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Metamath Proof Explorer

Theorem dalawlem3

Description: Lemma for dalaw . First piece of dalawlem5 . (Contributed by NM, 4-Oct-2012)

Ref Expression
Hypotheses dalawlem.l 
|- .<_ = ( le ` K )
dalawlem.j 
|- .\/ = ( join ` K )
dalawlem.m 
|- ./\ = ( meet ` K )
dalawlem.a 
|- A = ( Atoms ` K )
Assertion dalawlem3 
|- ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ./\ ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 simp22 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 )
9 simp32 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 )
10 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> ( Q .\/ T ) e. ( Base ` K ) )
11 6 8 9 10 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
12 simp21 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 )
13 5 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
14 12 13 syl 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
15 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ P e. ( Base ` K ) ) -> ( ( Q .\/ T ) .\/ P ) e. ( Base ` K ) )
16 7 11 14 15 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) e. ( Base ` K ) )
17 simp31 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 /\ ( ( Q .\/ T ) .\/ P ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( ( Q .\/ T ) .\/ P ) ./\ S ) e. ( Base ` K ) )
21 7 16 19 20 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
22 simp23 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 8 22 23 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 12 30 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 9 25 38 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
45 8 44 syl 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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. ( Base ` K ) )
46 5 3 latmcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> ( Q ./\ U ) e. ( Base ` K ) )
47 7 45 27 46 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) e. ( Base ` K ) )
48 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) )
49 6 12 17 48 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
50 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ Q ) e. ( Base ` K ) )
51 7 49 45 50 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) e. ( Base ` K ) )
52 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( Q ./\ U ) e. ( Base ` K ) /\ ( ( P .\/ S ) ./\ Q ) e. ( Base ` K ) ) -> ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) e. ( Base ` K ) )
53 7 47 51 52 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) e. ( Base ` K ) )
54 5 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) e. ( Base ` K ) ) -> ( P .\/ ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) ) e. ( Base ` K ) )
55 7 14 53 54 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) ) 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 14 59 60 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 2 latjcl 
 |-  ( ( K e. Lat /\ ( Q ./\ U ) e. ( Base ` K ) /\ P e. ( Base ` K ) ) -> ( ( Q ./\ U ) .\/ P ) e. ( Base ` K ) )
63 7 47 14 62 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ P ) e. ( Base ` K ) )
64 5 1 2 3 latmlej22 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ ( ( Q .\/ T ) .\/ P ) e. ( Base ` K ) /\ ( ( Q ./\ U ) .\/ P ) e. ( Base ` K ) ) ) -> ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( ( ( Q ./\ U ) .\/ P ) .\/ S ) )
65 7 19 16 63 64 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ P ) .\/ S ) )
66 5 2 latjass 
 |-  ( ( K e. Lat /\ ( ( Q ./\ U ) e. ( Base ` K ) /\ P e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( ( Q ./\ U ) .\/ P ) .\/ S ) = ( ( Q ./\ U ) .\/ ( P .\/ S ) ) )
67 7 47 14 19 66 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ P ) .\/ S ) = ( ( Q ./\ U ) .\/ ( P .\/ S ) ) )
68 65 67 breqtrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( P .\/ S ) ) )
69 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) e. ( Base ` K ) )
70 7 11 49 69 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
71 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( ( Q .\/ T ) ./\ ( P .\/ S ) ) e. ( Base ` K ) /\ P e. ( Base ` K ) ) -> ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .\/ P ) e. ( Base ` K ) )
72 7 70 14 71 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) e. ( Base ` K ) )
73 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
