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Theorem dalem44 30513
Description: Lemma for dath 30533. Dummy center of perspectivity  c lies outside of plane  G H I. (Contributed by NM, 16-Aug-2012.)
Hypotheses
Ref Expression
dalem.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 ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem44.m  |-  ./\  =  ( meet `  K )
dalem44.o  |-  O  =  ( LPlanes `  K )
dalem44.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem44.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem44.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem44.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem44.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
Assertion
Ref Expression
dalem44  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( ( G  .\/  H ) 
.\/  I ) )

Proof of Theorem dalem44
StepHypRef Expression
1 dalem.ph . . . 4  |-  ( 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 ) ) ) ) )
2 dalem.l . . . 4  |-  .<_  =  ( le `  K )
3 dalem.j . . . 4  |-  .\/  =  ( join `  K )
4 dalem.a . . . 4  |-  A  =  ( Atoms `  K )
5 dalem.ps . . . 4  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
6 dalem44.m . . . 4  |-  ./\  =  ( meet `  K )
7 dalem44.o . . . 4  |-  O  =  ( LPlanes `  K )
8 dalem44.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
9 dalem44.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
10 dalem44.g . . . 4  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
11 dalem44.h . . . 4  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
12 dalem44.i . . . 4  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem43 30512 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  =/=  Y )
1413necomd 2687 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  =/=  ( ( G 
.\/  H )  .\/  I ) )
151dalemkelat 30421 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
16153ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
175, 4dalemcceb 30486 . . . . . . 7  |-  ( ps 
->  c  e.  ( Base `  K ) )
18173ad2ant3 980 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem42 30511 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  O )
20 eqid 2436 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2120, 7lplnbase 30331 . . . . . . 7  |-  ( ( ( G  .\/  H
)  .\/  I )  e.  O  ->  ( ( G  .\/  H ) 
.\/  I )  e.  ( Base `  K
) )
2219, 21syl 16 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
) )
2320, 2, 3latleeqj1 14492 . . . . . 6  |-  ( ( K  e.  Lat  /\  c  e.  ( Base `  K )  /\  (
( G  .\/  H
)  .\/  I )  e.  ( Base `  K
) )  ->  (
c  .<_  ( ( G 
.\/  H )  .\/  I )  <->  ( c  .\/  ( ( G  .\/  H )  .\/  I ) )  =  ( ( G  .\/  H ) 
.\/  I ) ) )
2416, 18, 22, 23syl3anc 1184 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  ( ( G  .\/  H ) 
.\/  I )  <->  ( c  .\/  ( ( G  .\/  H )  .\/  I ) )  =  ( ( G  .\/  H ) 
.\/  I ) ) )
251, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem28 30497 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  .<_  ( G  .\/  c ) )
261dalemkehl 30420 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  HL )
27263ad2ant1 978 . . . . . . . . . . . . 13  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
285dalemccea 30480 . . . . . . . . . . . . . 14  |-  ( ps 
->  c  e.  A
)
29283ad2ant3 980 . . . . . . . . . . . . 13  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
301, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 30493 . . . . . . . . . . . . 13  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
313, 4hlatjcom 30165 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  c  e.  A  /\  G  e.  A )  ->  ( c  .\/  G
)  =  ( G 
.\/  c ) )
3227, 29, 30, 31syl3anc 1184 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  G
)  =  ( G 
.\/  c ) )
3325, 32breqtrrd 4238 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  .<_  ( c  .\/  G ) )
341, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem33 30502 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  .<_  ( H  .\/  c ) )
351, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 30498 . . . . . . . . . . . . 13  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
363, 4hlatjcom 30165 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  c  e.  A  /\  H  e.  A )  ->  ( c  .\/  H
)  =  ( H 
.\/  c ) )
3727, 29, 35, 36syl3anc 1184 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  H
)  =  ( H 
.\/  c ) )
3834, 37breqtrrd 4238 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  .<_  ( c  .\/  H ) )
391, 4dalempeb 30436 . . . . . . . . . . . . 13  |-  ( ph  ->  P  e.  ( Base `  K ) )
40393ad2ant1 978 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  ( Base `  K ) )
4120, 3, 4hlatjcl 30164 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  c  e.  A  /\  G  e.  A )  ->  ( c  .\/  G
)  e.  ( Base `  K ) )
4227, 29, 30, 41syl3anc 1184 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  G
)  e.  ( Base `  K ) )
431, 4dalemqeb 30437 . . . . . . . . . . . . 13  |-  ( ph  ->  Q  e.  ( Base `  K ) )
44433ad2ant1 978 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  e.  ( Base `  K ) )
4520, 3, 4hlatjcl 30164 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  c  e.  A  /\  H  e.  A )  ->  ( c  .\/  H
)  e.  ( Base `  K ) )
4627, 29, 35, 45syl3anc 1184 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  H
)  e.  ( Base `  K ) )
4720, 2, 3latjlej12 14496 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  ( c  .\/  G
)  e.  ( Base `  K ) )  /\  ( Q  e.  ( Base `  K )  /\  ( c  .\/  H
)  e.  ( Base `  K ) ) )  ->  ( ( P 
.<_  ( c  .\/  G
)  /\  Q  .<_  ( c  .\/  H ) )  ->  ( P  .\/  Q )  .<_  ( ( c  .\/  G ) 
.\/  ( c  .\/  H ) ) ) )
4816, 40, 42, 44, 46, 47syl122anc 1193 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .<_  ( c  .\/  G )  /\  Q  .<_  ( c 
.\/  H ) )  ->  ( P  .\/  Q )  .<_  ( (
c  .\/  G )  .\/  ( c  .\/  H
) ) ) )
4933, 38, 48mp2and 661 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  .<_  ( ( c 
.\/  G )  .\/  ( c  .\/  H
) ) )
5020, 4atbase 30087 . . . . . . . . . . . 12  |-  ( G  e.  A  ->  G  e.  ( Base `  K
) )
5130, 50syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  ( Base `  K ) )
5220, 4atbase 30087 . . . . . . . . . . . 12  |-  ( H  e.  A  ->  H  e.  ( Base `  K
) )
5335, 52syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  ( Base `  K ) )
5420, 3latjjdi 14532 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( c  e.  (
Base `  K )  /\  G  e.  ( Base `  K )  /\  H  e.  ( Base `  K ) ) )  ->  ( c  .\/  ( G  .\/  H ) )  =  ( ( c  .\/  G ) 
.\/  ( c  .\/  H ) ) )
5516, 18, 51, 53, 54syl13anc 1186 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  ( G  .\/  H ) )  =  ( ( c 
.\/  G )  .\/  ( c  .\/  H
) ) )
5649, 55breqtrrd 4238 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  .<_  ( c  .\/  ( G  .\/  H ) ) )
571, 2, 3, 4, 5, 6, 7, 8, 9, 12dalem37 30506 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  .<_  ( I  .\/  c ) )
581, 2, 3, 4, 5, 6, 7, 8, 9, 12dalem34 30503 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
593, 4hlatjcom 30165 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  c  e.  A  /\  I  e.  A )  ->  ( c  .\/  I
)  =  ( I 
.\/  c ) )
6027, 29, 58, 59syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  I
)  =  ( I 
.\/  c ) )
6157, 60breqtrrd 4238 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  .<_  ( c  .\/  I ) )
621, 3, 4dalempjqeb 30442 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
63623ad2ant1 978 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
6420, 3, 4hlatjcl 30164 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
6527, 30, 35, 64syl3anc 1184 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
6620, 3latjcl 14479 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  c  e.  ( Base `  K )  /\  ( G  .\/  H )  e.  ( Base `  K
