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Theorem dalem-cly 30468
Description: Lemma for dalem9 30469. Center of perspectivity  C is not in plane  Y (when  Y and  Z are different planes). (Contributed by NM, 13-Aug-2012.)
Hypotheses
Ref Expression
dalema.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalem-cly.o  |-  O  =  ( LPlanes `  K )
dalem-cly.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem-cly.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
Assertion
Ref Expression
dalem-cly  |-  ( (
ph  /\  Y  =/=  Z )  ->  -.  C  .<_  Y )

Proof of Theorem dalem-cly
StepHypRef Expression
1 dalema.ph . . . . . . 7  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
21dalemkelat 30421 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . . . . 7  |-  A  =  ( Atoms `  K )
41, 3dalemceb 30435 . . . . . 6  |-  ( ph  ->  C  e.  ( Base `  K ) )
5 dalem-cly.o . . . . . . 7  |-  O  =  ( LPlanes `  K )
61, 5dalemyeb 30446 . . . . . 6  |-  ( ph  ->  Y  e.  ( Base `  K ) )
7 eqid 2436 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 dalemc.l . . . . . . 7  |-  .<_  =  ( le `  K )
9 dalemc.j . . . . . . 7  |-  .\/  =  ( join `  K )
107, 8, 9latleeqj1 14492 . . . . . 6  |-  ( ( K  e.  Lat  /\  C  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( C  .<_  Y  <->  ( C  .\/  Y )  =  Y ) )
112, 4, 6, 10syl3anc 1184 . . . . 5  |-  ( ph  ->  ( C  .<_  Y  <->  ( C  .\/  Y )  =  Y ) )
121dalemclpjs 30431 . . . . . . . . . . . . 13  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
131dalemkehl 30420 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  HL )
14 dalem-cly.y . . . . . . . . . . . . . . 15  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
151, 8, 9, 3, 5, 14dalemcea 30457 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  A )
161dalemsea 30426 . . . . . . . . . . . . . 14  |-  ( ph  ->  S  e.  A )
171dalempea 30423 . . . . . . . . . . . . . 14  |-  ( ph  ->  P  e.  A )
181dalemqea 30424 . . . . . . . . . . . . . . 15  |-  ( ph  ->  Q  e.  A )
191dalem-clpjq 30434 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
208, 9, 3atnlej1 30176 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  C  .<_  ( P  .\/  Q
) )  ->  C  =/=  P )
2113, 15, 17, 18, 19, 20syl131anc 1197 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/=  P )
228, 9, 3hlatexch1 30192 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  S  e.  A  /\  P  e.  A
)  /\  C  =/=  P )  ->  ( C  .<_  ( P  .\/  S
)  ->  S  .<_  ( P  .\/  C ) ) )
2313, 15, 16, 17, 21, 22syl131anc 1197 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C  .<_  ( P 
.\/  S )  ->  S  .<_  ( P  .\/  C ) ) )
2412, 23mpd 15 . . . . . . . . . . . 12  |-  ( ph  ->  S  .<_  ( P  .\/  C ) )
259, 3hlatjcom 30165 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  P  e.  A )  ->  ( C  .\/  P
)  =  ( P 
.\/  C ) )
2613, 15, 17, 25syl3anc 1184 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  P
)  =  ( P 
.\/  C ) )
2724, 26breqtrrd 4238 . . . . . . . . . . 11  |-  ( ph  ->  S  .<_  ( C  .\/  P ) )
281dalemclqjt 30432 . . . . . . . . . . . . 13  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
291dalemtea 30427 . . . . . . . . . . . . . 14  |-  ( ph  ->  T  e.  A )
301dalemrea 30425 . . . . . . . . . . . . . . 15  |-  ( ph  ->  R  e.  A )
31 simp312 1105 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) )  ->  -.  C  .<_  ( Q  .\/  R ) )
321, 31sylbi 188 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  C  .<_  ( Q 
.\/  R ) )
338, 9, 3atnlej1 30176 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  C  .<_  ( Q  .\/  R
) )  ->  C  =/=  Q )
3413, 15, 18, 30, 32, 33syl131anc 1197 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/=  Q )
