Step |
Hyp |
Ref |
Expression |
1 |
|
3at.l |
|- .<_ = ( le ` K ) |
2 |
|
3at.j |
|- .\/ = ( join ` K ) |
3 |
|
3at.a |
|- A = ( Atoms ` K ) |
4 |
|
simp11 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> K e. HL ) |
5 |
|
simp12 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( P e. A /\ Q e. A /\ R e. A ) ) |
6 |
|
simp13l |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> S e. A ) |
7 |
|
simp13r |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> T e. A ) |
8 |
|
simp123 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> R e. A ) |
9 |
6 7 8
|
3jca |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( S e. A /\ T e. A /\ R e. A ) ) |
10 |
|
simp2l |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> -. R .<_ ( P .\/ Q ) ) |
11 |
4
|
hllatd |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> K e. Lat ) |
12 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
13 |
12 3
|
atbase |
|- ( R e. A -> R e. ( Base ` K ) ) |
14 |
8 13
|
syl |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> R e. ( Base ` K ) ) |
15 |
|
simp121 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> P e. A ) |
16 |
12 3
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
17 |
15 16
|
syl |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> P e. ( Base ` K ) ) |
18 |
|
simp122 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> Q e. A ) |
19 |
12 3
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
20 |
18 19
|
syl |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> Q e. ( Base ` K ) ) |
21 |
12 1 2
|
latnlej1l |
|- ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= P ) |
22 |
11 14 17 20 10 21
|
syl131anc |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> R =/= P ) |
23 |
22
|
necomd |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> P =/= R ) |
24 |
|
simp2r |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> P =/= Q ) |
25 |
24
|
necomd |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> Q =/= P ) |
26 |
1 2 3
|
hlatexch1 |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ P e. A ) /\ Q =/= P ) -> ( Q .<_ ( P .\/ R ) -> R .<_ ( P .\/ Q ) ) ) |
27 |
4 18 8 15 25 26
|
syl131anc |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( Q .<_ ( P .\/ R ) -> R .<_ ( P .\/ Q ) ) ) |
28 |
10 27
|
mtod |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> -. Q .<_ ( P .\/ R ) ) |
29 |
|
simp3 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) |
30 |
1 2 3
|
3atlem3 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ R e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= R /\ -. Q .<_ ( P .\/ R ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ R ) ) |
31 |
4 5 9 10 23 28 29 30
|
syl331anc |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ R ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ R ) ) |