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Theorem cdlemg18c 31539
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 982 . 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 1075 . 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 983 . . 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 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  e.  A  /\  -.  P  .<_  W ) )
5 simp22l 1077 . . 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 994 . . 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 31070 . . 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 1195 . 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 958 . . 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 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  e.  T )
17 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
187, 10, 11, 17ltrnat 30999 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
1915, 16, 5, 18syl3anc 1185 . 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 30999 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
2115, 16, 2, 20syl3anc 1185 . 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 31538 . . 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 995 . . . 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 2689 . . 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 520 . 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 996 . . 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 31537 . . 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 1203 . 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 30235 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
311, 2, 5, 30syl3anc 1185 . . . 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 31073 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )
331, 3, 4, 5, 32syl22anc 1186 . . . 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 4240 . . 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 30235 . . . . 5  |-  ( ( K  e.  HL  /\  ( F `  Q )  e.  A  /\  ( F `  P )  e.  A )  ->  ( F `  P )  .<_  ( ( F `  Q )  .\/  ( F `  P )
) )
361, 19, 21, 35syl3anc 1185 . . . 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 992 . . . . . 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 31461 . . . . . 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 1190 . . . . 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 30227 . . . . . 6  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  ( F `  Q )  e.  A )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  Q )  .\/  ( F `  P )
) )
411, 21, 19, 40syl3anc 1185 . . . . 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 30227 . . . . . 6  |-  ( ( K  e.  HL  /\  ( F `  Q )  e.  A  /\  U  e.  A )  ->  (
( F `  Q
)  .\/  U )  =  ( U  .\/  ( F `  Q ) ) )
431, 19, 14, 42syl3anc 1185 . . . . 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 2478 . . . 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 4238 . . 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 520 . 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 30387 . 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 1217 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30123   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   trLctrl 31017
This theorem is referenced by:  cdlemg18d  31540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359  df-lines 30360  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018
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