Step |
Hyp |
Ref |
Expression |
1 |
|
2atm.l |
|- .<_ = ( le ` K ) |
2 |
|
2atm.j |
|- .\/ = ( join ` K ) |
3 |
|
2atm.m |
|- ./\ = ( meet ` K ) |
4 |
|
2atm.a |
|- A = ( Atoms ` K ) |
5 |
|
simp31 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T .<_ ( P .\/ Q ) ) |
6 |
|
simp32 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T .<_ ( R .\/ S ) ) |
7 |
|
simp11 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> K e. HL ) |
8 |
7
|
hllatd |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> K e. Lat ) |
9 |
|
simp23 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T e. A ) |
10 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
11 |
10 4
|
atbase |
|- ( T e. A -> T e. ( Base ` K ) ) |
12 |
9 11
|
syl |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T e. ( Base ` K ) ) |
13 |
|
simp12 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> P e. A ) |
14 |
10 4
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
15 |
13 14
|
syl |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> P e. ( Base ` K ) ) |
16 |
|
simp13 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> Q e. A ) |
17 |
10 4
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
18 |
16 17
|
syl |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> Q e. ( Base ` K ) ) |
19 |
10 2
|
latjcl |
|- ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
20 |
8 15 18 19
|
syl3anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
21 |
|
simp21 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> R e. A ) |
22 |
|
simp22 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> S e. A ) |
23 |
10 2 4
|
hlatjcl |
|- ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) ) |
24 |
7 21 22 23
|
syl3anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( R .\/ S ) e. ( Base ` K ) ) |
25 |
10 1 3
|
latlem12 |
|- ( ( K e. Lat /\ ( T e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( R .\/ S ) e. ( Base ` K ) ) ) -> ( ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) ) <-> T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) ) |
26 |
8 12 20 24 25
|
syl13anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) ) <-> T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) ) |
27 |
5 6 26
|
mpbi2and |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) |
28 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
29 |
7 28
|
syl |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> K e. AtLat ) |
30 |
10 3
|
latmcl |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( R .\/ S ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. ( Base ` K ) ) |
31 |
8 20 24 30
|
syl3anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. ( Base ` K ) ) |
32 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
33 |
10 1 32 4
|
atlen0 |
|- ( ( ( K e. AtLat /\ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. ( Base ` K ) /\ T e. A ) /\ T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) =/= ( 0. ` K ) ) |
34 |
29 31 9 27 33
|
syl31anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) =/= ( 0. ` K ) ) |
35 |
34
|
neneqd |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> -. ( ( P .\/ Q ) ./\ ( R .\/ S ) ) = ( 0. ` K ) ) |
36 |
|
simp33 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( P .\/ Q ) =/= ( R .\/ S ) ) |
37 |
2 3 32 4
|
2atmat0 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A \/ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) = ( 0. ` K ) ) ) |
38 |
7 13 16 21 22 36 37
|
syl33anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A \/ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) = ( 0. ` K ) ) ) |
39 |
38
|
ord |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( -. ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) = ( 0. ` K ) ) ) |
40 |
35 39
|
mt3d |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A ) |
41 |
1 4
|
atcmp |
|- ( ( K e. AtLat /\ T e. A /\ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) e. A ) -> ( T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) <-> T = ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) ) |
42 |
29 9 40 41
|
syl3anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> ( T .<_ ( ( P .\/ Q ) ./\ ( R .\/ S ) ) <-> T = ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) ) |
43 |
27 42
|
mpbid |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( T .<_ ( P .\/ Q ) /\ T .<_ ( R .\/ S ) /\ ( P .\/ Q ) =/= ( R .\/ S ) ) ) -> T = ( ( P .\/ Q ) ./\ ( R .\/ S ) ) ) |