Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem56 Structured version   Unicode version

Theorem dalem56 33094
Description: Lemma for dath 33102. Analog of dalem55 33093 for line  S T. (Contributed by NM, 8-Aug-2012.)
Hypotheses
Ref Expression
dalem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem54.m  |-  ./\  =  ( meet `  K )
dalem54.o  |-  O  =  ( LPlanes `  K )
dalem54.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem54.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem54.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem54.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem54.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
dalem54.b1  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
Assertion
Ref Expression
dalem56  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  =  ( ( G  .\/  H )  ./\  B )
)

Proof of Theorem dalem56
StepHypRef Expression
1 dalem.ph . . . . 5  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
2 dalem.l . . . . 5  |-  .<_  =  ( le `  K )
3 dalem.j . . . . 5  |-  .\/  =  ( join `  K )
4 dalem.a . . . . 5  |-  A  =  ( Atoms `  K )
51, 2, 3, 4dalemswapyz 33022 . . . 4  |-  ( ph  ->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) ) )
653ad2ant1 1004 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) ) )
7 simp2 984 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  =  Z )
87eqcomd 2446 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Z  =  Y )
9 dalem.ps . . . 4  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
101, 2, 3, 4, 9dalemswapyzps 33056 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( d  e.  A  /\  c  e.  A )  /\  -.  d  .<_  Z  /\  (
c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) ) )
11 biid 236 . . . 4  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) )  <-> 
( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) ) )
12 biid 236 . . . 4  |-  ( ( ( d  e.  A  /\  c  e.  A
)  /\  -.  d  .<_  Z  /\  ( c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c
) ) )  <->  ( (
d  e.  A  /\  c  e.  A )  /\  -.  d  .<_  Z  /\  ( c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) ) )
13 dalem54.m . . . 4  |-  ./\  =  ( meet `  K )
14 dalem54.o . . . 4  |-  O  =  ( LPlanes `  K )
15 dalem54.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
16 dalem54.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
17 eqid 2441 . . . 4  |-  ( ( d  .\/  S ) 
./\  ( c  .\/  P ) )  =  ( ( d  .\/  S
)  ./\  ( c  .\/  P ) )
18 eqid 2441 . . . 4  |-  ( ( d  .\/  T ) 
./\  ( c  .\/  Q ) )  =  ( ( d  .\/  T
)  ./\  ( c  .\/  Q ) )
19 eqid 2441 . . . 4  |-  ( ( d  .\/  U ) 
./\  ( c  .\/  R ) )  =  ( ( d  .\/  U
)  ./\  ( c  .\/  R ) )
20 eqid 2441 . . . 4  |-  ( ( ( ( ( d 
.\/  S )  ./\  ( c  .\/  P
) )  .\/  (
( d  .\/  T
)  ./\  ( c  .\/  Q ) ) ) 
.\/  ( ( d 
.\/  U )  ./\  ( c  .\/  R
) ) )  ./\  Z )  =  ( ( ( ( ( d 
.\/  S )  ./\  ( c  .\/  P
) )  .\/  (
( d  .\/  T
)  ./\  ( c  .\/  Q ) ) ) 
.\/  ( ( d 
.\/  U )  ./\  ( c  .\/  R
) ) )  ./\  Z )
2111, 2, 3, 4, 12, 13, 14, 15, 16, 17, 18, 19, 20dalem55 33093 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) )  /\  Z  =  Y  /\  ( ( d  e.  A  /\  c  e.  A )  /\  -.  d  .<_  Z  /\  (
c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) ) )  ->  ( (
( ( d  .\/  S )  ./\  ( c  .\/  P ) )  .\/  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) ) 
./\  ( S  .\/  T ) )  =  ( ( ( ( d 
.\/  S )  ./\  ( c  .\/  P
) )  .\/  (
( d  .\/  T
)  ./\  ( c  .\/  Q ) ) ) 
./\  ( ( ( ( ( d  .\/  S )  ./\  ( c  .\/  P ) )  .\/  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) ) 
.\/  ( ( d 
