This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dalawlem12

Description: Lemma for dalaw . Second part of dalawlem13 . (Contributed by NM, 17-Sep-2012)

Ref Expression
Hypotheses dalawlem.l 
|- .<_ = ( le ` K )
dalawlem.j 
|- .\/ = ( join ` K )
dalawlem.m 
|- ./\ = ( meet ` K )
dalawlem.a 
|- A = ( Atoms ` K )
Assertion dalawlem12 
|- ( ( ( K e. HL /\ 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 ) ./\ ( 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 eqid 
 |-  ( Base ` K ) = ( Base ` K )
6 simp11 
 |-  ( ( ( K e. HL /\ 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 /\ 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 /\ 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 /\ 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 /\ 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 simp31 
 |-  ( ( ( K e. HL /\ 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 )
13 simp32 
 |-  ( ( ( K e. HL /\ 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 )
14 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) )
15 6 12 13 14 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ T ) e. ( Base ` K ) )
16 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) e. ( Base ` K ) )
17 7 11 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) ./\ ( S .\/ T ) ) e. ( Base ` K ) )
18 5 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
19 12 18 syl 
 |-  ( ( ( K e. HL /\ 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 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) )
21 7 11 19 20 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) e. ( Base ` K ) )
22 5 4 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
23 13 22 syl 
 |-  ( ( ( K e. HL /\ 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 ) )
24 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> ( ( ( P .\/ Q ) .\/ S ) ./\ T ) e. ( Base ` K ) )
25 7 21 23 24 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) ./\ T ) e. ( Base ` K ) )
26 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .\/ S ) e. ( Base ` K ) )
27 7 25 19 26 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) ./\ T ) .\/ S ) e. ( Base ` K ) )
28 5 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
29 9 28 syl 
 |-  ( ( ( K e. HL /\ 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. ( Base ` K ) )
30 simp33 
 |-  ( ( ( K e. HL /\ 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 )
31 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) e. ( Base ` K ) )
32 6 13 30 31 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) )
33 5 3 latmcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) -> ( Q ./\ ( T .\/ U ) ) e. ( Base ` K ) )
34 7 29 32 33 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ U ) ) e. ( Base ` K ) )
35 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ U e. A /\ S e. A ) -> ( U .\/ S ) e. ( Base ` K ) )
36 6 30 12 35 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) )
37 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( Q ./\ ( T .\/ U ) ) e. ( Base ` K ) /\ ( U .\/ S ) e. ( Base ` K ) ) -> ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) e. ( Base ` K ) )
38 7 34 36 37 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ U ) ) .\/ ( U .\/ S ) ) e. ( Base ` K ) )
39 5 1 2 latlej1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ S ) )
40 7 11 19 39 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) .<_ ( ( P .\/ Q ) .\/ S ) )
41 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ T e. A /\ S e. A ) -> ( T .\/ S ) e. ( Base ` K ) )
42 6 13 12 41 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ S ) e. ( Base ` K ) )
43 5 1 3 latmlem1 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) /\ ( T .\/ S ) e. ( Base ` K ) ) ) -> ( ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ S ) -> ( ( P .\/ Q ) ./\ ( T .\/ S ) ) .<_ ( ( ( P .\/ Q ) .\/ S ) ./\ ( T .\/ S ) ) ) )
