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 /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> K e. HL ) |
5 |
|
simp21 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> R e. A ) |
6 |
|
simp22 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> V e. A ) |
7 |
4
|
hllatd |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> K e. Lat ) |
8 |
|
simp1 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> ( K e. HL /\ P e. A /\ Q e. A ) ) |
9 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
10 |
9 2 3
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
11 |
8 10
|
syl |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
12 |
|
simp23 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> W e. A ) |
13 |
9 3
|
atbase |
|- ( W e. A -> W e. ( Base ` K ) ) |
14 |
12 13
|
syl |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> W e. ( Base ` K ) ) |
15 |
9 2
|
latjcl |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ W ) e. ( Base ` K ) ) |
16 |
7 11 14 15
|
syl3anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> ( ( P .\/ Q ) .\/ W ) e. ( Base ` K ) ) |
17 |
|
simp3 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> -. R .<_ ( ( P .\/ Q ) .\/ W ) ) |
18 |
9 1 2 3
|
hlexchb2 |
|- ( ( K e. HL /\ ( R e. A /\ V e. A /\ ( ( P .\/ Q ) .\/ W ) e. ( Base ` K ) ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> ( R .<_ ( V .\/ ( ( P .\/ Q ) .\/ W ) ) <-> ( R .\/ ( ( P .\/ Q ) .\/ W ) ) = ( V .\/ ( ( P .\/ Q ) .\/ W ) ) ) ) |
19 |
4 5 6 16 17 18
|
syl131anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> ( R .<_ ( V .\/ ( ( P .\/ Q ) .\/ W ) ) <-> ( R .\/ ( ( P .\/ Q ) .\/ W ) ) = ( V .\/ ( ( P .\/ Q ) .\/ W ) ) ) ) |
20 |
1 2 3
|
4atlem4c |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( V e. A /\ W e. A ) ) -> ( ( P .\/ Q ) .\/ ( V .\/ W ) ) = ( V .\/ ( ( P .\/ Q ) .\/ W ) ) ) |
21 |
8 6 12 20
|
syl12anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> ( ( P .\/ Q ) .\/ ( V .\/ W ) ) = ( V .\/ ( ( P .\/ Q ) .\/ W ) ) ) |
22 |
21
|
breq2d |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) <-> R .<_ ( V .\/ ( ( P .\/ Q ) .\/ W ) ) ) ) |
23 |
1 2 3
|
4atlem4c |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ W e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ W ) ) = ( R .\/ ( ( P .\/ Q ) .\/ W ) ) ) |
24 |
8 5 12 23
|
syl12anc |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ W ) ) = ( R .\/ ( ( P .\/ Q ) .\/ W ) ) ) |
25 |
24 21
|
eqeq12d |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ W ) ) = ( ( P .\/ Q ) .\/ ( V .\/ W ) ) <-> ( R .\/ ( ( P .\/ Q ) .\/ W ) ) = ( V .\/ ( ( P .\/ Q ) .\/ W ) ) ) ) |
26 |
19 22 25
|
3bitr4d |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ W ) ) = ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) |