Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemk26b-3 Unicode version

Theorem cdlemk26b-3 31094
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 958 . . 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 30755 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. x  e.  T  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x )  =/=  ( R `  G
) ) )
71, 6syl 15 . 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 984 . . . . . 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 985 . . . . . . 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 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 ) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
11 simp131 1090 . . . . . . 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 1087 . . . . . . 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 983 . . . . . . 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 1089 . . . . . . 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 1064 . . . . . . . 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 1065 . . . . . . . 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 518 . . . . . . 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 1088 . . . . . . . 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 1091 . . . . . . . 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 1063 . . . . . . . 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 1132 . . . . . . 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 956 . . . . . . 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 31087 . . . . . . 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 1214 . . . . . 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 518 . . . . 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 1153 . . . 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 425 . . 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 2653 . 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 14 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 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023    e. cmpt 4077    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemk28-3  31097
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
  Copyright terms: Public domain W3C validator