) )  ->  (
c  .\/  ( G  .\/  H ) )  e.  ( Base `  K
) )
6716, 18, 65, 66syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  ( G  .\/  H ) )  e.  ( Base `  K
) )
681, 4dalemreb 30438 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  ( Base `  K ) )
69683ad2ant1 978 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  e.  ( Base `  K ) )
7020, 3, 4hlatjcl 30164 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  c  e.  A  /\  I  e.  A )  ->  ( c  .\/  I
)  e.  ( Base `  K ) )
7127, 29, 58, 70syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  I
)  e.  ( Base `  K ) )
7220, 2, 3latjlej12 14496 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  (
c  .\/  ( G  .\/  H ) )  e.  ( Base `  K
) )  /\  ( R  e.  ( Base `  K )  /\  (
c  .\/  I )  e.  ( Base `  K
) ) )  -> 
( ( ( P 
.\/  Q )  .<_  ( c  .\/  ( G  .\/  H ) )  /\  R  .<_  ( c 
.\/  I ) )  ->  ( ( P 
.\/  Q )  .\/  R )  .<_  ( (
c  .\/  ( G  .\/  H ) )  .\/  ( c  .\/  I
) ) ) )
7316, 63, 67, 69, 71, 72syl122anc 1193 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( P 
.\/  Q )  .<_  ( c  .\/  ( G  .\/  H ) )  /\  R  .<_  ( c 
.\/  I ) )  ->  ( ( P 
.\/  Q )  .\/  R )  .<_  ( (
c  .\/  ( G  .\/  H ) )  .\/  ( c  .\/  I
) ) ) )
7456, 61, 73mp2and 661 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( c  .\/  ( G  .\/  H ) )  .\/  ( c 
.\/  I ) ) )
7520, 4atbase 30087 . . . . . . . . . 10  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
7658, 75syl 16 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
7720, 3latjjdi 14532 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( c  e.  (
Base `  K )  /\  ( G  .\/  H
)  e.  ( Base `  K )  /\  I  e.  ( Base `  K
) ) )  -> 
( c  .\/  (
( G  .\/  H
)  .\/  I )
)  =  ( ( c  .\/  ( G 
.\/  H ) ) 
.\/  ( c  .\/  I ) ) )
7816, 18, 65, 76, 77syl13anc 1186 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  (
( G  .\/  H
)  .\/  I )
)  =  ( ( c  .\/  ( G 
.\/  H ) ) 
.\/  ( c  .\/  I ) ) )
7974, 78breqtrrd 4238 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( c  .\/  (
( G  .\/  H
)  .\/  I )
) )
808, 79syl5eqbr 4245 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  .<_  ( c  .\/  ( ( G  .\/  H )  .\/  I ) ) )
81 breq2 4216 . . . . . 6  |-  ( ( c  .\/  ( ( G  .\/  H ) 
.\/  I ) )  =  ( ( G 
.\/  H )  .\/  I )  ->  ( Y  .<_  ( c  .\/  ( ( G  .\/  H )  .\/  I ) )  <->  Y  .<_  ( ( G  .\/  H ) 
.\/  I ) ) )
8280, 81syl5ibcom 212 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  ( ( G  .\/  H )  .\/  I ) )  =  ( ( G  .\/  H ) 
.\/  I )  ->  Y  .<_  ( ( G 
.\/  H )  .\/  I ) ) )
8324, 82sylbid 207 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  ( ( G  .\/  H ) 
.\/  I )  ->  Y  .<_  ( ( G 
.\/  H )  .\/  I ) ) )
841dalemyeo 30429 . . . . . 6  |-  ( ph  ->  Y  e.  O )
85843ad2ant1 978 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  O )
862, 7lplncmp 30359 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  O  /\  ( ( G  .\/  H )  .\/  I )  e.  O )  -> 
( Y  .<_  ( ( G  .\/  H ) 
.\/  I )  <->  Y  =  ( ( G  .\/  H )  .\/  I ) ) )
8727, 85, 19, 86syl3anc 1184 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( Y  .<_  ( ( G  .\/  H ) 
.\/  I )  <->  Y  =  ( ( G  .\/  H )  .\/  I ) ) )
8883, 87sylibd 206 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  ( ( G  .\/  H ) 
.\/  I )  ->  Y  =  ( ( G  .\/  H )  .\/  I ) ) )
8988necon3ad 2637 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( Y  =/=  (
( G  .\/  H
)  .\/  I )  ->  -.  c  .<_  ( ( G  .\/  H ) 
.\/  I ) ) )
9014, 89mpd 15 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( ( G  .\/  H ) 
.\/  I ) )
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   HLchlt 30148   LPlanesclpl 30289
This theorem is referenced by:  dalem45  30514
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  df-lvols 30297
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