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Metamath Proof Explorer

Theorem dalawlem1

Description: Lemma for dalaw . Special case of dath2 , where C is replaced by ( ( P .\/ S ) ./\ ( Q .\/ T ) ) . The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 . (Contributed by NM, 6-Oct-2012)

Ref Expression
Hypotheses dalawlem.l 
|- .<_ = ( le ` K )
dalawlem.j 
|- .\/ = ( join ` K )
dalawlem.m 
|- ./\ = ( meet ` K )
dalawlem.a 
|- A = ( Atoms ` K )
dalawlem.o 
|- O = ( LPlanes ` K )
Assertion 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) ) )

Proof

Step Hyp Ref Expression
1 dalawlem.l 
 |-  .<_ = ( le ` K )
2 dalawlem.j 
 |-  .\/ = ( join ` K )
3 dalawlem.m 
 |-  ./\ = ( meet ` K )
4 dalawlem.a 
 |-  A = ( Atoms ` K )
5 dalawlem.o 
 |-  O = ( LPlanes ` K )
6 simp11 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) ) -> K e. HL )
7 6 hllatd 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) ) -> K e. Lat )
8 simp121 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 e. A )
9 simp131 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) ) -> S e. A )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) )
12 6 8 9 11 syl3anc 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 .\/ S ) e. ( Base ` K ) )
13 simp122 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) ) -> Q e. A )
14 simp132 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) ) -> T e. A )
15 10 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> ( Q .\/ T ) e. ( Base ` K ) )
16 6 13 14 15 syl3anc 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) ) -> ( Q .\/ T ) e. ( Base ` K ) )
17 10 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
18 7 12 16 17 syl3anc 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
19 6 18 jca 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) ) -> ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) ) )
20 simp12 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 e. A /\ Q e. A /\ R e. A ) )
21 simp13 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) ) -> ( S e. A /\ T e. A /\ U e. A ) )
22 simp2l 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) .\/ R ) e. O )
23 simp2r 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) ) -> ( ( S .\/ T ) .\/ U ) e. O )
24 simp31 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) )
25 simp32 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) )
26 10 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ S ) )
27 7 12 16 26 syl3anc 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ S ) )
28 10 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ T ) )
29 7 12 16 28 syl3anc 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ T ) )
30 simp33 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) )
31 27 29 30 3jca 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ S ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ T ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) )
32 eqid 
 |-  ( ( P .\/ Q ) ./\ ( S .\/ T ) ) = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
33 eqid 
 |-  ( ( Q .\/ R ) ./\ ( T .\/ U ) ) = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
34 eqid 
 |-  ( ( R .\/ P ) ./\ ( U .\/ S ) ) = ( ( R .\/ P ) ./\ ( U .\/ S ) )
35 10 1 2 4 3 5 32 33 34 dath2 
 |-  ( ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) .<_ ( P .\/ S ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ T ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )
36 19 20 21 22 23 24 25 31 35 syl323anc 
 |-  ( ( ( 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. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( 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 ) ) ) )