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Theorem cdlemk26b-3 29998
Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 14-Jul-2013.)
Hypotheses
Ref Expression
cdlemk3.b  |-  B  =  ( Base `  K
)
cdlemk3.l  |-  .<_  =  ( le `  K )
cdlemk3.j  |-  .\/  =  ( join `  K )
cdlemk3.m  |-  ./\  =  ( meet `  K )
cdlemk3.a  |-  A  =  ( Atoms `  K )
cdlemk3.h  |-  H  =  ( LHyp `  K
)
cdlemk3.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk3.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk3.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk3.u1  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
Assertion
Ref Expression
cdlemk26b-3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. x  e.  T  ( (
x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F
)  /\  ( R `  x )  =/=  ( R `  G )
)  /\  ( x Y G )  e.  T
) )
Distinct variable groups:    e, d,
f, i,  ./\    .<_ , i    .\/ , d, e, f, i    A, i    j, d, e, f, i, F    G, d,
e, j    i, H    i, K    f, N, i    P, d, e, f, i    R, d, e, f, i    T, d, e, f, i    W, d, e, f, i    ./\ , j    .<_ , j    .\/ , j    A, j    j, F    j, H    j, K    j, N    P, j    R, j    S, d, e, j    T, j   
j, W    F, d,
e    .<_ , e    f, G, i    x, d, e, f, i, j    x,  .<_    x, A    x, B    x, F    x, G    x, H    x, K    x, N    x, P    x, R    x, T    x, Y    x, W
Allowed substitution hints:    A( e, f, d)    B( e, f, i, j, d)    S( x, f, i)    H( e, f, d)    .\/ ( x)    K( e, f, d)    .<_ ( f, d)    ./\ (
x)    N( e, d)    Y( e, f, i, j, d)

Proof of Theorem cdlemk26b-3
StepHypRef Expression
1 simpl1 963 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 cdlemk3.b . . . 4  |-  B  =  ( Base `  K
)
3 cdlemk3.h . . . 4  |-  H  =  ( LHyp `  K
)
4 cdlemk3.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 cdlemk3.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
62, 3, 4, 5cdlemftr2 29659 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. x  e.  T  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x )  =/=  ( R `  G
) ) )
71, 6syl 17 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. x  e.  T  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) )
8 simp3r 989 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  (
x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F
)  /\  ( R `  x )  =/=  ( R `  G )
) )
9 simp11 990 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
10 simp133 1097 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
11 simp131 1095 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  G  e.  T )
12 simp121 1092 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  F  e.  T )
13 simp3l 988 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  x  e.  T )
14 simp123 1094 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  N  e.  T )
15 simp3r2 1069 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  x )  =/=  ( R `  F
) )
16 simp3r3 1070 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  x )  =/=  ( R `  G
) )
1715, 16jca 520 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  (
( R `  x
)  =/=  ( R `
 F )  /\  ( R `  x )  =/=  ( R `  G ) ) )
18 simp122 1093 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  F  =/=  (  _I  |`  B ) )
19 simp132 1096 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  G  =/=  (  _I  |`  B ) )
20 simp3r1 1068 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  x  =/=  (  _I  |`  B ) )
2118, 19, 203jca 1137 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )
22 simp2 961 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
23 cdlemk3.l . . . . . . . 8  |-  .<_  =  ( le `  K )
24 cdlemk3.j . . . . . . . 8  |-  .\/  =  ( join `  K )
25 cdlemk3.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
26 cdlemk3.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
27 cdlemk3.s . . . . . . . 8  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
28 cdlemk3.u1 . . . . . . . 8  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
292, 23, 24, 25, 26, 3, 4, 5, 27, 28cdlemkuel-3 29991 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  x  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 x )  =/=  ( R `  F
)  /\  ( R `  x )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( x Y G )  e.  T
)
309, 10, 11, 12, 13, 14, 17, 21, 22, 29syl333anc 1219 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  (
x Y G )  e.  T )
318, 30jca 520 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) ) ) )  ->  (
( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x )  =/=  ( R `  G
) )  /\  (
x Y G )  e.  T ) )
32313expia 1158 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( (
x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x )  =/=  ( R `  G
) ) )  -> 
( ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G ) )  /\  ( x Y G )  e.  T
) ) )
3332exp3a 427 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( x  e.  T  ->  ( ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F
)  /\  ( R `  x )  =/=  ( R `  G )
)  ->  ( (
x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F
)  /\  ( R `  x )  =/=  ( R `  G )
)  /\  ( x Y G )  e.  T
) ) ) )
3433reximdvai 2615 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( E. x  e.  T  (
x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F
)  /\  ( R `  x )  =/=  ( R `  G )
)  ->  E. x  e.  T  ( (
x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F
)  /\  ( R `  x )  =/=  ( R `  G )
)  /\  ( x Y G )  e.  T
) ) )
357, 34mpd 16 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. x  e.  T  ( (
x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F
)  /\  ( R `  x )  =/=  ( R `  G )
)  /\  ( x Y G )  e.  T
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510   class class class wbr 3920    e. cmpt 3974    _I cid 4197   `'ccnv 4579    |` cres 4582    o. ccom 4584   ` cfv 4592  (class class class)co 5710    e. cmpt2 5712   iota_crio 6181   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28357   HLchlt 28444   LHypclh 29077   LTrncltrn 29194   trLctrl 29251
This theorem is referenced by:  cdlemk28-3  30001
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-llines 28591  df-lplanes 28592  df-lvols 28593  df-lines 28594  df-psubsp 28596  df-pmap 28597  df-padd 28889  df-lhyp 29081  df-laut 29082  df-ldil 29197  df-ltrn 29198  df-trl 29252
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