Metamath Proof Explorer

Theorem dalawlem6

Description: Lemma for dalaw . First piece of dalawlem8 . (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 )
Assertion dalawlem6 
|- ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( ( ( 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 eqid 
 |-  ( Base ` K ) = ( Base ` K )
6 simp11 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 )
7 6 hllatd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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. Lat )
8 simp21 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 )
9 simp22 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> Q e. A )
10 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
11 6 8 9 10 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) e. ( Base ` K ) )
12 simp32 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> T e. A )
13 5 4 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
14 12 13 syl 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> T e. ( Base ` K ) )
15 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) )
16 7 11 14 15 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) e. ( Base ` K ) )
17 simp31 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 )
18 5 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
19 17 18 syl 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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. ( Base ` K ) )
20 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( ( P .\/ Q ) .\/ T ) ./\ S ) e. ( Base ` K ) )
21 7 16 19 20 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) e. ( Base ` K ) )
22 simp23 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> R e. A )
23 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) )
24 6 9 22 23 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
25 simp33 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> U e. A )
26 5 4 atbase 
 |-  ( U e. A -> U e. ( Base ` K ) )
27 25 26 syl 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> U e. ( Base ` K ) )
28 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> ( ( Q .\/ R ) ./\ U ) e. ( Base ` K ) )
29 7 24 27 28 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ R ) ./\ U ) e. ( Base ` K ) )
30 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ P e. A ) -> ( R .\/ P ) e. ( Base ` K ) )
31 6 22 8 30 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( R .\/ P ) e. ( Base ` K ) )
32 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ U e. A /\ S e. A ) -> ( U .\/ S ) e. ( Base ` K ) )
33 6 25 17 32 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( U .\/ S ) e. ( Base ` K ) )
34 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( R .\/ P ) e. ( Base ` K ) /\ ( U .\/ S ) e. ( Base ` K ) ) -> ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) )
35 7 31 33 34 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) )
36 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( ( Q .\/ R ) ./\ U ) e. ( Base ` K ) /\ ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) ) -> ( ( ( Q .\/ R ) ./\ U ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) e. ( Base ` K ) )
37 7 29 35 36 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( Q .\/ R ) ./\ U ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) e. ( Base ` K ) )
38 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) e. ( Base ` K ) )
39 6 12 25 38 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( T .\/ U ) e. ( Base ` K ) )
40 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) e. ( Base ` K ) )
41 7 24 39 40 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) e. ( Base ` K ) )
42 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) e. ( Base ` K ) /\ ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) ) -> ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) e. ( Base ` K ) )
43 7 41 35 42 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) e. ( Base ` K ) )
44 5 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
45 8 44 syl 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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. ( Base ` K ) )
46 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) )
47 6 8 17 46 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) e. ( Base ` K ) )
48 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> ( Q .\/ T ) e. ( Base ` K ) )
49 6 9 12 48 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( Q .\/ T ) e. ( Base ` K ) )
50 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
51 7 24 49 50 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
52 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) e. ( Base ` K ) )
53 7 47 51 52 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ R ) ./\ ( Q .\/ T ) ) ) e. ( Base ` K ) )
54 5 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) e. ( Base ` K ) ) -> ( P .\/ ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) e. ( Base ` K ) )
55 7 45 53 54 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) e. ( Base ` K ) )
56 5 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
57 22 56 syl 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> R e. ( Base ` K ) )
58 5 2 latjcl 
 |-  ( ( K e. Lat /\ R e. ( Base ` K ) /\ ( ( Q .\/ R ) ./\ U ) e. ( Base ` K ) ) -> ( R .\/ ( ( Q .\/ R ) ./\ U ) ) e. ( Base ` K ) )
59 7 57 29 58 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( R .\/ ( ( Q .\/ R ) ./\ U ) ) e. ( Base ` K ) )
60 5 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) e. ( Base ` K ) ) -> ( P .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) e. ( Base ` K ) )
61 7 45 59 60 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) e. ( Base ` K ) )
62 5 1 2 3 latmlej22 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) /\ P e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( P .\/ S ) )
63 7 19 16 45 62 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( P .\/ S ) )
64 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) e. ( Base ` K ) )
65 7 49 47 64 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) e. ( Base ` K ) )
66 5 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( ( Q .\/ T ) ./\ ( P .\/ S ) ) e. ( Base ` K ) ) -> ( P .\/ ( ( Q .\/ T ) ./\ ( P .\/ S ) ) ) e. ( Base ` K ) )
67 7 45 65 66 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ T ) ./\ ( P .\/ S ) ) ) e. ( Base ` K ) )
68 5 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) ) -> ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) e. ( Base ` K ) )
69 7 45 51 68 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) ./\ ( Q .\/ T ) ) ) e. ( Base ` K ) )
70 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> S .<_ ( P .\/ S ) )
71 6 8 17 70 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .<_ ( P .\/ S ) )
72 5 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( P .\/ ( Q .\/ T ) ) e. ( Base ` K ) )
73 7 45 49 72 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ T ) ) e. ( Base ` K ) )
74 5 1 3 latmlem2 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) /\ ( P .\/ ( Q .\/ T ) ) e. ( Base ` K ) ) ) -> ( S .<_ ( P .\/ S ) -> ( ( P .\/ ( Q .\/ T ) ) ./\ S ) .<_ ( ( P .\/ ( Q .\/ T ) ) ./\ ( P .\/ S ) ) ) )
75 7 19 47 73 74 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .<_ ( P .\/ S ) -> ( ( P .\/ ( Q .\/ T ) ) ./\ S ) .<_ ( ( P .\/ ( Q .\/ T ) ) ./\ ( P .\/ S ) ) ) )
76 71 75 mpd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ T ) ) ./\ S ) .<_ ( ( P .\/ ( Q .\/ T ) ) ./\ ( P .\/ S ) ) )
77 2 4 hlatjass 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ T e. A ) ) -> ( ( P .\/ Q ) .\/ T ) = ( P .\/ ( Q .\/ T ) ) )
78 6 8 9 12 77 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) = ( P .\/ ( Q .\/ T ) ) )
79 78 oveq1d 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) = ( ( P .\/ ( Q .\/ T ) ) ./\ S ) )
80 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> P .<_ ( P .\/ S ) )
81 6 8 17 80 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .<_ ( P .\/ S ) )
82 5 1 2 3 4 atmod1i1 
 |-  ( ( K e. HL /\ ( P e. A /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) /\ P .<_ ( P .\/ S ) ) -> ( P .\/ ( ( Q .\/ T ) ./\ ( P .\/ S ) ) ) = ( ( P .\/ ( Q .\/ T ) ) ./\ ( P .\/ S ) ) )
83 6 8 49 47 81 82 syl131anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ T ) ./\ ( P .\/ S ) ) ) = ( ( P .\/ ( Q .\/ T ) ) ./\ ( P .\/ S ) ) )
84 76 79 83 3brtr4d 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( P .\/ ( ( Q .\/ T ) ./\ ( P .\/ S ) ) ) )
85 5 3 latmcom 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = ( ( P .\/ S ) ./\ ( Q .\/ T ) ) )
86 7 49 47 85 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = ( ( P .\/ S ) ./\ ( Q .\/ T ) ) )
87 simp12 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) ) .<_ ( Q .\/ R ) )
88 86 87 eqbrtrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ R ) )
89 5 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) )
90 7 49 47 89 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) )
91 5 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) ) -> ( ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ R ) /\ ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) ) <-> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) )
92 7 65 24 49 91 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ R ) /\ ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) ) <-> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) )
93 88 90 92 mpbi2and 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) )
94 5 1 2 latjlej2 
 |-  ( ( K e. Lat /\ ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) e. ( Base ` K ) /\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) /\ P e. ( Base ` K ) ) ) -> ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) -> ( P .\/ ( ( Q .\/ T ) ./\ ( P .\/ S ) ) ) .<_ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) )
95 7 65 51 45 94 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) -> ( P .\/ ( ( Q .\/ T ) ./\ ( P .\/ S ) ) ) .<_ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) )
96 93 95 mpd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ T ) ./\ ( P .\/ S ) ) ) .<_ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) )
97 5 1 7 21 67 69 84 96 lattrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) )
98 5 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) /\ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) e. ( Base ` K ) ) ) -> ( ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( P .\/ S ) /\ ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) <-> ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( ( P .\/ S ) ./\ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) ) )
99 7 21 47 69 98 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( P .\/ S ) /\ ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) <-> ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( ( P .\/ S ) ./\ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) ) )
100 63 97 99 mpbi2and 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( ( P .\/ S ) ./\ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) )
101 5 1 2 3 4 atmod3i1 
 |-  ( ( K e. HL /\ ( P e. A /\ ( P .\/ S ) e. ( Base ` K ) /\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) ) /\ P .<_ ( P .\/ S ) ) -> ( P .\/ ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) = ( ( P .\/ S ) ./\ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) )
102 6 8 47 51 81 101 syl131anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) = ( ( P .\/ S ) ./\ ( P .\/ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) )
103 100 102 breqtrrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( P .\/ ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) )
104 simp13 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) )
105 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
106 7 47 49 105 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) ) e. ( Base ` K ) )
107 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ U e. A ) -> ( R .\/ U ) e. ( Base ` K ) )
108 6 22 25 107 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( R .\/ U ) e. ( Base ` K ) )
109 5 1 3 latmlem2 
 |-  ( ( K e. Lat /\ ( ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) /\ ( R .\/ U ) e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) ) -> ( ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) -> ( ( Q .\/ R ) ./\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) .<_ ( ( Q .\/ R ) ./\ ( R .\/ U ) ) ) )
110 7 106 108 24 109 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) -> ( ( Q .\/ R ) ./\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) .<_ ( ( Q .\/ R ) ./\ ( R .\/ U ) ) ) )
111 104 110 mpd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ R ) ./\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) .<_ ( ( Q .\/ R ) ./\ ( R .\/ U ) ) )
112 hlol 
 |-  ( K e. HL -> K e. OL )
113 6 112 syl 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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. OL )
114 5 3 latm12 
 |-  ( ( K e. OL /\ ( ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) ) -> ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) = ( ( Q .\/ R ) ./\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) )
115 113 47 24 49 114 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ R ) ./\ ( Q .\/ T ) ) ) = ( ( Q .\/ R ) ./\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) )
116 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> R .<_ ( Q .\/ R ) )
117 6 9 22 116 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> R .<_ ( Q .\/ R ) )
118 5 1 2 3 4 atmod3i1 
 |-  ( ( K e. HL /\ ( R e. A /\ ( Q .\/ R ) e. ( Base ` K ) /\ U e. ( Base ` K ) ) /\ R .<_ ( Q .\/ R ) ) -> ( R .\/ ( ( Q .\/ R ) ./\ U ) ) = ( ( Q .\/ R ) ./\ ( R .\/ U ) ) )
119 6 22 24 27 117 118 syl131anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( R .\/ ( ( Q .\/ R ) ./\ U ) ) = ( ( Q .\/ R ) ./\ ( R .\/ U ) ) )
120 111 115 119 3brtr4d 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ R ) ./\ ( Q .\/ T ) ) ) .<_ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) )
121 5 1 2 latjlej2 
 |-  ( ( K e. Lat /\ ( ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) e. ( Base ` K ) /\ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) e. ( Base ` K ) /\ P e. ( Base ` K ) ) ) -> ( ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) .<_ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) -> ( P .\/ ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) .<_ ( P .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) ) )
122 7 53 59 45 121 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ R ) ./\ ( Q .\/ T ) ) ) .<_ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) -> ( P .\/ ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) .<_ ( P .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) ) )
123 120 122 mpd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ ( ( P .\/ S ) ./\ ( ( Q .\/ R ) ./\ ( Q .\/ T ) ) ) ) .<_ ( P .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) )
124 5 1 7 21 55 61 103 123 lattrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( P .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) )
125 5 2 latj13 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ R e. ( Base ` K ) /\ ( ( Q .\/ R ) ./\ U ) e. ( Base ` K ) ) ) -> ( P .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) = ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) )
126 7 45 57 29 125 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) = ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) )
127 124 126 breqtrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) )
128 5 1 2 3 latmlej22 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) /\ U e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( U .\/ S ) )
129 7 19 16 27 128 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( U .\/ S ) )
130 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( ( Q .\/ R ) ./\ U ) e. ( Base ` K ) /\ ( R .\/ P ) e. ( Base ` K ) ) -> ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) e. ( Base ` K ) )
131 7 29 31 130 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) e. ( Base ` K ) )
132 5 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) e. ( Base ` K ) /\ ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) e. ( Base ` K ) /\ ( U .\/ S ) e. ( Base ` K ) ) ) -> ( ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) /\ ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( U .\/ S ) ) <-> ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) ./\ ( U .\/ S ) ) ) )
133 7 21 131 33 132 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) /\ ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( U .\/ S ) ) <-> ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) ./\ ( U .\/ S ) ) ) )
134 127 129 133 mpbi2and 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) ./\ ( U .\/ S ) ) )
135 5 1 2 3 latmlej21 
 |-  ( ( K e. Lat /\ ( U e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( Q .\/ R ) ./\ U ) .<_ ( U .\/ S ) )
136 7 27 24 19 135 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ R ) ./\ U ) .<_ ( U .\/ S ) )
137 5 1 2 3 4 atmod1i1m 
 |-  ( ( ( K e. HL /\ U e. A ) /\ ( ( Q .\/ R ) e. ( Base ` K ) /\ ( R .\/ P ) e. ( Base ` K ) /\ ( U .\/ S ) e. ( Base ` K ) ) /\ ( ( Q .\/ R ) ./\ U ) .<_ ( U .\/ S ) ) -> ( ( ( Q .\/ R ) ./\ U ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) = ( ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) ./\ ( U .\/ S ) ) )
138 6 25 24 31 33 136 137 syl231anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( Q .\/ R ) ./\ U ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) = ( ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) ./\ ( U .\/ S ) ) )
139 134 138 breqtrrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( ( ( Q .\/ R ) ./\ U ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )
140 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> U .<_ ( T .\/ U ) )
141 6 12 25 140 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> U .<_ ( T .\/ U ) )
142 5 1 3 latmlem2 
 |-  ( ( K e. Lat /\ ( U e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) ) -> ( U .<_ ( T .\/ U ) -> ( ( Q .\/ R ) ./\ U ) .<_ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) ) )
143 7 27 39 24 142 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( U .<_ ( T .\/ U ) -> ( ( Q .\/ R ) ./\ U ) .<_ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) ) )
144 141 143 mpd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ R ) ./\ U ) .<_ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) )
145 5 1 2 latjlej1 
 |-  ( ( K e. Lat /\ ( ( ( Q .\/ R ) ./\ U ) e. ( Base ` K ) /\ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) e. ( Base ` K ) /\ ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) ) ) -> ( ( ( Q .\/ R ) ./\ U ) .<_ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) -> ( ( ( Q .\/ R ) ./\ U ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) ) )
146 7 29 41 35 145 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( Q .\/ R ) ./\ U ) .<_ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) -> ( ( ( Q .\/ R ) ./\ U ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) ) )
147 144 146 mpd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( Q .\/ R ) ./\ U ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )
148 5 1 7 21 37 43 139 147 lattrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( 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 ) .\/ T ) ./\ S ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )