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Theorem cdlemk26b-3 31764
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 961 . . 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 31425 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. x  e.  T  ( x  =/=  (  _I  |`  B )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  x )  =/=  ( R `  G
) ) )
71, 6syl 16 . 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 987 . . . . . 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 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 ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
10 simp133 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 ) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
11 simp131 1093 . . . . . . 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 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 ) ) ) )  ->  F  e.  T )
13 simp3l 986 . . . . . . 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 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 ) ) ) )  ->  N  e.  T )
15 simp3r2 1067 . . . . . . . 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 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 ) ) ) )  ->  ( 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 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 ) ) ) )  ->  F  =/=  (  _I  |`  B ) )
19 simp132 1094 . . . . . . . 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 1066 . . . . . . . 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 1135 . . . . . . 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 959 . . . . . . 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 31757 . . . . . . 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 1217 . . . . . 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 1156 . . . 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 2818 . 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 15 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4214    e. cmpt 4268    _I cid 4495   `'ccnv 4879    |` cres 4882    o. ccom 4884   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   iota_crio 6544   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30123   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   trLctrl 31017
This theorem is referenced by:  cdlemk28-3  31767
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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