74 6 12 8 73 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
75 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> S .<_ ( P .\/ S ) )
76 6 12 17 75 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
77 5 1 3 latmlem2 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) /\ ( ( Q .\/ T ) .\/ P ) e. ( Base ` K ) ) ) -> ( S .<_ ( P .\/ S ) -> ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( ( ( Q .\/ T ) .\/ P ) ./\ ( P .\/ S ) ) ) )
78 7 19 49 16 77 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) -> ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( ( ( Q .\/ T ) .\/ P ) ./\ ( P .\/ S ) ) ) )
79 76 78 mpd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) .\/ P ) ./\ ( P .\/ S ) ) )
80 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> P .<_ ( P .\/ S ) )
81 6 12 17 80 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 atmod4i1 
 |-  ( ( K e. HL /\ ( P e. A /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) /\ P .<_ ( P .\/ S ) ) -> ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .\/ P ) = ( ( ( Q .\/ T ) .\/ P ) ./\ ( P .\/ S ) ) )
83 6 12 11 49 81 82 syl131anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) = ( ( ( Q .\/ T ) .\/ P ) ./\ ( P .\/ S ) ) )
84 79 83 breqtrrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ./\ ( P .\/ S ) ) .\/ P ) )
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 11 49 85 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
88 86 87 eqbrtrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .\/ Q ) )
89 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> P .<_ ( P .\/ Q ) )
90 6 12 8 89 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .\/ Q ) )
91 5 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) e. ( Base ` K ) /\ P e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( P .\/ Q ) /\ P .<_ ( P .\/ Q ) ) <-> ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .\/ P ) .<_ ( P .\/ Q ) ) )
92 7 70 14 74 91 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .\/ Q ) /\ P .<_ ( P .\/ Q ) ) <-> ( ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .\/ P ) .<_ ( P .\/ Q ) ) )
93 88 90 92 mpbi2and 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) .<_ ( P .\/ Q ) )
94 5 1 7 21 72 74 84 93 lattrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .\/ Q ) )
95 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( Q ./\ U ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q ./\ U ) .\/ ( P .\/ S ) ) e. ( Base ` K ) )
96 7 47 49 95 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( P .\/ S ) ) e. ( Base ` K ) )
97 5 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( ( ( ( Q .\/ T ) .\/ P ) ./\ S ) e. ( Base ` K ) /\ ( ( Q ./\ U ) .\/ ( P .\/ S ) ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( ( Q ./\ U ) .\/ ( P .\/ S ) ) /\ ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( P .\/ Q ) ) <-> ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( ( ( Q ./\ U ) .\/ ( P .\/ S ) ) ./\ ( P .\/ Q ) ) ) )
98 7 21 96 74 97 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( P .\/ S ) ) /\ ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( P .\/ Q ) ) <-> ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( ( ( Q ./\ U ) .\/ ( P .\/ S ) ) ./\ ( P .\/ Q ) ) ) )
99 68 94 98 mpbi2and 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( P .\/ S ) ) ./\ ( P .\/ Q ) ) )
100 5 1 2 3 4 atmod3i1 
 |-  ( ( K e. HL /\ ( P e. A /\ ( P .\/ S ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ P .<_ ( P .\/ S ) ) -> ( P .\/ ( ( P .\/ S ) ./\ Q ) ) = ( ( P .\/ S ) ./\ ( P .\/ Q ) ) )
101 6 12 49 45 81 100 syl131anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) = ( ( P .\/ S ) ./\ ( P .\/ Q ) ) )
102 101 oveq2d 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( P .\/ ( ( P .\/ S ) ./\ Q ) ) ) = ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ ( P .\/ Q ) ) ) )
103 5 2 latj12 
 |-  ( ( K e. Lat /\ ( ( Q ./\ U ) e. ( Base ` K ) /\ P e. ( Base ` K ) /\ ( ( P .\/ S ) ./\ Q ) e. ( Base ` K ) ) ) -> ( ( Q ./\ U ) .\/ ( P .\/ ( ( P .\/ S ) ./\ Q ) ) ) = ( P .\/ ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) ) )
104 7 47 14 51 103 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( P .\/ ( ( P .\/ S ) ./\ Q ) ) ) = ( P .\/ ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) ) )
105 5 1 2 3 latmlej12 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ U e. ( Base ` K ) /\ P e. ( Base ` K ) ) ) -> ( Q ./\ U ) .<_ ( P .\/ Q ) )