44 7 11 21 42 43 syl13anc 
 |-  ( ( ( K e. HL /\ 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 ) .<_ ( ( P .\/ Q ) .\/ S ) -> ( ( P .\/ Q ) ./\ ( T .\/ S ) ) .<_ ( ( ( P .\/ Q ) .\/ S ) ./\ ( T .\/ S ) ) ) )
45 40 44 mpd 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) ./\ ( T .\/ S ) ) )
46 2 4 hlatjcom 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) = ( T .\/ S ) )
47 6 12 13 46 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ T ) = ( T .\/ S ) )
48 47 oveq2d 
 |-  ( ( ( K e. HL /\ 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 ) ./\ ( S .\/ T ) ) = ( ( P .\/ Q ) ./\ ( T .\/ S ) ) )
49 5 1 2 latlej2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> S .<_ ( ( P .\/ Q ) .\/ S ) )
50 7 11 19 49 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ Q ) .\/ S ) )
51 5 1 2 3 4 atmod2i2 
 |-  ( ( K e. HL /\ ( T e. A /\ ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) /\ S .<_ ( ( P .\/ Q ) .\/ S ) ) -> ( ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .\/ S ) = ( ( ( P .\/ Q ) .\/ S ) ./\ ( T .\/ S ) ) )
52 6 13 21 19 50 51 syl131anc 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) ./\ T ) .\/ S ) = ( ( ( P .\/ Q ) .\/ S ) ./\ ( T .\/ S ) ) )
53 45 48 52 3brtr4d 
 |-  ( ( ( K e. HL /\ 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 ) ./\ ( S .\/ T ) ) .<_ ( ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .\/ S ) )
54 hlol 
 |-  ( K e. HL -> K e. OL )
55 6 54 syl 
 |-  ( ( ( K e. HL /\ 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 )
56 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) )
57 6 8 12 56 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) )
58 5 2 latjcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( Q .\/ ( P .\/ S ) ) e. ( Base ` K ) )
59 7 29 57 58 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ ( P .\/ S ) ) e. ( Base ` K ) )
60 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> ( Q .\/ T ) e. ( Base ` K ) )
61 6 9 13 60 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) )
62 5 3 latmassOLD 
 |-  ( ( K e. OL /\ ( ( Q .\/ ( P .\/ S ) ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) ) -> ( ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) ./\ T ) = ( ( Q .\/ ( P .\/ S ) ) ./\ ( ( Q .\/ T ) ./\ T ) ) )
63 55 59 61 23 62 syl13anc 
 |-  ( ( ( K e. HL /\ 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 .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) ./\ T ) = ( ( Q .\/ ( P .\/ S ) ) ./\ ( ( Q .\/ T ) ./\ T ) ) )
64 2 4 hlatjass 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ S ) = ( P .\/ ( Q .\/ S ) ) )
65 6 8 9 12 64 syl13anc 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) = ( P .\/ ( Q .\/ S ) ) )
66 2 4 hlatj12 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> ( P .\/ ( Q .\/ S ) ) = ( Q .\/ ( P .\/ S ) ) )
67 6 8 9 12 66 syl13anc 
 |-  ( ( ( K e. HL /\ 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 .\/ S ) ) = ( Q .\/ ( P .\/ S ) ) )
68 65 67 eqtr2d 
 |-  ( ( ( K e. HL /\ 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 .\/ ( P .\/ S ) ) = ( ( P .\/ Q ) .\/ S ) )
69 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> T .<_ ( Q .\/ T ) )
70 6 9 13 69 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .<_ ( Q .\/ T ) )
71 5 1 3 latleeqm2 
 |-  ( ( K e. Lat /\ T e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( T .<_ ( Q .\/ T ) <-> ( ( Q .\/ T ) ./\ T ) = T ) )
72 7 23 61 71 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .<_ ( Q .\/ T ) <-> ( ( Q .\/ T ) ./\ T ) = T ) )
73 70 72 mpbid 
 |-  ( ( ( K e. HL /\ 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 ) ./\ T ) = T )
74 68 73 oveq12d 
 |-  ( ( ( K e. HL /\ 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 .\/ ( P .\/ S ) ) ./\ ( ( Q .\/ T ) ./\ T ) ) = ( ( ( P .\/ Q ) .\/ S ) ./\ T ) )
