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Theorem cdlemk26b-3 31716
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 31377 . . 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 31709 . . . . . . 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 2666 . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039    e. cmpt 4093    _I cid 4320   `'ccnv 4704    |` cres 4707    o. ccom 4709   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   iota_crio 6313   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969
This theorem is referenced by:  cdlemk28-3  31719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970
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