Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg28b Structured version   Unicode version

Theorem cdlemg28b 34239
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 34240.) (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 1035 . 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 1037 . 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 1039 . 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 1126 . 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 1127 . 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 1005 . . 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 1124 . . 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 1038 . . 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 1152 . . . 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 534 . . 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 1130 . . 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 1154 . . 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 1132 . . 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 34232 . . 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 1287 . 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 1153 . . . 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 534 . . 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 1131 . . 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 1133 . . 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 34232 . . 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 1287 . 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 34227 . 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 1287 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423   ` cfv 5601  (class class class)co 6305   lecple 15196   joincjn 16188   meetcmee 16189   Atomscatm 32798   HLchlt 32885   LHypclh 33518   LTrncltrn 33635   trLctrl 33693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-riotaBAD 32494
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-undef 7031  df-map 7485  df-preset 16172  df-poset 16190  df-plt 16203  df-lub 16219  df-glb 16220  df-join 16221  df-meet 16222  df-p0 16284  df-p1 16285  df-lat 16291  df-clat 16353  df-oposet 32711  df-ol 32713  df-oml 32714  df-covers 32801  df-ats 32802  df-atl 32833  df-cvlat 32857  df-hlat 32886  df-llines 33032  df-lplanes 33033  df-lvols 33034  df-lines 33035  df-psubsp 33037  df-pmap 33038  df-padd 33330  df-lhyp 33522  df-laut 33523  df-ldil 33638  df-ltrn 33639  df-trl 33694
This theorem is referenced by:  cdlemg28  34240
  Copyright terms: Public domain W3C validator