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Theorem cdlemk26b-3 30361
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 960 . . 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 30022 . . 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 986 . . . . . 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 987 . . . . . . 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 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 ) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
11 simp131 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 ) ) ) )  ->  G  e.  T )
12 simp121 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 ) ) ) )  ->  F  e.  T )
13 simp3l 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 ) ) ) )  ->  x  e.  T )
14 simp123 1091 . . . . . . 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 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 ) ) ) )  ->  ( R `  x )  =/=  ( R `  F
) )
16 simp3r3 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 `  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 1090 . . . . . . . 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 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 ) ) ) )  ->  G  =/=  (  _I  |`  B ) )
20 simp3r1 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 ) ) ) )  ->  x  =/=  (  _I  |`  B ) )
2118, 19, 203jca 1134 . . . . . . 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 958 . . . . . . 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 30354 . . . . . . 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 1216 . . . . . 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 1155 . . . 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 2654 . 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 936    = wceq 1624    e. wcel 1685    =/= wne 2447   E.wrex 2545   class class class wbr 4024    e. cmpt 4078    _I cid 4303   `'ccnv 4687    |` cres 4690    o. ccom 4692   ` cfv 5221  (class class class)co 5819    e. cmpt2 5821   iota_crio 6290   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614
This theorem is referenced by:  cdlemk28-3  30364
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-map 6769  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615
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