P ) )
3227, 31mpbid 210 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  .<_  X )  ->  R  =  P )
3332ex 434 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( R  .<_  X  ->  R  =  P ) )
3433necon3ad 2639 . 2  |-  ( ( ( 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 ) )
35 simp31 1024 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  ->  P  .<_  X )
36 nbrne2 4305 . . . . 5  |-  ( ( P  .<_  X  /\  -.  R  .<_  X )  ->  P  =/=  R
)
3736necomd 2690 . . . 4  |-  ( ( P  .<_  X  /\  -.  R  .<_  X )  ->  R  =/=  P
)
3837ex 434 . . 3  |-  ( P 
.<_  X  ->  ( -.  R  .<_  X  ->  R  =/=  P ) )
3935, 38syl 16 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( -.  R  .<_  X  ->  R  =/=  P ) )
4034, 39impbid 191 1  |-  ( ( ( 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 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   meetcmee 15107   Latclat 15207   Atomscatm 32748   AtLatcal 32749   HLchlt 32835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836
This theorem is referenced by:  atbtwnexOLDN  32931  atbtwnex  32932
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