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Metamath Proof Explorer

Theorem 3atlem2

Description: Lemma for 3at . (Contributed by NM, 22-Jun-2012)

Ref Expression
Hypotheses 3at.l 
|- .<_ = ( le ` K )
3at.j 
|- .\/ = ( join ` K )
3at.a 
|- A = ( Atoms ` K )
Assertion 3atlem2 
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )

Proof

Step Hyp Ref Expression
1 3at.l 
 |-  .<_ = ( le ` K )
2 3at.j 
 |-  .\/ = ( join ` K )
3 3at.a 
 |-  A = ( Atoms ` K )
4 simp3 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) )
5 simp11 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> K e. HL )
6 5 hllatd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> K e. Lat )
7 simp121 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> P e. A )
8 simp122 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> Q e. A )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 9 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
11 5 7 8 10 syl3anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
12 simp123 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> R e. A )
13 9 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
14 12 13 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> R e. ( Base ` K ) )
15 simp131 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> S e. A )
16 simp132 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> T e. A )
17 9 2 3 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) )
18 5 15 16 17 syl3anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( S .\/ T ) e. ( Base ` K ) )
19 simp133 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> U e. A )
20 9 3 atbase 
 |-  ( U e. A -> U e. ( Base ` K ) )
21 19 20 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> U e. ( Base ` K ) )
22 9 2 latjcl 
 |-  ( ( K e. Lat /\ ( S .\/ T ) e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) )
23 6 18 21 22 syl3anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) )
24 9 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) /\ ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .<_ ( ( S .\/ T ) .\/ U ) /\ R .<_ ( ( S .\/ T ) .\/ U ) ) <-> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) )
25 6 11 14 23 24 syl13anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( ( P .\/ Q ) .<_ ( ( S .\/ T ) .\/ U ) /\ R .<_ ( ( S .\/ T ) .\/ U ) ) <-> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) )
26 4 25 mpbird 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .<_ ( ( S .\/ T ) .\/ U ) /\ R .<_ ( ( S .\/ T ) .\/ U ) ) )
27 26 simprd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> R .<_ ( ( S .\/ T ) .\/ U ) )
28 2 3 hlatjass 
 |-  ( ( K e. HL /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( S .\/ T ) .\/ U ) = ( S .\/ ( T .\/ U ) ) )
29 5 15 16 19 28 syl13anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( S .\/ T ) .\/ U ) = ( S .\/ ( T .\/ U ) ) )
30 simp22r 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> P .<_ ( T .\/ U ) )
31 simp22l 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> P =/= U )
32 1 2 3 hlatexchb2 
 |-  ( ( K e. HL /\ ( P e. A /\ T e. A /\ U e. A ) /\ P =/= U ) -> ( P .<_ ( T .\/ U ) <-> ( P .\/ U ) = ( T .\/ U ) ) )
33 5 7 16 19 31 32 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( P .<_ ( T .\/ U ) <-> ( P .\/ U ) = ( T .\/ U ) ) )
34 30 33 mpbid 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( P .\/ U ) = ( T .\/ U ) )
35 34 oveq2d 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( S .\/ ( P .\/ U ) ) = ( S .\/ ( T .\/ U ) ) )
36 29 35 eqtr4d 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( S .\/ T ) .\/ U ) = ( S .\/ ( P .\/ U ) ) )
37 2 3 hlatjass 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ U e. A ) ) -> ( ( P .\/ Q ) .\/ U ) = ( P .\/ ( Q .\/ U ) ) )
38 5 7 8 19 37 syl13anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ U ) = ( P .\/ ( Q .\/ U ) ) )
39 2 3 hlatj12 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ U e. A ) ) -> ( P .\/ ( Q .\/ U ) ) = ( Q .\/ ( P .\/ U ) ) )
