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Theorem dalaw 30757
Description: Desargues' law, derived from Desargues' theorem dath 30607 and with no conditions on the atoms. If triples  <. P ,  Q ,  R >. and  <. S ,  T ,  U >. are centrally perspective, i.e.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ), then they are axially perspective. Theorem 13.3 of [Crawley] p. 110. (Contributed by NM, 7-Oct-2012.)
Hypotheses
Ref Expression
dalaw.l  |-  .<_  =  ( le `  K )
dalaw.j  |-  .\/  =  ( join `  K )
dalaw.m  |-  ./\  =  ( meet `  K )
dalaw.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
dalaw  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )

Proof of Theorem dalaw
StepHypRef Expression
1 dalaw.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
2 dalaw.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
3 dalaw.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
4 dalaw.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
5 eqid 2438 . . . . . . . . 9  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
61, 2, 3, 4, 5dalawlem14 30755 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
763expib 1157 . . . . . . 7  |-  ( ( K  e.  HL  /\  -.  ( ( ( P 
.\/  Q )  .\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( Q 
.\/  R )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  ->  ( (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )
873exp 1153 . . . . . 6  |-  ( K  e.  HL  ->  ( -.  ( ( ( P 
.\/  Q )  .\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( Q 
.\/  R )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P ) ) )  ->  (
( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  ( ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) ) ) )
91, 2, 3, 4, 5dalawlem15 30756 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( ( S  .\/  T ) 
.\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) )  /\  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
1093expib 1157 . . . . . . 7  |-  ( ( K  e.  HL  /\  -.  ( ( ( S 
.\/  T )  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( S  .\/  T )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( T 
.\/  U )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( U  .\/  S ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  ->  ( (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )
11103exp 1153 . . . . . 6  |-  ( K  e.  HL  ->  ( -.  ( ( ( S 
.\/  T )  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( S  .\/  T )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( T 
.\/  U )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( U  .\/  S ) ) )  ->  (
( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  ( ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) ) ) )
12 simp11 988 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  K  e.  HL )
13 simp2 959 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )
14 simp3 960 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )
15 simp2ll 1025 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( LPlanes `  K ) )
16153ad2ant1 979 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( LPlanes `  K ) )
17 simp2rl 1027 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  ->  ( ( S  .\/  T )  .\/  U )  e.  ( LPlanes `  K ) )
18173ad2ant1 979 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( S  .\/  T )  .\/  U )  e.  ( LPlanes `  K ) )
19 simp2lr 1026 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  ->  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( Q 
.\/  R )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P ) ) )
20193ad2ant1 979 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( Q 
.\/  R )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P ) ) )
21 simp2rr 1028 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  ->  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( S  .\/  T )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( T 
.\/  U )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( U  .\/  S ) ) )
22213ad2ant1 979 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( S  .\/  T )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( T 
.\/  U )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( U  .\/  S ) ) )
23 simp13 990 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )
241, 2, 3, 4, 5dalawlem1 30742 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( ( ( P 
.\/  Q )  .\/  R )  e.  ( LPlanes `  K )  /\  (
( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )
)  /\  ( ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( Q 
.\/  R )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P ) )  /\  ( -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( S  .\/  T )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( T 
.\/  U )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( U  .\/  S ) )  /\  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  U ) ) )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
2512, 13, 14, 16, 18, 20, 22, 23, 24syl323anc 1215 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
26253expib 1157 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  ->  ( (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )
27263exp 1153 . . . . . 6  |-  ( K  e.  HL  ->  (
( ( ( ( P  .\/  Q ) 
.\/  R )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( P 
.\/  Q )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) ) )  /\  ( ( ( S  .\/  T
)  .\/  U )  e.  ( LPlanes `  K )  /\  ( -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( S 
.\/  T )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( T  .\/  U )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  S ) ) ) )  ->  (
( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  ( ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) ) ) )
288, 11, 27ecased 912 . . . . 5  |-  ( K  e.  HL  ->  (
( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  ( ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) ) )
2928exp4a 591 . . . 4  |-  ( K  e.  HL  ->  (
( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  ->  (
( S  e.  A  /\  T  e.  A  /\  U  e.  A
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) ) ) ) )
3029com34 80 . . 3  |-  ( K  e.  HL  ->  (
( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  ( ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  ->  (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) ) ) ) )
3130com24 84 . 2  |-  ( K  e.  HL  ->  (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  ->  ( ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  ->  ( ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) ) ) )
32313imp 1148 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   lecple 13541   joincjn 14406   meetcmee 14407   Atomscatm 30135   HLchlt 30222   LPlanesclpl 30363
This theorem is referenced by:  cdleme14  31144  cdleme20f  31185  cdlemg9  31505  cdlemg12c  31516  cdlemk6  31708  cdlemk6u  31733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371  df-psubsp 30374  df-pmap 30375  df-padd 30667
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