75 63 74 eqtr2d 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) ./\ T ) = ( ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) ./\ T ) )
76 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> Q .<_ ( Q .\/ T ) )
77 6 9 13 76 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .<_ ( Q .\/ T ) )
78 5 1 2 3 4 atmod1i1 
 |-  ( ( K e. HL /\ ( Q e. A /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) /\ Q .<_ ( Q .\/ T ) ) -> ( Q .\/ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) = ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) )
79 6 9 57 61 77 78 syl131anc 
 |-  ( ( ( K e. HL /\ 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 .\/ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) = ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) )
80 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ U e. A /\ Q e. A ) -> Q .<_ ( U .\/ Q ) )
81 6 30 9 80 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .<_ ( U .\/ Q ) )
82 simp13 
 |-  ( ( ( K e. HL /\ 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 ) )
83 simp12 
 |-  ( ( ( K e. HL /\ 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 )
84 83 oveq1d 
 |-  ( ( ( K e. HL /\ 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 .\/ U ) = ( R .\/ U ) )
85 2 4 hlatjcom 
 |-  ( ( K e. HL /\ Q e. A /\ U e. A ) -> ( Q .\/ U ) = ( U .\/ Q ) )
86 6 9 30 85 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ U ) = ( U .\/ Q ) )
87 84 86 eqtr3d 
 |-  ( ( ( K e. HL /\ 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 ) = ( U .\/ Q ) )
88 82 87 breqtrd 
 |-  ( ( ( K e. HL /\ 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 ) ) .<_ ( U .\/ Q ) )
89 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
90 7 57 61 89 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) )
91 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ U e. A /\ Q e. A ) -> ( U .\/ Q ) e. ( Base ` K ) )
92 6 30 9 91 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ Q ) e. ( Base ` K ) )
93 5 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) /\ ( U .\/ Q ) e. ( Base ` K ) ) ) -> ( ( Q .<_ ( U .\/ Q ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ Q ) ) <-> ( Q .\/ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) .<_ ( U .\/ Q ) ) )
94 7 29 90 92 93 syl13anc 
 |-  ( ( ( K e. HL /\ 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 .<_ ( U .\/ Q ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ Q ) ) <-> ( Q .\/ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) .<_ ( U .\/ Q ) ) )
95 81 88 94 mpbi2and 
 |-  ( ( ( K e. HL /\ 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 .\/ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) .<_ ( U .\/ Q ) )
96 79 95 eqbrtrrd 
 |-  ( ( ( K e. HL /\ 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 .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ Q ) )
97 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> T .<_ ( T .\/ U ) )
98 6 13 30 97 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .<_ ( T .\/ U ) )
99 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( Q .\/ ( P .\/ S ) ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
100 7 59 61 99 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
101 5 1 3 latmlem12 
 |-  ( ( K e. Lat /\ ( ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) /\ ( U .\/ Q ) e. ( Base ` K ) ) /\ ( T e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) ) -> ( ( ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ Q ) /\ T .<_ ( T .\/ U ) ) -> ( ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) ./\ T ) .<_ ( ( U .\/ Q ) ./\ ( T .\/ U ) ) ) )
102 7 100 92 23 32 101 syl122anc 
 |-  ( ( ( K e. HL /\ 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 .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ Q ) /\ T .<_ ( T .\/ U ) ) -> ( ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) ./\ T ) .<_ ( ( U .\/ Q ) ./\ ( T .\/ U ) ) ) )
