Metamath Proof Explorer

Theorem dalaw

Description: Desargues's law, derived from Desargues's theorem dath 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)

Ref Expression
Hypotheses dalaw.l 
|- .<_ = ( le ` K )
dalaw.j 
|- .\/ = ( join ` K )
dalaw.m 
|- ./\ = ( meet ` K )
dalaw.a 
|- A = ( Atoms ` K )
Assertion 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

Step Hyp Ref Expression
1 dalaw.l 
 |-  .<_ = ( le ` K )
2 dalaw.j 
 |-  .\/ = ( join ` K )
3 dalaw.m 
 |-  ./\ = ( meet ` K )
4 dalaw.a 
 |-  A = ( Atoms ` K )
5 eqid 
 |-  ( LPlanes ` K ) = ( LPlanes ` K )
6 1 2 3 4 5 dalawlem14 
 |-  ( ( ( 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 ) ) ) )
7 6 3expib 
 |-  ( ( 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 ) ) ) ) )
8 7 3exp 
 |-  ( 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 ) ) ) ) ) ) )
9 1 2 3 4 5 dalawlem15 
 |-  ( ( ( 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 ) ) ) )
10 9 3expib 
 |-  ( ( 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 ) ) ) ) )
11 10 3exp 
 |-  ( 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 
 |-  ( ( ( 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 
 |-  ( ( ( 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 
 |-  ( ( ( 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 
 |-  ( ( 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 ) )
16 15 3ad2ant1 
 |-  ( ( ( 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 
 |-  ( ( 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 ) )
18 17 3ad2ant1 
 |-  ( ( ( 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 
 |-  ( ( 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 ) ) )
20 19 3ad2ant1 
 |-  ( ( ( 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 
 |-  ( ( 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 ) ) )
22 21 3ad2ant1 
 |-  ( ( ( 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 
 |-  ( ( ( 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 ) )
24 1 2 3 4 5 dalawlem1 
 |-  ( ( ( 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 ) ) ) )
25 12 13 14 16 18 20 22 23 24 syl323anc 
 |-  ( ( ( 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 ) ) ) )
26 25 3expib 
 |-  ( ( 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 ) ) ) ) )
27 26 3exp 
 |-  ( 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 ) ) ) ) ) ) )
28 8 11 27 ecased 
 |-  ( 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 ) ) ) ) ) )
29 28 exp4a 
 |-  ( 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 ) ) ) ) ) ) )
30 29 com34 
 |-  ( 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 ) ) ) ) ) ) )
31 30 com24 
 |-  ( 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 ) ) ) ) ) ) )
32 31 3imp 
 |-  ( ( 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 ) ) ) ) )