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Theorem dalaw 33535
Description: Desargues' law, derived from Desargues' theorem dath 33385 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 2443 . . . . . . . . 9  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
61, 2, 3, 4, 5dalawlem14 33533 . . . . . . . 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 1190 . . . . . . 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 1186 . . . . . 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 33534 . . . . . . . 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 1190 . . . . . . 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 1186 . . . . . 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 1018 . . . . . . . . 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 989 . . . . . . . . 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 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 )
)  ->  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )
15 simp2ll 1055 . . . . . . . . . 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 1009 . . . . . . . . 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 1057 . . . . . . . . . 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 1009 . . . . . . . . 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 1056 . . . . . . . . . 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 1009 . . . . . . . . 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 1058 . . . . . . . . . 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 1009 . . . . . . . . 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 1020 . . . . . . . . 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 33520 . . . . . . . . 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 1248 . . . . . . . 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 1190 . . . . . . 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 1186 . . . . . 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 935 . . . . 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 606 . . . 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 83 . . 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 87 . 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 1181 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   lecple 14250   joincjn 15119   meetcmee 15120   Atomscatm 32913   HLchlt 33000   LPlanesclpl 33141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-clat 15283  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001  df-llines 33147  df-lplanes 33148  df-lvols 33149  df-psubsp 33152  df-pmap 33153  df-padd 33445
This theorem is referenced by:  cdleme14  33922  cdleme20f  33963  cdlemg9  34283  cdlemg12c  34294  cdlemk6  34486  cdlemk6u  34511
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