358, 9, 3hlatexch1 30192 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  T  e.  A  /\  Q  e.  A
)  /\  C  =/=  Q )  ->  ( C  .<_  ( Q  .\/  T
)  ->  T  .<_  ( Q  .\/  C ) ) )
3613, 15, 29, 18, 34, 35syl131anc 1197 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C  .<_  ( Q 
.\/  T )  ->  T  .<_  ( Q  .\/  C ) ) )
3728, 36mpd 15 . . . . . . . . . . . 12  |-  ( ph  ->  T  .<_  ( Q  .\/  C ) )
389, 3hlatjcom 30165 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  Q  e.  A )  ->  ( C  .\/  Q
)  =  ( Q 
.\/  C ) )
3913, 15, 18, 38syl3anc 1184 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  Q
)  =  ( Q 
.\/  C ) )
4037, 39breqtrrd 4238 . . . . . . . . . . 11  |-  ( ph  ->  T  .<_  ( C  .\/  Q ) )
411, 3dalemseb 30439 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  ( Base `  K ) )
427, 9, 3hlatjcl 30164 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  P  e.  A )  ->  ( C  .\/  P
)  e.  ( Base `  K ) )
4313, 15, 17, 42syl3anc 1184 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  P
)  e.  ( Base `  K ) )
441, 3dalemteb 30440 . . . . . . . . . . . 12  |-  ( ph  ->  T  e.  ( Base `  K ) )
457, 9, 3hlatjcl 30164 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  Q  e.  A )  ->  ( C  .\/  Q
)  e.  ( Base `  K ) )
4613, 15, 18, 45syl3anc 1184 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  Q
)  e.  ( Base `  K ) )
477, 8, 9latjlej12 14496 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( C  .\/  P )  e.  ( Base `  K
) )  /\  ( T  e.  ( Base `  K )  /\  ( C  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( S  .<_  ( C  .\/  P )  /\  T  .<_  ( C 
.\/  Q ) )  ->  ( S  .\/  T )  .<_  ( ( C  .\/  P )  .\/  ( C  .\/  Q ) ) ) )
482, 41, 43, 44, 46, 47syl122anc 1193 . . . . . . . . . . 11  |-  ( ph  ->  ( ( S  .<_  ( C  .\/  P )  /\  T  .<_  ( C 
.\/  Q ) )  ->  ( S  .\/  T )  .<_  ( ( C  .\/  P )  .\/  ( C  .\/  Q ) ) ) )
4927, 40, 48mp2and 661 . . . . . . . . . 10  |-  ( ph  ->  ( S  .\/  T
)  .<_  ( ( C 
.\/  P )  .\/  ( C  .\/  Q ) ) )
501, 3dalempeb 30436 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  ( Base `  K ) )
511, 3dalemqeb 30437 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  ( Base `  K ) )
527, 9latjjdi 14532 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) ) )  -> 
( C  .\/  ( P  .\/  Q ) )  =  ( ( C 
.\/  P )  .\/  ( C  .\/  Q ) ) )
532, 4, 50, 51, 52syl13anc 1186 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  ( P  .\/  Q ) )  =  ( ( C 
.\/  P )  .\/  ( C  .\/  Q ) ) )
5449, 53breqtrrd 4238 . . . . . . . . 9  |-  ( ph  ->  ( S  .\/  T
)  .<_  ( C  .\/  ( P  .\/  Q ) ) )
551dalemclrju 30433 . . . . . . . . . . 11  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
561dalemuea 30428 . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  A )
57 simp313 1106 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) )  ->  -.  C  .<_  ( R  .\/  P ) )
581, 57sylbi 188 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  C  .<_  ( R 
.\/  P ) )
598, 9, 3atnlej1 30176 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  -.  C  .<_  ( R  .\/  P
) )  ->  C  =/=  R )
6013, 15, 30, 17, 58, 59syl131anc 1197 . . . . . . . . . . . 12  |-  ( ph  ->  C  =/=  R )
618, 9, 3hlatexch1 30192 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  U  e.  A  /\  R  e.  A
)  /\  C  =/=  R )  ->  ( C  .<_  ( R  .\/  U
)  ->  U  .<_  ( R  .\/  C ) ) )
6213, 15, 56, 30, 60, 61syl131anc 1197 . . . . . . . . . . 11  |-  ( ph  ->  ( C  .<_  ( R 
.\/  U )  ->  U  .<_  ( R  .\/  C ) ) )
6355, 62mpd 15 . . . . . . . . . 10  |-  ( ph  ->  U  .<_  ( R  .\/  C ) )
649, 3hlatjcom 30165 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  C  e.  A  /\  R  e.  A )  ->  ( C  .\/  R
)  =  ( R 
.\/  C ) )
6513, 15, 30, 64syl3anc 1184 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  R