.\/  U )  ./\  ( c  .\/  R
) ) )  ./\  Z ) ) )
226, 8, 10, 21syl3anc 1213 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( ( d  .\/  S ) 
./\  ( c  .\/  P ) )  .\/  (
( d  .\/  T
)  ./\  ( c  .\/  Q ) ) ) 
./\  ( S  .\/  T ) )  =  ( ( ( ( d 
.\/  S )  ./\  ( c  .\/  P
) )  .\/  (
( d  .\/  T
)  ./\  ( c  .\/  Q ) ) ) 
./\  ( ( ( ( ( d  .\/  S )  ./\  ( c  .\/  P ) )  .\/  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) ) 
.\/  ( ( d 
.\/  U )  ./\  ( c  .\/  R
) ) )  ./\  Z ) ) )
23 dalem54.g . . . . 5  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
241dalemkelat 32990 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
25243ad2ant1 1004 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
261dalemkehl 32989 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
27263ad2ant1 1004 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
289dalemccea 33049 . . . . . . . 8  |-  ( ps 
->  c  e.  A
)
29283ad2ant3 1006 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
301dalempea 32992 . . . . . . . 8  |-  ( ph  ->  P  e.  A )
31303ad2ant1 1004 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
32 eqid 2441 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
3332, 3, 4hlatjcl 32733 . . . . . . 7  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  e.  ( Base `  K ) )
3427, 29, 31, 33syl3anc 1213 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( Base `  K ) )
359dalemddea 33050 . . . . . . . 8  |-  ( ps 
->  d  e.  A
)
36353ad2ant3 1006 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
371dalemsea 32995 . . . . . . . 8  |-  ( ph  ->  S  e.  A )
38373ad2ant1 1004 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
3932, 3, 4hlatjcl 32733 . . . . . . 7  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  e.  ( Base `  K ) )
4027, 36, 38, 39syl3anc 1213 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( Base `  K ) )
4132, 13latmcom 15241 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( c  .\/  P
)  e.  ( Base `  K )  /\  (
d  .\/  S )  e.  ( Base `  K
) )  ->  (
( c  .\/  P
)  ./\  ( d  .\/  S ) )  =  ( ( d  .\/  S )  ./\  ( c  .\/  P ) ) )
4225, 34, 40, 41syl3anc 1213 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  =  ( ( d  .\/  S )  ./\  ( c  .\/  P ) ) )
4323, 42syl5eq 2485 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  =  ( (
d  .\/  S )  ./\  ( c  .\/  P
) ) )
44 dalem54.h . . . . 5  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
451dalemqea 32993 . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
46453ad2ant1 1004 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  e.  A )
4732, 3, 4hlatjcl 32733 . . . . . . 7  |-  ( ( K  e.  HL  /\  c  e.  A  /\  Q  e.  A )  ->  ( c  .\/  Q
)  e.  ( Base `  K ) )
4827, 29, 46, 47syl3anc 1213 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  Q
)  e.  ( Base `  K ) )
491dalemtea 32996 . . . . . . . 8  |-  ( ph  ->  T  e.  A )
50493ad2ant1 1004 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  T  e.  A )
5132, 3, 4hlatjcl 32733 . . . . . . 7  |-  ( ( K  e.  HL  /\  d  e.  A  /\  T  e.  A )  ->  ( d  .\/  T
)  e.  ( Base `  K ) )
5227, 36, 50, 51syl3anc 1213 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  T
)  e.  ( Base `  K ) )
5332, 13latmcom 15241 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( c  .\/  Q
)  e.  ( Base `  K )  /\  (
d  .\/  T )  e.  ( Base `  K
) )  ->  (
( c  .\/  Q
)  ./\  ( d  .\/  T ) )  =  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) )
5425, 48, 52, 53syl3anc 1213 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  Q )  ./\  ( d  .\/  T ) )  =  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) )
