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Theorem dalaw 29226
Description: Desargues' law, derived from Desargues' theorem dath 29076 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 2256 . . . . . . . . 9  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
61, 2, 3, 4, 5dalawlem14 29224 . . . . . . . 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 1159 . . . . . . 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 1155 . . . . . 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 29225 . . . . . . . 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 1159 . . . . . . 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 1155 . . . . . 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 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 )
)  ->  K  e.  HL )
13 simp2 961 . . . . . . . . 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 962 . . . . . . . . 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 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 ) )  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( LPlanes `  K ) )
16153ad2ant1 981 . . . . . . . . 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 1029 . . . . . . . . . 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 981 . . . . . . . . 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 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 ) )  .<_  ( P  .\/  Q )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( Q 
.\/  R )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P ) ) )
20193ad2ant1 981 . . . . . . . . 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 1030 . . . . . . . . . 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 981 . . . . . . . . 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 992 . . . . . . . . 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 29211 . . . . . . . . 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 1217 . . . . . . . 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 1159 . . . . . . 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 1155 . . . . . 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 915 . . . . 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 592 . . . 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 79 . . 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 83 . 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 1150 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   lecple 13163   joincjn 14026   meetcmee 14027   Atomscatm 28604   HLchlt 28691   LPlanesclpl 28832
This theorem is referenced by:  cdleme14  29613  cdleme20f  29654  cdlemg9  29974  cdlemg12c  29985  cdlemk6  30177  cdlemk6u  30202
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-psubsp 28843  df-pmap 28844  df-padd 29136
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