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Theorem dalem56 30210
Description: Lemma for dath 30218. Analog of dalem55 30209 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 30138 . . . 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 978 . . 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 958 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  =  Z )
87eqcomd 2409 . . 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 30172 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( d  e.  A  /\  c  e.  A )  /\  -.  d  .<_  Z  /\  (
c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) ) )
11 biid 228 . . . 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 228 . . . 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 2404 . . . 4  |-  ( ( d  .\/  S ) 
./\  ( c  .\/  P ) )  =  ( ( d  .\/  S
)  ./\  ( c  .\/  P ) )
18 eqid 2404 . . . 4  |-  ( ( d  .\/  T ) 
./\  ( c  .\/  Q ) )  =  ( ( d  .\/  T
)  ./\  ( c  .\/  Q ) )
19 eqid 2404 . . . 4  |-  ( ( d  .\/  U ) 
./\  ( c  .\/  R ) )  =  ( ( d  .\/  U
)  ./\  ( c  .\/  R ) )
20 eqid 2404 . . . 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 30209 . . 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 1184 . 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 30106 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
25243ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
261dalemkehl 30105 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
27263ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
289dalemccea 30165 . . . . . . . 8  |-  ( ps 
->  c  e.  A
)
29283ad2ant3 980 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
301dalempea 30108 . . . . . . . 8  |-  ( ph  ->  P  e.  A )
31303ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
32 eqid 2404 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
3332, 3, 4hlatjcl 29849 . . . . . . 7  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  e.  ( Base `  K ) )
3427, 29, 31, 33syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( Base `  K ) )
359dalemddea 30166 . . . . . . . 8  |-  ( ps 
->  d  e.  A
)
36353ad2ant3 980 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
371dalemsea 30111 . . . . . . . 8  |-  ( ph  ->  S  e.  A )
38373ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
3932, 3, 4hlatjcl 29849 . . . . . . 7  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  e.  ( Base `  K ) )
4027, 36, 38, 39syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( Base `  K ) )
4132, 13latmcom 14459 . . . . . 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 1184 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  =  ( ( d  .\/  S )  ./\  ( c  .\/  P ) ) )
4323, 42syl5eq 2448 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  =  ( (
d  .\/  S )  ./\  ( c  .\/  P
) ) )
44 dalem54.h . . . . 5  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
451dalemqea 30109 . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
46453ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  e.  A )
4732, 3, 4hlatjcl 29849 . . . . . . 7  |-  ( ( K  e.  HL  /\  c  e.  A  /\  Q  e.  A )  ->  ( c  .\/  Q
)  e.  ( Base `  K ) )
4827, 29, 46, 47syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  Q
)  e.  ( Base `  K ) )
491dalemtea 30112 . . . . . . . 8  |-  ( ph  ->  T  e.  A )
50493ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  T  e.  A )
5132, 3, 4hlatjcl 29849 . . . . . . 7  |-  ( ( K  e.  HL  /\  d  e.  A  /\  T  e.  A )  ->  ( d  .\/  T
)  e.  ( Base `  K ) )
5227, 36, 50, 51syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  T
)  e.  ( Base `  K ) )
5332, 13latmcom 14459 . . . . . 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 1184 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  Q )  ./\  ( d  .\/  T ) )  =  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) )
5544, 54syl5eq 2448 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  =  ( (
d  .\/  T )  ./\  ( c  .\/  Q
) ) )
5643, 55oveq12d 6058 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  =  ( ( ( d  .\/  S
)  ./\  ( c  .\/  P ) )  .\/  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) ) )
5756oveq1d 6055 . 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 30110 . . . . . . . . . 10  |-  ( ph  ->  R  e.  A )
61603ad2ant1 978 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  e.  A )
6232, 3, 4hlatjcl 29849 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  c  e.  A  /\  R  e.  A )  ->  ( c  .\/  R
)  e.  ( Base `  K ) )
6327, 29, 61, 62syl3anc 1184 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  R
)  e.  ( Base `  K ) )
641dalemuea 30113 . . . . . . . . . 10  |-  ( ph  ->  U  e.  A )
65643ad2ant1 978 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  U  e.  A )
6632, 3, 4hlatjcl 29849 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  d  e.  A  /\  U  e.  A )  ->  ( d  .\/  U
)  e.  ( Base `  K ) )
6727, 36, 65, 66syl3anc 1184 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  U
)  e.  ( Base `  K ) )
6832, 13latmcom 14459 . . . . . . . 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 1184 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  R )  ./\  ( d  .\/  U ) )  =  ( ( d  .\/  U )  ./\  ( c  .\/  R ) ) )
7059, 69syl5eq 2448 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  =  ( (
d  .\/  U )  ./\  ( c  .\/  R
) ) )
7156, 70oveq12d 6058 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  =  ( ( ( ( d  .\/  S
)  ./\  ( c  .\/  P ) )  .\/  ( ( d  .\/  T )  ./\  ( c  .\/  Q ) ) ) 
.\/  ( ( d 
.\/  U )  ./\  ( c  .\/  R
) ) ) )
7271, 7oveq12d 6058 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)  =  ( ( ( ( ( d 
.\/  S )  ./\  ( c  .\/  P
) )  .\/  (
( d  .\/  T
)  ./\  ( c  .\/  Q ) ) ) 
.\/  ( ( d 
.\/  U )  ./\  ( c  .\/  R
) ) )  ./\  Z ) )
7358, 72syl5eq 2448 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  =  ( (
( ( ( d 
.\/  S )  ./\  ( c  .\/  P
) )  .\/  (
( d  .\/  T
)  ./\  ( c  .\/  Q ) ) ) 
.\/  ( ( d 
.\/  U )  ./\  ( c  .\/  R
) ) )  ./\  Z ) )
7456, 73oveq12d 6058 . 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 2446 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  =  ( ( G  .\/  H )  ./\  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   Atomscatm 29746   HLchlt 29833   LPlanesclpl 29974
This theorem is referenced by:  dalem57  30211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982
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