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Theorem cdlemg18c 30869
Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg18b.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdlemg18c  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( F `  Q ) )  ./\  ( Q  .\/  ( F `
 P ) ) )  e.  A )

Proof of Theorem cdlemg18c
StepHypRef Expression
1 simp1l 979 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  K  e.  HL )
2 simp21l 1072 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  P  e.  A )
3 simp1r 980 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  W  e.  H )
4 simp21 988 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5 simp22l 1074 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  Q  e.  A )
6 simp31 991 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  P  =/=  Q )
7 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
8 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
9 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
10 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
11 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
12 cdlemg18b.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
137, 8, 9, 10, 11, 12cdleme0a 30400 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
141, 3, 4, 5, 6, 13syl212anc 1192 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  U  e.  A )
15 simp1 955 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
16 simp23 990 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  F  e.  T )
17 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
187, 10, 11, 17ltrnat 30329 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
1915, 16, 5, 18syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( F `  Q )  e.  A
)
207, 10, 11, 17ltrnat 30329 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
2115, 16, 2, 20syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( F `  P )  e.  A
)
22 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
237, 8, 9, 10, 11, 17, 22, 12cdlemg18b 30868 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  -.  P  .<_  ( U  .\/  ( F `  Q )
) )
24 simp32 992 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( F `  P )  =/=  Q
)
2524necomd 2529 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  Q  =/=  ( F `  P ) )
2623, 25jca 518 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( -.  P  .<_  ( U  .\/  ( F `  Q ) )  /\  Q  =/=  ( F `  P
) ) )
27 simp33 993 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) )
287, 8, 9, 10, 11, 17, 22cdlemg18a 30867 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  (
( F `  Q
)  .\/  ( F `  P ) )  =/=  ( P  .\/  Q
) ) )  -> 
( P  .\/  ( F `  Q )
)  =/=  ( Q 
.\/  ( F `  P ) ) )
2915, 2, 5, 16, 6, 27, 28syl132anc 1200 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( P  .\/  ( F `  Q
) )  =/=  ( Q  .\/  ( F `  P ) ) )
307, 8, 10hlatlej2 29565 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
311, 2, 5, 30syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  Q  .<_  ( P  .\/  Q ) )
327, 8, 9, 10, 11, 12cdleme0cp 30403 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )
331, 3, 4, 5, 32syl22anc 1183 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )
3431, 33breqtrrd 4049 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  Q  .<_  ( P  .\/  U ) )
357, 8, 10hlatlej2 29565 . . . . 5  |-  ( ( K  e.  HL  /\  ( F `  Q )  e.  A  /\  ( F `  P )  e.  A )  ->  ( F `  P )  .<_  ( ( F `  Q )  .\/  ( F `  P )
) )
361, 19, 21, 35syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( F `  P )  .<_  ( ( F `  Q ) 
.\/  ( F `  P ) ) )
37 simp22 989 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3811, 17, 7, 8, 10, 9, 12cdlemg2kq 30791 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  Q )  .\/  U
) )
3915, 4, 37, 16, 38syl121anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( F `  P )  .\/  ( F `  Q
) )  =  ( ( F `  Q
)  .\/  U )
)
408, 10hlatjcom 29557 . . . . . 6  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  ( F `  Q )  e.  A )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  Q )  .\/  ( F `  P )
) )
411, 21, 19, 40syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( F `  P )  .\/  ( F `  Q
) )  =  ( ( F `  Q
)  .\/  ( F `  P ) ) )
428, 10hlatjcom 29557 . . . . . 6  |-  ( ( K  e.  HL  /\  ( F `  Q )  e.  A  /\  U  e.  A )  ->  (
( F `  Q
)  .\/  U )  =  ( U  .\/  ( F `  Q ) ) )
431, 19, 14, 42syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( F `  Q )  .\/  U )  =  ( U  .\/  ( F `
 Q ) ) )
4439, 41, 433eqtr3d 2323 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( F `  Q )  .\/  ( F `  P
) )  =  ( U  .\/  ( F `
 Q ) ) )
4536, 44breqtrd 4047 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( F `  P )  .<_  ( U 
.\/  ( F `  Q ) ) )
4634, 45jca 518 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( Q  .<_  ( P  .\/  U
)  /\  ( F `  P )  .<_  ( U 
.\/  ( F `  Q ) ) ) )
477, 8, 9, 10ps-2c 29717 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  /\  ( ( F `  Q )  e.  A  /\  Q  e.  A  /\  ( F `  P
)  e.  A )  /\  ( ( -.  P  .<_  ( U  .\/  ( F `  Q
) )  /\  Q  =/=  ( F `  P
) )  /\  ( P  .\/  ( F `  Q ) )  =/=  ( Q  .\/  ( F `  P )
)  /\  ( Q  .<_  ( P  .\/  U
)  /\  ( F `  P )  .<_  ( U 
.\/  ( F `  Q ) ) ) ) )  ->  (
( P  .\/  ( F `  Q )
)  ./\  ( Q  .\/  ( F `  P
) ) )  e.  A )
481, 2, 14, 19, 5, 21, 26, 29, 46, 47syl333anc 1214 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( F `  Q ) )  ./\  ( Q  .\/  ( F `
 P ) ) )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemg18d  30870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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