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Theorem cdlemg36 34677
Description: Use cdlemg35 to eliminate  v from cdlemg34 34675. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
Hypotheses
Ref Expression
cdlemg35.l  |-  .<_  =  ( le `  K )
cdlemg35.j  |-  .\/  =  ( join `  K )
cdlemg35.m  |-  ./\  =  ( meet `  K )
cdlemg35.a  |-  A  =  ( Atoms `  K )
cdlemg35.h  |-  H  =  ( LHyp `  K
)
cdlemg35.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg35.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg36  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
Distinct variable groups:    A, r    F, r    G, r    H, r    .\/ , r    K, r    .<_ , r    ./\ , r    P, r    Q, r    R, r    W, r
Allowed substitution hint:    T( r)

Proof of Theorem cdlemg36
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 simp11 1018 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 1019 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp21 1021 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  F  e.  T )
4 simp22 1022 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  G  e.  T )
5 simp31l 1111 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( F `  P )  =/=  P
)
6 simp31r 1112 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( G `  P )  =/=  P
)
7 simp32 1025 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( R `  F )  =/=  ( R `  G )
)
8 cdlemg35.l . . . 4  |-  .<_  =  ( le `  K )
9 cdlemg35.j . . . 4  |-  .\/  =  ( join `  K )
10 cdlemg35.m . . . 4  |-  ./\  =  ( meet `  K )
11 cdlemg35.a . . . 4  |-  A  =  ( Atoms `  K )
12 cdlemg35.h . . . 4  |-  H  =  ( LHyp `  K
)
13 cdlemg35.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
14 cdlemg35.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
158, 9, 10, 11, 12, 13, 14cdlemg35 34676 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  E. v  e.  A  ( v  .<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
) ) )
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1242 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  E. v  e.  A  ( v  .<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )
17 simp11 1018 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
18 simp2 989 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  v  e.  A
)
19 simp3l 1016 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  v  .<_  W )
2018, 19jca 532 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  ( v  e.  A  /\  v  .<_  W ) )
21 simp121 1120 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  F  e.  T
)
22 simp122 1121 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  G  e.  T
)
2321, 22jca 532 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  ( F  e.  T  /\  G  e.  T ) )
24 simp123 1122 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  P  =/=  Q
)
25 simp3rl 1061 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  v  =/=  ( R `  F )
)
26 simp3rr 1062 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  v  =/=  ( R `  G )
)
27 simp133 1125 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )
28 eqid 2452 . . . . 5  |-  ( ( P  .\/  v ) 
./\  ( Q  .\/  ( R `  F ) ) )  =  ( ( P  .\/  v
)  ./\  ( Q  .\/  ( R `  F
) ) )
29 eqid 2452 . . . . 5  |-  ( ( P  .\/  v ) 
./\  ( Q  .\/  ( R `  G ) ) )  =  ( ( P  .\/  v
)  ./\  ( Q  .\/  ( R `  G
) ) )
308, 9, 10, 11, 12, 13, 14, 28, 29cdlemg34 34675 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
3117, 20, 23, 24, 25, 26, 27, 30syl133anc 1242 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  v  e.  A  /\  ( v 
.<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G
) ) ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
3231rexlimdv3a 2943 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( E. v  e.  A  (
v  .<_  W  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G ) ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) ) )
3316, 32mpd 15 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   E.wrex 2797   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   lecple 14359   joincjn 15228   meetcmee 15229   Atomscatm 33227   HLchlt 33314   LHypclh 33947   LTrncltrn 34064   trLctrl 34121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-riotaBAD 32923
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-undef 6897  df-map 7321  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-lplanes 33462  df-lvols 33463  df-lines 33464  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122
This theorem is referenced by:  cdlemg38  34678
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