5544, 54syl5eq 2485 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  =  ( (
d  .\/  T )  ./\  ( c  .\/  Q
) ) )
5643, 55oveq12d 6108 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  =  ( ( ( d  .\/  S
)  ./\  ( c  .\/  P ) )  .\/  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) ) )
5756oveq1d 6105 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  =  ( ( ( ( d  .\/  S ) 
./\  ( c  .\/  P ) )  .\/  (
( d  .\/  T
)  ./\  ( c  .\/  Q ) ) ) 
./\  ( S  .\/  T ) ) )
58 dalem54.b1 . . . 4  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
59 dalem54.i . . . . . . 7  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
601dalemrea 32994 . . . . . . . . . 10  |-  ( ph  ->  R  e.  A )
61603ad2ant1 1004 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  e.  A )
6232, 3, 4hlatjcl 32733 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  c  e.  A  /\  R  e.  A )  ->  ( c  .\/  R
)  e.  ( Base `  K ) )
6327, 29, 61, 62syl3anc 1213 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  R
)  e.  ( Base `  K ) )
641dalemuea 32997 . . . . . . . . . 10  |-  ( ph  ->  U  e.  A )
65643ad2ant1 1004 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  U  e.  A )
6632, 3, 4hlatjcl 32733 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  d  e.  A  /\  U  e.  A )  ->  ( d  .\/  U
)  e.  ( Base `  K ) )
6727, 36, 65, 66syl3anc 1213 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  U
)  e.  ( Base `  K ) )
6832, 13latmcom 15241 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( c  .\/  R
)  e.  ( Base `  K )  /\  (
d  .\/  U )  e.  ( Base `  K
) )  ->  (
( c  .\/  R
)  ./\  ( d  .\/  U ) )  =  ( ( d  .\/  U )  ./\  ( c  .\/  R ) ) )
6925, 63, 67, 68syl3anc 1213 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  R )  ./\  ( d  .\/  U ) )  =  ( ( d  .\/  U )  ./\  ( c  .\/  R ) ) )
7059, 69syl5eq 2485 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  =  ( (
d  .\/  U )  ./\  ( c  .\/  R
) ) )
7156, 70oveq12d 6108 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  =  ( ( ( ( d  .\/  S
)  ./\  ( c  .\/  P ) )  .\/  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) ) 
.\/  ( ( d 
.\/  U )  ./\  ( c  .\/  R
) ) ) )
7271, 7oveq12d 6108 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)  =  ( ( ( ( ( d 
.\/  S )  ./\  ( c  .\/  P
) )  .\/  (
( d  .\/  T
)  ./\  ( c  .\/  Q ) ) ) 
.\/  ( ( d 
.\/  U )  ./\  ( c  .\/  R
) ) )  ./\  Z ) )
7358, 72syl5eq 2485 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  =  ( (
( ( ( d 
.\/  S )  ./\  ( c  .\/  P
) )  .\/  (
( d  .\/  T
)  ./\  ( c  .\/  Q ) ) ) 
.\/  ( ( d 
.\/  U )  ./\  ( c  .\/  R
) ) )  ./\  Z ) )
7456, 73oveq12d 6108 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  =  ( ( ( ( d  .\/  S
)  ./\  ( c  .\/  P ) )  .\/  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) ) 
./\  ( ( ( ( ( d  .\/  S )  ./\  ( c  .\/  P ) )  .\/  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) ) 
.\/  ( ( d 
.\/  U )  ./\  ( c  .\/  R
) ) )  ./\  Z ) ) )
7522, 57, 743eqtr4d 2483 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  =  ( ( G  .\/  H )  ./\  B )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   Basecbs 14170   lecple 14241   joincjn 15110   meetcmee 15111   Latclat 15211   Atomscatm 32630   HLchlt 32717   LPlanesclpl 32858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-lat 15212  df-clat 15274  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718  df-llines 32864  df-lplanes 32865  df-lvols 32866
This theorem is referenced by:  dalem57  33095
  Copyright terms: Public domain W3C validator