103 96 98 102 mp2and 
 |-  ( ( ( K e. HL /\ 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 .\/ ( P .\/ S ) ) ./\ ( Q .\/ T ) ) ./\ T ) .<_ ( ( U .\/ Q ) ./\ ( T .\/ U ) ) )
104 75 103 eqbrtrd 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) ./\ T ) .<_ ( ( U .\/ Q ) ./\ ( T .\/ U ) ) )
105 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> U .<_ ( T .\/ U ) )
106 6 13 30 105 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) )
107 5 1 2 3 4 atmod1i1 
 |-  ( ( K e. HL /\ ( U e. A /\ Q e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) /\ U .<_ ( T .\/ U ) ) -> ( U .\/ ( Q ./\ ( T .\/ U ) ) ) = ( ( U .\/ Q ) ./\ ( T .\/ U ) ) )
108 6 30 29 32 106 107 syl131anc 
 |-  ( ( ( K e. HL /\ 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 .\/ ( Q ./\ ( T .\/ U ) ) ) = ( ( U .\/ Q ) ./\ ( T .\/ U ) ) )
109 5 4 atbase 
 |-  ( U e. A -> U e. ( Base ` K ) )
110 30 109 syl 
 |-  ( ( ( K e. HL /\ 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 ) )
111 5 2 latjcom 
 |-  ( ( K e. Lat /\ U e. ( Base ` K ) /\ ( Q ./\ ( T .\/ U ) ) e. ( Base ` K ) ) -> ( U .\/ ( Q ./\ ( T .\/ U ) ) ) = ( ( Q ./\ ( T .\/ U ) ) .\/ U ) )
112 7 110 34 111 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ ( Q ./\ ( T .\/ U ) ) ) = ( ( Q ./\ ( T .\/ U ) ) .\/ U ) )
113 108 112 eqtr3d 
 |-  ( ( ( K e. HL /\ 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 .\/ Q ) ./\ ( T .\/ U ) ) = ( ( Q ./\ ( T .\/ U ) ) .\/ U ) )
114 104 113 breqtrd 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) ./\ T ) .<_ ( ( Q ./\ ( T .\/ U ) ) .\/ U ) )
115 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( Q ./\ ( T .\/ U ) ) e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> ( ( Q ./\ ( T .\/ U ) ) .\/ U ) e. ( Base ` K ) )
116 7 34 110 115 syl3anc 
 |-  ( ( ( K e. HL /\ 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 .\/ U ) ) .\/ U ) e. ( Base ` K ) )
117 5 1 2 latjlej1 
 |-  ( ( K e. Lat /\ ( ( ( ( P .\/ Q ) .\/ S ) ./\ T ) e. ( Base ` K ) /\ ( ( Q ./\ ( T .\/ U ) ) .\/ U ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .<_ ( ( Q ./\ ( T .\/ U ) ) .\/ U ) -> ( ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .\/ S ) .<_ ( ( ( Q ./\ ( T .\/ U ) ) .\/ U ) .\/ S ) ) )
118 7 25 116 19 117 syl13anc 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) ./\ T ) .<_ ( ( Q ./\ ( T .\/ U ) ) .\/ U ) -> ( ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .\/ S ) .<_ ( ( ( Q ./\ ( T .\/ U ) ) .\/ U ) .\/ S ) ) )
119 114 118 mpd 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) ./\ T ) .\/ S ) .<_ ( ( ( Q ./\ ( T .\/ U ) ) .\/ U ) .\/ S ) )
120 5 2 latjass 
 |-  ( ( K e. Lat /\ ( ( Q ./\ ( T .\/ U ) ) e. ( Base ` K ) /\ U e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( ( Q ./\ ( T .\/ U ) ) .\/ U ) .\/ S ) = ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) )
121 7 34 110 19 120 syl13anc 
 |-  ( ( ( K e. HL /\ 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 .\/ U ) ) .\/ U ) .\/ S ) = ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) )
122 119 121 breqtrd 
 |-  ( ( ( K e. HL /\ 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 ) .\/ S ) ./\ T ) .\/ S ) .<_ ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) )
123 5 1 7 17 27 38 53 122 lattrd 
 |-  ( ( ( K e. HL /\ 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 ) ./\ ( S .\/ T ) ) .<_ ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) )
124 5 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( P .\/ Q ) )
125 7 11 15 124 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) ./\ ( S .\/ T ) ) .<_ ( P .\/ Q ) )
126 5 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( ( ( P .\/ Q ) ./\ ( S .\/ T ) ) e. ( Base ` K ) /\ ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) /\ ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( P .\/ Q ) ) <-> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) ./\ ( P .\/ Q ) ) ) )
127 7 17 38 11 126 syl13anc 
 |-  ( ( ( K e. HL /\ 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 ) ./\ ( S .\/ T ) ) .<_ ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) /\ ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( P .\/ Q ) ) <-> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) ./\ ( P .\/ Q ) ) ) )
128 123 125 127 mpbi2and 
 |-  ( ( ( K e. HL /\ 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 ) ./\ ( S .\/ T ) ) .<_ ( ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) ./\ ( P .\/ Q ) ) )
129 5 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
130 8 129 syl 
 |-  ( ( ( K e. HL /\ 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 ) )
131 5 1 2 3 latmlej12 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) /\ P e. ( Base ` K ) ) ) -> ( Q ./\ ( T .\/ U ) ) .<_ ( P .\/ Q ) )
132 7 29 32 130 131 syl13anc 
 |-  ( ( ( K e. HL /\ 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 .\/ U ) ) .<_ ( P .\/ Q ) )
133 5 1 2 3 4 llnmod1i2 
 |-  ( ( ( K e. HL /\ ( Q ./\ ( T .\/ U ) ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) /\ ( U e. A /\ S e. A ) /\ ( Q ./\ ( T .\/ U ) ) .<_ ( P .\/ Q ) ) -> ( ( Q ./\ ( T .\/ U ) ) .\/ ( ( U .\/ S ) ./\ ( P .\/ Q ) ) ) = ( ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) ./\ ( P .\/ Q ) ) )
134 6 34 11 30 12 132 133 syl321anc 
 |-  ( ( ( K e. HL /\ 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 .\/ U ) ) .\/ ( ( U .\/ S ) ./\ ( P .\/ Q ) ) ) = ( ( ( Q ./\ ( T .\/ U ) ) .\/ ( U .\/ S ) ) ./\ ( P .\/ Q ) ) )
135 2 4 hlatjidm 
 |-  ( ( K e. HL /\ Q e. A ) -> ( Q .\/ Q ) = Q )
136 6 9 135 syl2anc 
 |-  ( ( ( K e. HL /\ 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 .\/ Q ) = Q )
137 83 oveq2d 
 |-  ( ( ( K e. HL /\ 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 .\/ Q ) = ( Q .\/ R ) )
138 136 137 eqtr3d 
 |-  ( ( ( K e. HL /\ 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 = ( Q .\/ R ) )
139 138 oveq1d 
 |-  ( ( ( K e. HL /\ 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 .\/ U ) ) = ( ( Q .\/ R ) ./\ ( T .\/ U ) ) )
140 5 3 latmcom 
 |-  ( ( K e. Lat /\ ( U .\/ S ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( U .\/ S ) ./\ ( P .\/ Q ) ) = ( ( P .\/ Q ) ./\ ( U .\/ S ) ) )
141 7 36 11 140 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) ./\ ( P .\/ Q ) ) = ( ( P .\/ Q ) ./\ ( U .\/ S ) ) )
142 2 4 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
143 6 8 9 142 syl3anc 
 |-  ( ( ( K e. HL /\ 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 ) = ( Q .\/ P ) )
144 83 oveq1d 
 |-  ( ( ( K e. HL /\ 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 .\/ P ) = ( R .\/ P ) )
145 143 144 eqtrd 
 |-  ( ( ( K e. HL /\ 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 .\/ P ) )
146 145 oveq1d 
 |-  ( ( ( K e. HL /\ 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 ) ./\ ( U .\/ S ) ) = ( ( R .\/ P ) ./\ ( U .\/ S ) ) )
147 141 146 eqtrd 
 |-  ( ( ( K e. HL /\ 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 ) ./\ ( P .\/ Q ) ) = ( ( R .\/ P ) ./\ ( U .\/ S ) ) )
148 139 147 oveq12d 
 |-  ( ( ( K e. HL /\ 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 .\/ U ) ) .\/ ( ( U .\/ S ) ./\ ( P .\/ Q ) ) ) = ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )
149 134 148 eqtr3d 
 |-  ( ( ( K e. HL /\ 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 .\/ U ) ) .\/ ( U .\/ S ) ) ./\ ( P .\/ Q ) ) = ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )
150 128 149 breqtrd 
 |-  ( ( ( K e. HL /\ 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 ) ./\ ( S .\/ T ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )