Step |
Hyp |
Ref |
Expression |
1 |
|
4at.l |
|- .<_ = ( le ` K ) |
2 |
|
4at.j |
|- .\/ = ( join ` K ) |
3 |
|
4at.a |
|- A = ( Atoms ` K ) |
4 |
1 2 3
|
4atlem12 |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) ) |
5 |
|
simp11 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> 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 /\ V e. A /\ W e. A ) ) -> K e. Lat ) |
7 |
|
simp23 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> T e. A ) |
8 |
|
simp31 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> U e. A ) |
9 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
10 |
9 2 3
|
hlatjcl |
|- ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) 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 /\ V e. A /\ W e. A ) ) -> ( T .\/ U ) e. ( Base ` K ) ) |
12 |
|
simp32 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> V e. A ) |
13 |
|
simp33 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> W e. A ) |
14 |
9 2 3
|
hlatjcl |
|- ( ( K e. HL /\ V e. A /\ W e. A ) -> ( V .\/ W ) e. ( Base ` K ) ) |
15 |
5 12 13 14
|
syl3anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> ( V .\/ W ) e. ( Base ` K ) ) |
16 |
9 2
|
latjcl |
|- ( ( K e. Lat /\ ( T .\/ U ) e. ( Base ` K ) /\ ( V .\/ W ) e. ( Base ` K ) ) -> ( ( T .\/ U ) .\/ ( V .\/ W ) ) e. ( Base ` K ) ) |
17 |
6 11 15 16
|
syl3anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> ( ( T .\/ U ) .\/ ( V .\/ W ) ) e. ( Base ` K ) ) |
18 |
9 1
|
latref |
|- ( ( K e. Lat /\ ( ( T .\/ U ) .\/ ( V .\/ W ) ) e. ( Base ` K ) ) -> ( ( T .\/ U ) .\/ ( V .\/ W ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) |
19 |
6 17 18
|
syl2anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> ( ( T .\/ U ) .\/ ( V .\/ W ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) |
20 |
|
breq1 |
|- ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) <-> ( ( T .\/ U ) .\/ ( V .\/ W ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) ) |
21 |
19 20
|
syl5ibrcom |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) ) |
22 |
21
|
adantr |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) ) |
23 |
4 22
|
impbid |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) ) |