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Theorem dalaw 30414
Description: Desargues' law, derived from Desargues' theorem dath 30264 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 2430 . . . . . . . . 9  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
61, 2, 3, 4, 5dalawlem14 30412 . . . . . . . 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 1156 . . . . . . 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 1152 . . . . . 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 30413 . . . . . . . 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 1156 . . . . . . 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 1152 . . . . . 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 987 . . . . . . . . 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 958 . . . . . . . . 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 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 )
)  ->  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )
15 simp2ll 1024 . . . . . . . . . 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 978 . . . . . . . . 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 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 ) )  ->  ( ( S  .\/  T )  .\/  U )  e.  ( LPlanes `  K ) )
18173ad2ant1 978 . . . . . . . . 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 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  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( Q 
.\/  R )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P ) ) )
20193ad2ant1 978 . . . . . . . . 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 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  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( S  .\/  T )  /\  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( T 
.\/  U )  /\  -.  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( U  .\/  S ) ) )
22213ad2ant1 978 . . . . . . . . 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 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  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )
241, 2, 3, 4, 5dalawlem1 30399 . . . . . . . . 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 1214 . . . . . . . 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 1156 . . . . . . 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 1152 . . . . . 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 911 . . . . 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 590 . . . 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 1147 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   lecple 13519   joincjn 14384   meetcmee 14385   Atomscatm 29792   HLchlt 29879   LPlanesclpl 30020
This theorem is referenced by:  cdleme14  30801  cdleme20f  30842  cdlemg9  31162  cdlemg12c  31173  cdlemk6  31365  cdlemk6u  31390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-llines 30026  df-lplanes 30027  df-lvols 30028  df-psubsp 30031  df-pmap 30032  df-padd 30324
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