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Theorem cdlemg18c 34329
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 1012 . 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 1105 . 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 1013 . . 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 1021 . . 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 1107 . . 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 1024 . . 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 33860 . . 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 1228 . 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 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 ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
16 simp23 1023 . . 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 33789 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
1915, 16, 5, 18syl3anc 1218 . 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 33789 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
2115, 16, 2, 20syl3anc 1218 . 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 34328 . . 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 1025 . . . 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 2700 . . 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 532 . 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 1026 . . 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 34327 . . 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 1236 . 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 33025 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
311, 2, 5, 30syl3anc 1218 . . . 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 33863 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )
331, 3, 4, 5, 32syl22anc 1219 . . . 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 4323 . . 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 33025 . . . . 5  |-  ( ( K  e.  HL  /\  ( F `  Q )  e.  A  /\  ( F `  P )  e.  A )  ->  ( F `  P )  .<_  ( ( F `  Q )  .\/  ( F `  P )
) )
361, 19, 21, 35syl3anc 1218 . . . 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 1022 . . . . . 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 34251 . . . . . 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 1223 . . . . 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 33017 . . . . . 6  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  ( F `  Q )  e.  A )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  Q )  .\/  ( F `  P )
) )
411, 21, 19, 40syl3anc 1218 . . . . 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 33017 . . . . . 6  |-  ( ( K  e.  HL  /\  ( F `  Q )  e.  A  /\  U  e.  A )  ->  (
( F `  Q
)  .\/  U )  =  ( U  .\/  ( F `  Q ) ) )
431, 19, 14, 42syl3anc 1218 . . . . 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 2483 . . . 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 4321 . . 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 532 . 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 33177 . 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 1250 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   lecple 14250   joincjn 15119   meetcmee 15120   Atomscatm 32913   HLchlt 33000   LHypclh 33633   LTrncltrn 33750   trLctrl 33807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-riotaBAD 32609
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-undef 6797  df-map 7221  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-clat 15283  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001  df-llines 33147  df-lplanes 33148  df-lvols 33149  df-lines 33150  df-psubsp 33152  df-pmap 33153  df-padd 33445  df-lhyp 33637  df-laut 33638  df-ldil 33753  df-ltrn 33754  df-trl 33808
This theorem is referenced by:  cdlemg18d  34330
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