)  =  ( R 
.\/  C ) )
6663, 65breqtrrd 4238 . . . . . . . . 9  |-  ( ph  ->  U  .<_  ( C  .\/  R ) )
671, 9, 3dalemsjteb 30443 . . . . . . . . . 10  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
681, 9, 3dalempjqeb 30442 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
697, 9latjcl 14479 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  C  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( C  .\/  ( P  .\/  Q ) )  e.  (
Base `  K )
)
702, 4, 68, 69syl3anc 1184 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  ( P  .\/  Q ) )  e.  ( Base `  K
) )
711, 3dalemueb 30441 . . . . . . . . . 10  |-  ( ph  ->  U  e.  ( Base `  K ) )
727, 9, 3hlatjcl 30164 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  C  e.  A  /\  R  e.  A )  ->  ( C  .\/  R
)  e.  ( Base `  K ) )
7313, 15, 30, 72syl3anc 1184 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  R
)  e.  ( Base `  K ) )
747, 8, 9latjlej12 14496 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( ( S  .\/  T )  e.  ( Base `  K )  /\  ( C  .\/  ( P  .\/  Q ) )  e.  (
Base `  K )
)  /\  ( U  e.  ( Base `  K
)  /\  ( C  .\/  R )  e.  (
Base `  K )
) )  ->  (
( ( S  .\/  T )  .<_  ( C  .\/  ( P  .\/  Q
) )  /\  U  .<_  ( C  .\/  R
) )  ->  (
( S  .\/  T
)  .\/  U )  .<_  ( ( C  .\/  ( P  .\/  Q ) )  .\/  ( C 
.\/  R ) ) ) )
752, 67, 70, 71, 73, 74syl122anc 1193 . . . . . . . . 9  |-  ( ph  ->  ( ( ( S 
.\/  T )  .<_  ( C  .\/  ( P 
.\/  Q ) )  /\  U  .<_  ( C 
.\/  R ) )  ->  ( ( S 
.\/  T )  .\/  U )  .<_  ( ( C  .\/  ( P  .\/  Q ) )  .\/  ( C  .\/  R ) ) ) )
7654, 66, 75mp2and 661 . . . . . . . 8  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U ) 
.<_  ( ( C  .\/  ( P  .\/  Q ) )  .\/  ( C 
.\/  R ) ) )
771, 3dalemreb 30438 . . . . . . . . 9  |-  ( ph  ->  R  e.  ( Base `  K ) )
787, 9latjjdi 14532 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
) )  ->  ( C  .\/  ( ( P 
.\/  Q )  .\/  R ) )  =  ( ( C  .\/  ( P  .\/  Q ) ) 
.\/  ( C  .\/  R ) ) )
792, 4, 68, 77, 78syl13anc 1186 . . . . . . . 8  |-  ( ph  ->  ( C  .\/  (
( P  .\/  Q
)  .\/  R )
)  =  ( ( C  .\/  ( P 
.\/  Q ) ) 
.\/  ( C  .\/  R ) ) )
8076, 79breqtrrd 4238 . . . . . . 7  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U ) 
.<_  ( C  .\/  (
( P  .\/  Q
)  .\/  R )
) )
81 dalem-cly.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
8214oveq2i 6092 . . . . . . 7  |-  ( C 
.\/  Y )  =  ( C  .\/  (
( P  .\/  Q
)  .\/  R )
)
8380, 81, 823brtr4g 4244 . . . . . 6  |-  ( ph  ->  Z  .<_  ( C  .\/  Y ) )
84 breq2 4216 . . . . . 6  |-  ( ( C  .\/  Y )  =  Y  ->  ( Z  .<_  ( C  .\/  Y )  <->  Z  .<_  Y ) )
8583, 84syl5ibcom 212 . . . . 5  |-  ( ph  ->  ( ( C  .\/  Y )  =  Y  ->  Z  .<_  Y ) )
8611, 85sylbid 207 . . . 4  |-  ( ph  ->  ( C  .<_  Y  ->  Z  .<_  Y ) )
871dalemzeo 30430 . . . . . 6  |-  ( ph  ->  Z  e.  O )
881dalemyeo 30429 . . . . . 6  |-  ( ph  ->  Y  e.  O )
898, 5lplncmp 30359 . . . . . 6  |-  ( ( K  e.  HL  /\  Z  e.  O  /\  Y  e.  O )  ->  ( Z  .<_  Y  <->  Z  =  Y ) )
9013, 87, 88, 89syl3anc 1184 . . . . 5  |-  ( ph  ->  ( Z  .<_  Y  <->  Z  =  Y ) )
91 eqcom 2438 . . . . 5  |-  ( Z  =  Y  <->  Y  =  Z )
9290, 91syl6bb 253 . . . 4  |-  ( ph  ->  ( Z  .<_  Y  <->  Y  =  Z ) )
9386, 92sylibd 206 . . 3  |-  ( ph  ->  ( C  .<_  Y  ->  Y  =  Z )
)
9493necon3ad 2637 . 2  |-  ( ph  ->  ( Y  =/=  Z  ->  -.  C  .<_  Y ) )
9594imp 419 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  -.  C  .<_  Y )
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   Latclat 14474   Atomscatm 30061   HLchlt 30148   LPlanesclpl 30289
This theorem is referenced by:  dalem9  30469
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  df-llines 30295  df-lplanes 30296
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