40 5 7 8 19 39 syl13anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( P .\/ ( Q .\/ U ) ) = ( Q .\/ ( P .\/ U ) ) )
41 2 3 hlatj32 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( P .\/ R ) .\/ Q ) )
42 5 7 8 12 41 syl13anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( P .\/ R ) .\/ Q ) )
43 4 42 29 3brtr3d 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ R ) .\/ Q ) .<_ ( S .\/ ( T .\/ U ) ) )
44 9 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) e. ( Base ` K ) )
45 5 7 12 44 syl3anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( P .\/ R ) e. ( Base ` K ) )
46 9 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
47 8 46 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> Q e. ( Base ` K ) )
48 9 3 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
49 15 48 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> S e. ( Base ` K ) )
50 9 2 3 hlatjcl 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) e. ( Base ` K ) )
51 5 16 19 50 syl3anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( T .\/ U ) e. ( Base ` K ) )
52 9 2 latjcl 
 |-  ( ( K e. Lat /\ S e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) -> ( S .\/ ( T .\/ U ) ) e. ( Base ` K ) )
53 6 49 51 52 syl3anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( S .\/ ( T .\/ U ) ) e. ( Base ` K ) )
54 9 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( ( P .\/ R ) e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ ( S .\/ ( T .\/ U ) ) e. ( Base ` K ) ) ) -> ( ( ( P .\/ R ) .<_ ( S .\/ ( T .\/ U ) ) /\ Q .<_ ( S .\/ ( T .\/ U ) ) ) <-> ( ( P .\/ R ) .\/ Q ) .<_ ( S .\/ ( T .\/ U ) ) ) )
55 6 45 47 53 54 syl13anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( ( P .\/ R ) .<_ ( S .\/ ( T .\/ U ) ) /\ Q .<_ ( S .\/ ( T .\/ U ) ) ) <-> ( ( P .\/ R ) .\/ Q ) .<_ ( S .\/ ( T .\/ U ) ) ) )
56 43 55 mpbird 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ R ) .<_ ( S .\/ ( T .\/ U ) ) /\ Q .<_ ( S .\/ ( T .\/ U ) ) ) )
57 56 simprd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> Q .<_ ( S .\/ ( T .\/ U ) ) )
58 57 35 breqtrrd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> Q .<_ ( S .\/ ( P .\/ U ) ) )
59 9 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ U e. A ) -> ( P .\/ U ) e. ( Base ` K ) )
60 5 7 19 59 syl3anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( P .\/ U ) e. ( Base ` K ) )
61 simp23 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> -. Q .<_ ( P .\/ U ) )
62 9 1 2 3 hlexchb2 
 |-  ( ( K e. HL /\ ( Q e. A /\ S e. A /\ ( P .\/ U ) e. ( Base ` K ) ) /\ -. Q .<_ ( P .\/ U ) ) -> ( Q .<_ ( S .\/ ( P .\/ U ) ) <-> ( Q .\/ ( P .\/ U ) ) = ( S .\/ ( P .\/ U ) ) ) )
63 5 8 15 60 61 62 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( Q .<_ ( S .\/ ( P .\/ U ) ) <-> ( Q .\/ ( P .\/ U ) ) = ( S .\/ ( P .\/ U ) ) ) )
64 58 63 mpbid 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( Q .\/ ( P .\/ U ) ) = ( S .\/ ( P .\/ U ) ) )
65 38 40 64 3eqtrd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ U ) = ( S .\/ ( P .\/ U ) ) )
66 36 65 eqtr4d 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( S .\/ T ) .\/ U ) = ( ( P .\/ Q ) .\/ U ) )
67 27 66 breqtrd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> R .<_ ( ( P .\/ Q ) .\/ U ) )
68 simp21 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> -. R .<_ ( P .\/ Q ) )
69 9 1 2 3 hlexchb1 
 |-  ( ( K e. HL /\ ( R e. A /\ U e. A /\ ( P .\/ Q ) e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( R .<_ ( ( P .\/ Q ) .\/ U ) <-> ( ( P .\/ Q ) .\/ R ) = ( ( P .\/ Q ) .\/ U ) ) )
70 5 12 19 11 68 69 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( R .<_ ( ( P .\/ Q ) .\/ U ) <-> ( ( P .\/ Q ) .\/ R ) = ( ( P .\/ Q ) .\/ U ) ) )
71 67 70 mpbid 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( P .\/ Q ) .\/ U ) )
72 71 66 eqtr4d 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ ( P =/= U /\ P .<_ ( T .\/ U ) ) /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )