Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme0.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme0.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme0.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme0.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme0.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme0.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
cdleme0c.3 |
|- V = ( ( P .\/ R ) ./\ W ) |
8 |
6 7
|
oveq12i |
|- ( U ./\ V ) = ( ( ( P .\/ Q ) ./\ W ) ./\ ( ( P .\/ R ) ./\ W ) ) |
9 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. HL ) |
10 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
11 |
9 10
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. OL ) |
12 |
|
simp21l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P e. A ) |
13 |
|
simp22 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q e. A ) |
14 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
15 |
14 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
16 |
9 12 13 15
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
17 |
|
simp23l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> R e. A ) |
18 |
14 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) e. ( Base ` K ) ) |
19 |
9 12 17 18
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P .\/ R ) e. ( Base ` K ) ) |
20 |
|
simp1r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> W e. H ) |
21 |
14 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
22 |
20 21
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> W e. ( Base ` K ) ) |
23 |
14 3
|
latmmdir |
|- ( ( K e. OL /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ ( P .\/ R ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) ./\ ( P .\/ R ) ) ./\ W ) = ( ( ( P .\/ Q ) ./\ W ) ./\ ( ( P .\/ R ) ./\ W ) ) ) |
24 |
11 16 19 22 23
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( ( ( P .\/ Q ) ./\ ( P .\/ R ) ) ./\ W ) = ( ( ( P .\/ Q ) ./\ W ) ./\ ( ( P .\/ R ) ./\ W ) ) ) |
25 |
9
|
hllatd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. Lat ) |
26 |
14 4
|
atbase |
|- ( R e. A -> R e. ( Base ` K ) ) |
27 |
17 26
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> R e. ( Base ` K ) ) |
28 |
14 4
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
29 |
12 28
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P e. ( Base ` K ) ) |
30 |
14 4
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
31 |
13 30
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q e. ( Base ` K ) ) |
32 |
|
simp3r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. R .<_ ( P .\/ Q ) ) |
33 |
14 1 2
|
latnlej1r |
|- ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= Q ) |
34 |
33
|
necomd |
|- ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> Q =/= R ) |
35 |
25 27 29 31 32 34
|
syl131anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q =/= R ) |
36 |
|
simp3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) |
37 |
1 2 4
|
hlatcon3 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. P .<_ ( Q .\/ R ) ) |
38 |
9 12 13 17 36 37
|
syl131anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. P .<_ ( Q .\/ R ) ) |
39 |
1 2 3 4
|
2llnma2 |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( Q =/= R /\ -. P .<_ ( Q .\/ R ) ) ) -> ( ( P .\/ Q ) ./\ ( P .\/ R ) ) = P ) |
40 |
9 13 17 12 35 38 39
|
syl132anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( P .\/ R ) ) = P ) |
41 |
40
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( ( ( P .\/ Q ) ./\ ( P .\/ R ) ) ./\ W ) = ( P ./\ W ) ) |
42 |
24 41
|
eqtr3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( ( ( P .\/ Q ) ./\ W ) ./\ ( ( P .\/ R ) ./\ W ) ) = ( P ./\ W ) ) |
43 |
8 42
|
syl5eq |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( U ./\ V ) = ( P ./\ W ) ) |
44 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) ) |
45 |
|
simp21 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
46 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
47 |
1 3 46 4 5
|
lhpmat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = ( 0. ` K ) ) |
48 |
44 45 47
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P ./\ W ) = ( 0. ` K ) ) |
49 |
43 48
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( U ./\ V ) = ( 0. ` K ) ) |
50 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
51 |
9 50
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. AtLat ) |
52 |
|
simp3l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P =/= Q ) |
53 |
1 2 3 4 5 6
|
lhpat2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A ) |
54 |
44 45 13 52 53
|
syl112anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> U e. A ) |
55 |
14 1 2
|
latnlej1l |
|- ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= P ) |
56 |
55
|
necomd |
|- ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> P =/= R ) |
57 |
25 27 29 31 32 56
|
syl131anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P =/= R ) |
58 |
1 2 3 4 5 7
|
lhpat2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ P =/= R ) ) -> V e. A ) |
59 |
44 45 17 57 58
|
syl112anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> V e. A ) |
60 |
3 46 4
|
atnem0 |
|- ( ( K e. AtLat /\ U e. A /\ V e. A ) -> ( U =/= V <-> ( U ./\ V ) = ( 0. ` K ) ) ) |
61 |
51 54 59 60
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( U =/= V <-> ( U ./\ V ) = ( 0. ` K ) ) ) |
62 |
49 61
|
mpbird |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> U =/= V ) |