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Theorem cdlemg28b 30043
Description: Part of proof of Lemma G of [Crawley] p. 116. Second equality of the equation of line 14 on p. 117. Note that  -.  z  .<_  W is redundant here (but simplifies cdlemg28 30044.) (Contributed by NM, 29-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg31.n  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
cdlemg33.o  |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  G ) ) )
Assertion
Ref Expression
cdlemg28b  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
)  =  ( ( z  .\/  ( F `
 ( G `  z ) ) ) 
./\  W ) )
Distinct variable groups:    z, A    z, F    z, H    z,  .\/    z, K    z,  .<_    z, N    z, P    z, Q    z, R    z, T    z, W    z, v    z, G   
z, O
Allowed substitution hints:    A( v)    P( v)    Q( v)    R( v)    T( v)    F( v)    G( v)    H( v)    .\/ ( v)    K( v)   
.<_ ( v)    ./\ ( z, v)    N( v)    O( v)    W( v)

Proof of Theorem cdlemg28b
StepHypRef Expression
1 simp11 990 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp13 992 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3 simp22 994 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( z  e.  A  /\  -.  z  .<_  W ) )
4 simp23l 1081 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  F  e.  T
)
5 simp23r 1082 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  G  e.  T
)
6 simp1 960 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
7 simp22l 1079 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  z  e.  A
)
8 simp21 993 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( v  e.  A  /\  v  .<_  W ) )
9 simp311 1107 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  z  =/=  N
)
104, 9jca 520 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( F  e.  T  /\  z  =/= 
N ) )
11 simp32l 1085 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  v  =/=  ( R `  F )
)
12 simp313 1109 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  z  .<_  ( P 
.\/  v ) )
13 simp33l 1087 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( F `  P )  =/=  P
)
14 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
15 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
16 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
17 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
18 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
19 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
20 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
21 cdlemg31.n . . . 4  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
2214, 15, 16, 17, 18, 19, 20, 21cdlemg27b 30036 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  -.  ( R `  F )  .<_  ( Q 
.\/  z ) )
236, 7, 8, 10, 11, 12, 13, 22syl133anc 1210 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  -.  ( R `  F )  .<_  ( Q 
.\/  z ) )
24 simp312 1108 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  z  =/=  O
)
255, 24jca 520 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( G  e.  T  /\  z  =/= 
O ) )
26 simp32r 1086 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  v  =/=  ( R `  G )
)
27 simp33r 1088 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( G `  P )  =/=  P
)
28 cdlemg33.o . . . 4  |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  G ) ) )
2914, 15, 16, 17, 18, 19, 20, 28cdlemg27b 30036 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( G  e.  T  /\  z  =/= 
O ) )  /\  ( v  =/=  ( R `  G )  /\  z  .<_  ( P 
.\/  v )  /\  ( G `  P )  =/=  P ) )  ->  -.  ( R `  G )  .<_  ( Q 
.\/  z ) )
306, 7, 8, 25, 26, 12, 27, 29syl133anc 1210 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  -.  ( R `  G )  .<_  ( Q 
.\/  z ) )
3114, 15, 16, 17, 18, 19, 20cdlemg26zz 30031 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  ( R `  F )  .<_  ( Q  .\/  z
)  /\  -.  ( R `  G )  .<_  ( Q  .\/  z
) ) )  -> 
( ( Q  .\/  ( F `  ( G `
 Q ) ) )  ./\  W )  =  ( ( z 
.\/  ( F `  ( G `  z ) ) )  ./\  W
) )
321, 2, 3, 4, 5, 23, 30, 31syl133anc 1210 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
)  =  ( ( z  .\/  ( F `
 ( G `  z ) ) ) 
./\  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   lecple 13163   joincjn 14026   meetcmee 14027   Atomscatm 28604   HLchlt 28691   LHypclh 29324   LTrncltrn 29441   trLctrl 29498
This theorem is referenced by:  cdlemg28  30044
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 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-map 6728  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-lines 28841  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499
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