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 ) ) |