106 7 45 27 14 105 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .<_ ( P .\/ Q ) )
107 5 1 2 3 4 atmod1i1m 
 |-  ( ( ( K e. HL /\ U e. A ) /\ ( Q e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) /\ ( Q ./\ U ) .<_ ( P .\/ Q ) ) -> ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ ( P .\/ Q ) ) ) = ( ( ( Q ./\ U ) .\/ ( P .\/ S ) ) ./\ ( P .\/ Q ) ) )
108 6 25 45 49 74 106 107 syl231anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( ( P .\/ S ) ./\ ( P .\/ Q ) ) ) = ( ( ( Q ./\ U ) .\/ ( P .\/ S ) ) ./\ ( P .\/ Q ) ) )
109 102 104 108 3eqtr3rd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( P .\/ S ) ) ./\ ( P .\/ Q ) ) = ( P .\/ ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) ) )
110 99 109 breqtrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .\/ ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) ) )
111 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> Q .<_ ( Q .\/ R ) )
112 6 8 22 111 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .<_ ( Q .\/ R ) )
113 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ R e. A /\ U e. A ) -> U .<_ ( R .\/ U ) )
114 6 22 25 113 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .<_ ( R .\/ U ) )
115 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
116 7 49 11 115 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
117 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ U e. A ) -> ( R .\/ U ) e. ( Base ` K ) )
118 6 22 25 117 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
119 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> Q .<_ ( Q .\/ T ) )
120 6 8 9 119 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .<_ ( Q .\/ T ) )
121 5 1 3 latmlem2 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) ) -> ( Q .<_ ( Q .\/ T ) -> ( ( P .\/ S ) ./\ Q ) .<_ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) )
122 7 45 11 49 121 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .<_ ( Q .\/ T ) -> ( ( P .\/ S ) ./\ Q ) .<_ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) )
123 120 122 mpd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) .<_ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) )
124 simp13 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
125 5 1 7 51 116 118 123 124 lattrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .\/ U ) )
126 5 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( U e. ( Base ` K ) /\ ( ( P .\/ S ) ./\ Q ) e. ( Base ` K ) /\ ( R .\/ U ) e. ( Base ` K ) ) ) -> ( ( U .<_ ( R .\/ U ) /\ ( ( P .\/ S ) ./\ Q ) .<_ ( R .\/ U ) ) <-> ( U .\/ ( ( P .\/ S ) ./\ Q ) ) .<_ ( R .\/ U ) ) )
127 7 27 51 118 126 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .<_ ( R .\/ U ) /\ ( ( P .\/ S ) ./\ Q ) .<_ ( R .\/ U ) ) <-> ( U .\/ ( ( P .\/ S ) ./\ Q ) ) .<_ ( R .\/ U ) ) )
128 114 125 127 mpbi2and 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .\/ ( ( P .\/ S ) ./\ Q ) ) .<_ ( R .\/ U ) )
129 5 2 latjcl 
 |-  ( ( K e. Lat /\ U e. ( Base ` K ) /\ ( ( P .\/ S ) ./\ Q ) e. ( Base ` K ) ) -> ( U .\/ ( ( P .\/ S ) ./\ Q ) ) e. ( Base ` K ) )
130 7 27 51 129 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .\/ ( ( P .\/ S ) ./\ Q ) ) e. ( Base ` K ) )
131 5 1 3 latmlem12 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) /\ ( ( U .\/ ( ( P .\/ S ) ./\ Q ) ) e. ( Base ` K ) /\ ( R .\/ U ) e. ( Base ` K ) ) ) -> ( ( Q .<_ ( Q .\/ R ) /\ ( U .\/ ( ( P .\/ S ) ./\ Q ) ) .<_ ( R .\/ U ) ) -> ( Q ./\ ( U .\/ ( ( P .\/ S ) ./\ Q ) ) ) .<_ ( ( Q .\/ R ) ./\ ( R .\/ U ) ) ) )
132 7 45 24 130 118 131 syl122anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .<_ ( Q .\/ R ) /\ ( U .\/ ( ( P .\/ S ) ./\ Q ) ) .<_ ( R .\/ U ) ) -> ( Q ./\ ( U .\/ ( ( P .\/ S ) ./\ Q ) ) ) .<_ ( ( Q .\/ R ) ./\ ( R .\/ U ) ) ) )
133 112 128 132 mp2and 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ ( U .\/ ( ( P .\/ S ) ./\ Q ) ) ) .<_ ( ( Q .\/ R ) ./\ ( R .\/ U ) ) )
134 5 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ Q ) .<_ Q )
135 7 49 45 134 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) .<_ Q )
136 5 1 2 3 4 atmod2i2 
 |-  ( ( K e. HL /\ ( U e. A /\ Q e. ( Base ` K ) /\ ( ( P .\/ S ) ./\ Q ) e. ( Base ` K ) ) /\ ( ( P .\/ S ) ./\ Q ) .<_ Q ) -> ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) = ( Q ./\ ( U .\/ ( ( P .\/ S ) ./\ Q ) ) ) )
137 6 25 45 51 135 136 syl131anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) = ( Q ./\ ( U .\/ ( ( P .\/ S ) ./\ Q ) ) ) )
138 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> R .<_ ( Q .\/ R ) )
139 6 8 22 138 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
140 5 1 2 3 4 atmod3i2 
 |-  ( ( K e. HL /\ ( U e. A /\ R e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) /\ R .<_ ( Q .\/ R ) ) -> ( R .\/ ( ( Q .\/ R ) ./\ U ) ) = ( ( Q .\/ R ) ./\ ( R .\/ U ) ) )
141 6 25 57 24 139 140 syl131anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) )
142 133 137 141 3brtr4d 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) .<_ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) )
143 5 1 2 latjlej2 
 |-  ( ( K e. Lat /\ ( ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) e. ( Base ` K ) /\ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) e. ( Base ` K ) /\ P e. ( Base ` K ) ) ) -> ( ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) .<_ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) -> ( P .\/ ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) ) .<_ ( P .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) ) )
144 7 53 59 14 143 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) .<_ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) -> ( P .\/ ( ( Q ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) ) .<_ ( P .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) ) )
145 142 144 mpd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ./\ U ) .\/ ( ( P .\/ S ) ./\ Q ) ) ) .<_ ( P .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) )
146 5 1 7 21 55 61 110 145 lattrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 .\/ ( R .\/ ( ( Q .\/ R ) ./\ U ) ) ) )
147 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 ) ) )
148 7 14 57 29 147 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) )
149 146 148 breqtrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ./\ U ) .\/ ( R .\/ P ) ) )
150 5 1 2 3 latmlej22 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ ( ( Q .\/ T ) .\/ P ) e. ( Base ` K ) /\ U e. ( Base ` K ) ) ) -> ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( U .\/ S ) )
151 7 19 16 27 150 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) .<_ ( U .\/ S ) )
152 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 ) )
153 7 29 31 152 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
154 5 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( ( ( ( Q .\/ T ) .\/ P ) ./\ S ) e. ( Base ` K ) /\ ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) e. ( Base ` K ) /\ ( U .\/ S ) e. ( Base ` K ) ) ) -> ( ( ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) /\ ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( U .\/ S ) ) <-> ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) ./\ ( U .\/ S ) ) ) )
155 7 21 153 33 154 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ./\ U ) .\/ ( R .\/ P ) ) /\ ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( U .\/ S ) ) <-> ( ( ( Q .\/ T ) .\/ P ) ./\ S ) .<_ ( ( ( ( Q .\/ R ) ./\ U ) .\/ ( R .\/ P ) ) ./\ ( U .\/ S ) ) ) )
156 149 151 155 mpbi2and 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ./\ U ) .\/ ( R .\/ P ) ) ./\ ( U .\/ S ) ) )
157 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 ) )
158 7 27 24 19 157 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
159 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 ) ) )
160 6 25 24 31 33 158 159 syl231anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) )
161 156 160 breqtrrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ./\ U ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )
162 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> U .<_ ( T .\/ U ) )
163 6 9 25 162 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) )
164 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 ) ) ) )
165 7 27 39 24 164 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) ) )
166 163 165 mpd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) )
167 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 ) ) ) ) )
168 7 29 41 35 167 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) ) ) )
169 166 168 mpd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ) ) )
170 5 1 7 21 37 43 161 169 lattrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ ( ( 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 ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )