Step |
Hyp |
Ref |
Expression |
1 |
|
2lplnja.l |
|- .<_ = ( le ` K ) |
2 |
|
2lplnja.j |
|- .\/ = ( join ` K ) |
3 |
|
2lplnja.a |
|- A = ( Atoms ` K ) |
4 |
|
2lplnja.v |
|- V = ( LVols ` K ) |
5 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
6 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> K e. HL ) |
7 |
6
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> K e. Lat ) |
8 |
|
simp121 |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> P e. A ) |
9 |
|
simp122 |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> Q e. A ) |
10 |
5 2 3
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
11 |
6 8 9 10
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
12 |
|
simp123 |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> R e. A ) |
13 |
5 3
|
atbase |
|- ( R e. A -> R e. ( Base ` K ) ) |
14 |
12 13
|
syl |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> R e. ( Base ` K ) ) |
15 |
5 2
|
latjcl |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) ) |
16 |
7 11 14 15
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) ) |
17 |
|
simp2l1 |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> S e. A ) |
18 |
|
simp2l2 |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> T e. A ) |
19 |
5 2 3
|
hlatjcl |
|- ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) ) |
20 |
6 17 18 19
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( S .\/ T ) e. ( Base ` K ) ) |
21 |
|
simp2l3 |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> U e. A ) |
22 |
5 3
|
atbase |
|- ( U e. A -> U e. ( Base ` K ) ) |
23 |
21 22
|
syl |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> U e. ( Base ` K ) ) |
24 |
5 2
|
latjcl |
|- ( ( K e. Lat /\ ( S .\/ T ) e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) ) |
25 |
7 20 23 24
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) ) |
26 |
5 2
|
latjcl |
|- ( ( K e. Lat /\ ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) /\ ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) e. ( Base ` K ) ) |
27 |
7 16 25 26
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) e. ( Base ` K ) ) |
28 |
|
simp11r |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> W e. V ) |
29 |
5 4
|
lvolbase |
|- ( W e. V -> W e. ( Base ` K ) ) |
30 |
28 29
|
syl |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> W e. ( Base ` K ) ) |
31 |
|
simp31 |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ W ) |
32 |
|
simp32 |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( S .\/ T ) .\/ U ) .<_ W ) |
33 |
5 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) /\ ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W ) <-> ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) .<_ W ) ) |
34 |
7 16 25 30 33
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W ) <-> ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) .<_ W ) ) |
35 |
31 32 34
|
mpbi2and |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) .<_ W ) |
36 |
5 1 2
|
latlej2 |
|- ( ( K e. Lat /\ ( S .\/ T ) e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> U .<_ ( ( S .\/ T ) .\/ U ) ) |
37 |
7 20 23 36
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> U .<_ ( ( S .\/ T ) .\/ U ) ) |
38 |
5 1 7 23 25 30 37 32
|
lattrd |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> U .<_ W ) |
39 |
5 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) /\ U e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ U .<_ W ) <-> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) .<_ W ) ) |
40 |
7 16 23 30 39
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ U .<_ W ) <-> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) .<_ W ) ) |
41 |
31 38 40
|
mpbi2and |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) .<_ W ) |
42 |
41
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) .<_ W ) |
43 |
6
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> K e. HL ) |
44 |
6 8 9
|
3jca |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( K e. HL /\ P e. A /\ Q e. A ) ) |
45 |
44
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( K e. HL /\ P e. A /\ Q e. A ) ) |
46 |
12 21
|
jca |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( R e. A /\ U e. A ) ) |
47 |
46
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( R e. A /\ U e. A ) ) |
48 |
|
simp13l |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> P =/= Q ) |
49 |
48
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> P =/= Q ) |
50 |
|
simp13r |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> -. R .<_ ( P .\/ Q ) ) |
51 |
50
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> -. R .<_ ( P .\/ Q ) ) |
52 |
|
simp33 |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) |
53 |
52
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) |
54 |
|
simplr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> S .<_ ( ( P .\/ Q ) .\/ R ) ) |
55 |
|
simpr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> T .<_ ( ( P .\/ Q ) .\/ R ) ) |
56 |
5 3
|
atbase |
|- ( S e. A -> S e. ( Base ` K ) ) |
57 |
17 56
|
syl |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> S e. ( Base ` K ) ) |
58 |
5 3
|
atbase |
|- ( T e. A -> T e. ( Base ` K ) ) |
59 |
18 58
|
syl |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> T e. ( Base ` K ) ) |
60 |
5 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ T e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) ) ) -> ( ( S .<_ ( ( P .\/ Q ) .\/ R ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) <-> ( S .\/ T ) .<_ ( ( P .\/ Q ) .\/ R ) ) ) |
61 |
7 57 59 16 60
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( S .<_ ( ( P .\/ Q ) .\/ R ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) <-> ( S .\/ T ) .<_ ( ( P .\/ Q ) .\/ R ) ) ) |
62 |
61
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( S .<_ ( ( P .\/ Q ) .\/ R ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) <-> ( S .\/ T ) .<_ ( ( P .\/ Q ) .\/ R ) ) ) |
63 |
54 55 62
|
mpbi2and |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( S .\/ T ) .<_ ( ( P .\/ Q ) .\/ R ) ) |
64 |
63
|
adantr |
|- ( ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) /\ U .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( S .\/ T ) .<_ ( ( P .\/ Q ) .\/ R ) ) |
65 |
|
simpr |
|- ( ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) /\ U .<_ ( ( P .\/ Q ) .\/ R ) ) -> U .<_ ( ( P .\/ Q ) .\/ R ) ) |
66 |
5 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( ( S .\/ T ) e. ( Base ` K ) /\ U e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) ) ) -> ( ( ( S .\/ T ) .<_ ( ( P .\/ Q ) .\/ R ) /\ U .<_ ( ( P .\/ Q ) .\/ R ) ) <-> ( ( S .\/ T ) .\/ U ) .<_ ( ( P .\/ Q ) .\/ R ) ) ) |
67 |
7 20 23 16 66
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( S .\/ T ) .<_ ( ( P .\/ Q ) .\/ R ) /\ U .<_ ( ( P .\/ Q ) .\/ R ) ) <-> ( ( S .\/ T ) .\/ U ) .<_ ( ( P .\/ Q ) .\/ R ) ) ) |
68 |
67
|
ad3antrrr |
|- ( ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) /\ U .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( S .\/ T ) .<_ ( ( P .\/ Q ) .\/ R ) /\ U .<_ ( ( P .\/ Q ) .\/ R ) ) <-> ( ( S .\/ T ) .\/ U ) .<_ ( ( P .\/ Q ) .\/ R ) ) ) |
69 |
64 65 68
|
mpbi2and |
|- ( ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) /\ U .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( S .\/ T ) .\/ U ) .<_ ( ( P .\/ Q ) .\/ R ) ) |
70 |
|
simp2l |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( S e. A /\ T e. A /\ U e. A ) ) |
71 |
|
simp12 |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( P e. A /\ Q e. A /\ R e. A ) ) |
72 |
|
simp2rr |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> -. U .<_ ( S .\/ T ) ) |
73 |
|
simp2rl |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> S =/= T ) |
74 |
1 2 3
|
3at |
|- ( ( ( K e. HL /\ ( S e. A /\ T e. A /\ U e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( -. U .<_ ( S .\/ T ) /\ S =/= T ) ) -> ( ( ( S .\/ T ) .\/ U ) .<_ ( ( P .\/ Q ) .\/ R ) <-> ( ( S .\/ T ) .\/ U ) = ( ( P .\/ Q ) .\/ R ) ) ) |
75 |
6 70 71 72 73 74
|
syl32anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( S .\/ T ) .\/ U ) .<_ ( ( P .\/ Q ) .\/ R ) <-> ( ( S .\/ T ) .\/ U ) = ( ( P .\/ Q ) .\/ R ) ) ) |
76 |
75
|
ad3antrrr |
|- ( ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) /\ U .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( S .\/ T ) .\/ U ) .<_ ( ( P .\/ Q ) .\/ R ) <-> ( ( S .\/ T ) .\/ U ) = ( ( P .\/ Q ) .\/ R ) ) ) |
77 |
69 76
|
mpbid |
|- ( ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) /\ U .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( S .\/ T ) .\/ U ) = ( ( P .\/ Q ) .\/ R ) ) |
78 |
77
|
eqcomd |
|- ( ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) /\ U .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) ) |
79 |
78
|
ex |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( U .<_ ( ( P .\/ Q ) .\/ R ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) ) ) |
80 |
79
|
necon3ad |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) -> -. U .<_ ( ( P .\/ Q ) .\/ R ) ) ) |
81 |
53 80
|
mpd |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> -. U .<_ ( ( P .\/ Q ) .\/ R ) ) |
82 |
1 2 3 4
|
lvoli2 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ U e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. U .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) e. V ) |
83 |
45 47 49 51 81 82
|
syl113anc |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) e. V ) |
84 |
28
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> W e. V ) |
85 |
1 4
|
lvolcmp |
|- ( ( K e. HL /\ ( ( ( P .\/ Q ) .\/ R ) .\/ U ) e. V /\ W e. V ) -> ( ( ( ( P .\/ Q ) .\/ R ) .\/ U ) .<_ W <-> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) = W ) ) |
86 |
43 83 84 85
|
syl3anc |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .\/ U ) .<_ W <-> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) = W ) ) |
87 |
42 86
|
mpbid |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) = W ) |
88 |
5 1 2
|
latjlej2 |
|- ( ( K e. Lat /\ ( U e. ( Base ` K ) /\ ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) ) ) -> ( U .<_ ( ( S .\/ T ) .\/ U ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) ) |
89 |
7 23 25 16 88
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( U .<_ ( ( S .\/ T ) .\/ U ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) ) |
90 |
37 89
|
mpd |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) |
91 |
90
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ U ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) |
92 |
87 91
|
eqbrtrrd |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ T .<_ ( ( P .\/ Q ) .\/ R ) ) -> W .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) |
93 |
5 2 3
|
hlatjcl |
|- ( ( K e. HL /\ S e. A /\ U e. A ) -> ( S .\/ U ) e. ( Base ` K ) ) |
94 |
6 17 21 93
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( S .\/ U ) e. ( Base ` K ) ) |
95 |
5 1 2
|
latlej2 |
|- ( ( K e. Lat /\ ( S .\/ U ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> T .<_ ( ( S .\/ U ) .\/ T ) ) |
96 |
7 94 59 95
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> T .<_ ( ( S .\/ U ) .\/ T ) ) |
97 |
2 3
|
hlatj32 |
|- ( ( K e. HL /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( S .\/ T ) .\/ U ) = ( ( S .\/ U ) .\/ T ) ) |
98 |
6 17 18 21 97
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( S .\/ T ) .\/ U ) = ( ( S .\/ U ) .\/ T ) ) |
99 |
96 98
|
breqtrrd |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> T .<_ ( ( S .\/ T ) .\/ U ) ) |
100 |
5 1 7 59 25 30 99 32
|
lattrd |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> T .<_ W ) |
101 |
5 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) /\ T e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ T .<_ W ) <-> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) .<_ W ) ) |
102 |
7 16 59 30 101
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ T .<_ W ) <-> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) .<_ W ) ) |
103 |
31 100 102
|
mpbi2and |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) .<_ W ) |
104 |
103
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) .<_ W ) |
105 |
6
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> K e. HL ) |
106 |
44
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( K e. HL /\ P e. A /\ Q e. A ) ) |
107 |
12 18
|
jca |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( R e. A /\ T e. A ) ) |
108 |
107
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( R e. A /\ T e. A ) ) |
109 |
48
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> P =/= Q ) |
110 |
50
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> -. R .<_ ( P .\/ Q ) ) |
111 |
|
simpr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> -. T .<_ ( ( P .\/ Q ) .\/ R ) ) |
112 |
1 2 3 4
|
lvoli2 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ T e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) e. V ) |
113 |
106 108 109 110 111 112
|
syl113anc |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) e. V ) |
114 |
28
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> W e. V ) |
115 |
1 4
|
lvolcmp |
|- ( ( K e. HL /\ ( ( ( P .\/ Q ) .\/ R ) .\/ T ) e. V /\ W e. V ) -> ( ( ( ( P .\/ Q ) .\/ R ) .\/ T ) .<_ W <-> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) = W ) ) |
116 |
105 113 114 115
|
syl3anc |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .\/ T ) .<_ W <-> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) = W ) ) |
117 |
104 116
|
mpbid |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) = W ) |
118 |
5 1 2
|
latjlej2 |
|- ( ( K e. Lat /\ ( T e. ( Base ` K ) /\ ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) ) ) -> ( T .<_ ( ( S .\/ T ) .\/ U ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) ) |
119 |
7 59 25 16 118
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( T .<_ ( ( S .\/ T ) .\/ U ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) ) |
120 |
99 119
|
mpd |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) |
121 |
120
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ T ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) |
122 |
117 121
|
eqbrtrrd |
|- ( ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. T .<_ ( ( P .\/ Q ) .\/ R ) ) -> W .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) |
123 |
92 122
|
pm2.61dan |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ S .<_ ( ( P .\/ Q ) .\/ R ) ) -> W .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) |
124 |
5 2 3
|
hlatjcl |
|- ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) e. ( Base ` K ) ) |
125 |
6 18 21 124
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( T .\/ U ) e. ( Base ` K ) ) |
126 |
5 1 2
|
latlej1 |
|- ( ( K e. Lat /\ S e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) -> S .<_ ( S .\/ ( T .\/ U ) ) ) |
127 |
7 57 125 126
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> S .<_ ( S .\/ ( T .\/ U ) ) ) |
128 |
5 2
|
latjass |
|- ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ T e. ( Base ` K ) /\ U e. ( Base ` K ) ) ) -> ( ( S .\/ T ) .\/ U ) = ( S .\/ ( T .\/ U ) ) ) |
129 |
7 57 59 23 128
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( S .\/ T ) .\/ U ) = ( S .\/ ( T .\/ U ) ) ) |
130 |
127 129
|
breqtrrd |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> S .<_ ( ( S .\/ T ) .\/ U ) ) |
131 |
5 1 7 57 25 30 130 32
|
lattrd |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> S .<_ W ) |
132 |
5 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) /\ S e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ S .<_ W ) <-> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ W ) ) |
133 |
7 16 57 30 132
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ S .<_ W ) <-> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ W ) ) |
134 |
31 131 133
|
mpbi2and |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ W ) |
135 |
134
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ W ) |
136 |
6
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> K e. HL ) |
137 |
44
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( K e. HL /\ P e. A /\ Q e. A ) ) |
138 |
12 17
|
jca |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( R e. A /\ S e. A ) ) |
139 |
138
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( R e. A /\ S e. A ) ) |
140 |
48
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> P =/= Q ) |
141 |
50
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> -. R .<_ ( P .\/ Q ) ) |
142 |
|
simpr |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> -. S .<_ ( ( P .\/ Q ) .\/ R ) ) |
143 |
1 2 3 4
|
lvoli2 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) e. V ) |
144 |
137 139 140 141 142 143
|
syl113anc |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) e. V ) |
145 |
28
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> W e. V ) |
146 |
1 4
|
lvolcmp |
|- ( ( K e. HL /\ ( ( ( P .\/ Q ) .\/ R ) .\/ S ) e. V /\ W e. V ) -> ( ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ W <-> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = W ) ) |
147 |
136 144 145 146
|
syl3anc |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ W <-> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = W ) ) |
148 |
135 147
|
mpbid |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = W ) |
149 |
5 1 2
|
latjlej2 |
|- ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) ) ) -> ( S .<_ ( ( S .\/ T ) .\/ U ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) ) |
150 |
7 57 25 16 149
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( S .<_ ( ( S .\/ T ) .\/ U ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) ) |
151 |
130 150
|
mpd |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) |
152 |
151
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) |
153 |
148 152
|
eqbrtrrd |
|- ( ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> W .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) |
154 |
123 153
|
pm2.61dan |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> W .<_ ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) ) |
155 |
5 1 7 27 30 35 154
|
latasymd |
|- ( ( ( ( K e. HL /\ W e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( S e. A /\ T e. A /\ U e. A ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) /\ ( ( ( P .\/ Q ) .\/ R ) .<_ W /\ ( ( S .\/ T ) .\/ U ) .<_ W /\ ( ( P .\/ Q ) .\/ R ) =/= ( ( S .\/ T ) .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ ( ( S .\/ T ) .\/ U ) ) = W ) |