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Theorem atbtwn 30243
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 1040 . . . . . . 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 448 . . . . . . 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 1032 . . . . . . . . 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 30161 . . . . . . . . 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 1010 . . . . . . . . 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 30087 . . . . . . . . 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 960 . . . . . . . . 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 30164 . . . . . . . . 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 1011 . . . . . . . 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 2436 . . . . . . . . 9  |-  ( meet `  K )  =  (
meet `  K )
187, 16, 17latlem12 14507 . . . . . . . 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 1186 . . . . . . 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 888 . . . . . 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 1033 . . . . . . 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 1034 . . . . . . 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 1038 . . . . . . 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 1039 . . . . . . 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 30242 . . . . . . 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 1202 . . . . . 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 4236 . . . . 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 30158 . . . . . . 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 30109 . . . . . 6  |-  ( ( K  e.  AtLat  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .<_  P  <->  R  =  P ) )
3129, 6, 21, 30syl3anc 1184 . . . . 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 202 . . . 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 424 . . 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 2637 . 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 993 . . 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 4230 . . . . 5  |-  ( ( P  .<_  X  /\  -.  R  .<_  X )  ->  P  =/=  R
)
3736necomd 2687 . . . 4  |-  ( ( P  .<_  X  /\  -.  R  .<_  X )  ->  R  =/=  P
)
3837ex 424 . . 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 184 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   Latclat 14474   Atomscatm 30061   AtLatcal 30062   HLchlt 30148
This theorem is referenced by:  atbtwnexOLDN  30244  atbtwnex  30245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149
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