Step |
Hyp |
Ref |
Expression |
1 |
|
4at.l |
|- .<_ = ( le ` K ) |
2 |
|
4at.j |
|- .\/ = ( join ` K ) |
3 |
|
4at.a |
|- A = ( Atoms ` K ) |
4 |
|
simp11 |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> K e. HL ) |
5 |
|
simp12 |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> P e. A ) |
6 |
|
simp13 |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> T e. A ) |
7 |
4
|
hllatd |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> K e. Lat ) |
8 |
|
simp21 |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> U e. A ) |
9 |
|
simp22 |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> V e. A ) |
10 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
11 |
10 2 3
|
hlatjcl |
|- ( ( K e. HL /\ U e. A /\ V e. A ) -> ( U .\/ V ) e. ( Base ` K ) ) |
12 |
4 8 9 11
|
syl3anc |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> ( U .\/ V ) e. ( Base ` K ) ) |
13 |
|
simp23 |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> W e. A ) |
14 |
10 3
|
atbase |
|- ( W e. A -> W e. ( Base ` K ) ) |
15 |
13 14
|
syl |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> W e. ( Base ` K ) ) |
16 |
10 2
|
latjcl |
|- ( ( K e. Lat /\ ( U .\/ V ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( U .\/ V ) .\/ W ) e. ( Base ` K ) ) |
17 |
7 12 15 16
|
syl3anc |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> ( ( U .\/ V ) .\/ W ) e. ( Base ` K ) ) |
18 |
|
simp3 |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> -. P .<_ ( ( U .\/ V ) .\/ W ) ) |
19 |
10 1 2 3
|
hlexchb2 |
|- ( ( K e. HL /\ ( P e. A /\ T e. A /\ ( ( U .\/ V ) .\/ W ) e. ( Base ` K ) ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> ( P .<_ ( T .\/ ( ( U .\/ V ) .\/ W ) ) <-> ( P .\/ ( ( U .\/ V ) .\/ W ) ) = ( T .\/ ( ( U .\/ V ) .\/ W ) ) ) ) |
20 |
4 5 6 17 18 19
|
syl131anc |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> ( P .<_ ( T .\/ ( ( U .\/ V ) .\/ W ) ) <-> ( P .\/ ( ( U .\/ V ) .\/ W ) ) = ( T .\/ ( ( U .\/ V ) .\/ W ) ) ) ) |
21 |
1 2 3
|
4atlem4a |
|- ( ( ( K e. HL /\ T e. A /\ U e. A ) /\ ( V e. A /\ W e. A ) ) -> ( ( T .\/ U ) .\/ ( V .\/ W ) ) = ( T .\/ ( ( U .\/ V ) .\/ W ) ) ) |
22 |
4 6 8 9 13 21
|
syl32anc |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> ( ( T .\/ U ) .\/ ( V .\/ W ) ) = ( T .\/ ( ( U .\/ V ) .\/ W ) ) ) |
23 |
22
|
breq2d |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> ( P .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) <-> P .<_ ( T .\/ ( ( U .\/ V ) .\/ W ) ) ) ) |
24 |
1 2 3
|
4atlem4a |
|- ( ( ( K e. HL /\ P e. A /\ U e. A ) /\ ( V e. A /\ W e. A ) ) -> ( ( P .\/ U ) .\/ ( V .\/ W ) ) = ( P .\/ ( ( U .\/ V ) .\/ W ) ) ) |
25 |
4 5 8 9 13 24
|
syl32anc |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> ( ( P .\/ U ) .\/ ( V .\/ W ) ) = ( P .\/ ( ( U .\/ V ) .\/ W ) ) ) |
26 |
25 22
|
eqeq12d |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> ( ( ( P .\/ U ) .\/ ( V .\/ W ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) <-> ( P .\/ ( ( U .\/ V ) .\/ W ) ) = ( T .\/ ( ( U .\/ V ) .\/ W ) ) ) ) |
27 |
20 23 26
|
3bitr4d |
|- ( ( ( K e. HL /\ P e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. P .<_ ( ( U .\/ V ) .\/ W ) ) -> ( P .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) <-> ( ( P .\/ U ) .\/ ( V .\/ W ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) ) |