Step |
Hyp |
Ref |
Expression |
1 |
|
dalawlem.l |
|- .<_ = ( le ` K ) |
2 |
|
dalawlem.j |
|- .\/ = ( join ` K ) |
3 |
|
dalawlem.m |
|- ./\ = ( meet ` K ) |
4 |
|
dalawlem.a |
|- A = ( Atoms ` K ) |
5 |
|
dalawlem.o |
|- O = ( LPlanes ` K ) |
6 |
|
simp11 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> K e. HL ) |
7 |
6
|
hllatd |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> K e. Lat ) |
8 |
|
simp121 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> P e. A ) |
9 |
|
simp131 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> S e. A ) |
10 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
11 |
10 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) ) |
12 |
6 8 9 11
|
syl3anc |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( P .\/ S ) e. ( Base ` K ) ) |
13 |
|
simp122 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> Q e. A ) |
14 |
|
simp132 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> T e. A ) |
15 |
10 2 4
|
hlatjcl |
|- ( ( K e. HL /\ Q e. A /\ T e. A ) -> ( Q .\/ T ) e. ( Base ` K ) ) |
16 |
6 13 14 15
|
syl3anc |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( Q .\/ T ) e. ( Base ` K ) ) |
17 |
10 3
|
latmcl |
|- ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) ) |
18 |
7 12 16 17
|
syl3anc |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) ) |
19 |
6 18
|
jca |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) ) ) |
20 |
|
simp12 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( P e. A /\ Q e. A /\ R e. A ) ) |
21 |
|
simp13 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( S e. A /\ T e. A /\ U e. A ) ) |
22 |
|
simp2l |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( ( P .\/ Q ) .\/ R ) e. O ) |
23 |
|
simp2r |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( ( S .\/ T ) .\/ U ) e. O ) |
24 |
|
simp31 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) ) |
25 |
|
simp32 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) ) |
26 |
10 1 3
|
latmle1 |
|- ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ S ) ) |
27 |
7 12 16 26
|
syl3anc |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ S ) ) |
28 |
10 1 3
|
latmle2 |
|- ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ T ) ) |
29 |
7 12 16 28
|
syl3anc |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ T ) ) |
30 |
|
simp33 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) |
31 |
27 29 30
|
3jca |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ S ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ T ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) |
32 |
|
eqid |
|- ( ( P .\/ Q ) ./\ ( S .\/ T ) ) = ( ( P .\/ Q ) ./\ ( S .\/ T ) ) |
33 |
|
eqid |
|- ( ( Q .\/ R ) ./\ ( T .\/ U ) ) = ( ( Q .\/ R ) ./\ ( T .\/ U ) ) |
34 |
|
eqid |
|- ( ( R .\/ P ) ./\ ( U .\/ S ) ) = ( ( R .\/ P ) ./\ ( U .\/ S ) ) |
35 |
10 1 2 4 3 5 32 33 34
|
dath2 |
|- ( ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ S ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ T ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) ) |
36 |
19 20 21 22 23 24 25 31 35
|
syl323anc |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( ( ( P .\/ Q ) .\/ R ) e. O /\ ( ( S .\/ T ) .\/ U ) e. O ) /\ ( ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) ) /\ ( -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( S .\/ T ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( T .\/ U ) /\ -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( U .\/ S ) ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) ) |