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Metamath Proof Explorer

Theorem dalawlem7

Description: Lemma for dalaw . Second 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 dalawlem7 
|- ( ( ( 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 ) .\/ 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 /\ ( ( 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 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 )
13 5 4 atbase 
 |-  ( S e. A -> S 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 ) ) -> S e. ( Base ` K ) )
15 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ S ) 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 ) .\/ S ) e. ( Base ` K ) )
17 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 )
18 5 4 atbase 
 |-  ( T e. A -> T 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 ) ) -> T e. ( Base ` K ) )
20 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> ( ( ( P .\/ Q ) .\/ S ) ./\ T ) 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 ) .\/ S ) ./\ T ) 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 2 4 hlatjcl 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) e. ( Base ` K ) )
27 6 17 25 26 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 ) )
28 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) -> ( ( Q .\/ R ) ./\ ( T .\/ 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 ) ./\ ( T .\/ 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 12 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 ) ./\ ( T .\/ U ) ) e. ( Base ` K ) /\ ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) ) -> ( ( ( Q .\/ R ) ./\ ( T .\/ 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 ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) e. ( Base ` K ) )
38 hlol 
 |-  ( K e. HL -> K e. OL )
39 6 38 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 )
40 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) )
41 6 8 12 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 ) ) -> ( P .\/ S ) e. ( Base ` K ) )
42 5 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
43 9 42 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 ) ) -> Q e. ( Base ` K ) )
44 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( ( P .\/ S ) .\/ Q ) e. ( Base ` K ) )
45 7 41 43 44 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 ) e. ( Base ` K ) )
46 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> ( Q .\/ T ) e. ( Base ` K ) )
47 6 9 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 ) ) -> ( Q .\/ T ) e. ( Base ` K ) )
48 5 3 latmassOLD 
 |-  ( ( K e. OL /\ ( ( ( P .\/ S ) .\/ Q ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) ) -> ( ( ( ( P .\/ S ) .\/ Q ) ./\ ( Q .\/ T ) ) ./\ T ) = ( ( ( P .\/ S ) .\/ Q ) ./\ ( ( Q .\/ T ) ./\ T ) ) )
49 39 45 47 19 48 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 ) ./\ ( Q .\/ T ) ) ./\ T ) = ( ( ( P .\/ S ) .\/ Q ) ./\ ( ( Q .\/ T ) ./\ T ) ) )
50 2 4 hlatj32 
 |-  ( ( K e. HL /\ ( P e. A /\ S e. A /\ Q e. A ) ) -> ( ( P .\/ S ) .\/ Q ) = ( ( P .\/ Q ) .\/ S ) )
51 6 8 12 9 50 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 ) = ( ( P .\/ Q ) .\/ S ) )
52 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> T .<_ ( Q .\/ T ) )
53 6 9 17 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 ) ) -> T .<_ ( Q .\/ T ) )
54 5 1 3 latleeqm2 
 |-  ( ( K e. Lat /\ T e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( T .<_ ( Q .\/ T ) <-> ( ( Q .\/ T ) ./\ T ) = T ) )
55 7 19 47 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 ) ) -> ( T .<_ ( Q .\/ T ) <-> ( ( Q .\/ T ) ./\ T ) = T ) )
56 53 55 mpbid 
 |-  ( ( ( 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 ) ./\ T ) = T )
57 51 56 oveq12d 
 |-  ( ( ( 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 ) ./\ ( ( Q .\/ T ) ./\ T ) ) = ( ( ( P .\/ Q ) .\/ S ) ./\ T ) )
58 49 57 eqtr2d 
 |-  ( ( ( 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 ) .\/ S ) ./\ T ) = ( ( ( ( P .\/ S ) .\/ Q ) ./\ ( Q .\/ T ) ) ./\ T ) )
59 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 ) )
60 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
61 7 41 47 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 .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
62 5 1 2 latjlej1 
 |-  ( ( K e. Lat /\ ( ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) ) -> ( ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) -> ( ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .\/ Q ) .<_ ( ( Q .\/ R ) .\/ Q ) ) )
63 7 61 24 43 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 .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) -> ( ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .\/ Q ) .<_ ( ( Q .\/ R ) .\/ Q ) ) )
64 59 63 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 .\/ S ) ./\ ( Q .\/ T ) ) .\/ Q ) .<_ ( ( Q .\/ R ) .\/ Q ) )
65 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> Q .<_ ( Q .\/ T ) )
66 6 9 17 65 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 .<_ ( Q .\/ T ) )
67 5 1 2 3 4 atmod4i1 
 |-  ( ( K e. HL /\ ( Q e. A /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) /\ Q .<_ ( Q .\/ T ) ) -> ( ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .\/ Q ) = ( ( ( P .\/ S ) .\/ Q ) ./\ ( Q .\/ T ) ) )
68 6 9 41 47 66 67 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 .\/ S ) ./\ ( Q .\/ T ) ) .\/ Q ) = ( ( ( P .\/ S ) .\/ Q ) ./\ ( Q .\/ T ) ) )
69 2 4 hlatj32 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ Q e. A ) ) -> ( ( Q .\/ R ) .\/ Q ) = ( ( Q .\/ Q ) .\/ R ) )
70 6 9 22 9 69 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 ) .\/ Q ) = ( ( Q .\/ Q ) .\/ R ) )
71 5 2 latjidm 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) ) -> ( Q .\/ Q ) = Q )
72 7 43 71 syl2anc 
 |-  ( ( ( 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 .\/ Q ) = Q )
73 72 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 ) ) -> ( ( Q .\/ Q ) .\/ R ) = ( Q .\/ R ) )
74 70 73 eqtrd 
 |-  ( ( ( 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 ) = ( Q .\/ R ) )
75 64 68 74 3brtr3d 
 |-  ( ( ( 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 ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) )
76 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> T .<_ ( T .\/ U ) )
77 6 17 25 76 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 .<_ ( T .\/ U ) )
78 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( ( P .\/ S ) .\/ Q ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( ( P .\/ S ) .\/ Q ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
79 7 45 47 78 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 ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) )
80 5 1 3 latmlem12 
 |-  ( ( K e. Lat /\ ( ( ( ( P .\/ S ) .\/ Q ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) /\ ( T e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) ) -> ( ( ( ( ( P .\/ S ) .\/ Q ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ T .<_ ( T .\/ U ) ) -> ( ( ( ( P .\/ S ) .\/ Q ) ./\ ( Q .\/ T ) ) ./\ T ) .<_ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) ) )
81 7 79 24 19 27 80 syl122anc 
 |-  ( ( ( 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 ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ T .<_ ( T .\/ U ) ) -> ( ( ( ( P .\/ S ) .\/ Q ) ./\ ( Q .\/ T ) ) ./\ T ) .<_ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) ) )
82 75 77 81 mp2and 
 |-  ( ( ( 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 ) ./\ ( Q .\/ T ) ) ./\ T ) .<_ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) )
83 58 82 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 ) ) -> ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .<_ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) )
84 5 1 2 latlej1 
 |-  ( ( K e. Lat /\ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) e. ( Base ` K ) /\ ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) ) -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )
85 7 29 35 84 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 ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )
86 5 1 7 21 29 37 83 85 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 ) .\/ S ) ./\ T ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )