Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme12.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme12.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme12.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme12.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme12.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme12.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
cdleme12.f |
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) |
8 |
|
cdleme12.g |
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) ) |
9 |
|
cdleme15.c |
|- C = ( ( P .\/ S ) ./\ W ) |
10 |
|
cdleme15.x |
|- X = ( ( P .\/ T ) ./\ W ) |
11 |
9
|
oveq2i |
|- ( P .\/ C ) = ( P .\/ ( ( P .\/ S ) ./\ W ) ) |
12 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> K e. HL ) |
13 |
|
simp12l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> P e. A ) |
14 |
|
simp21l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> S e. A ) |
15 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
16 |
15 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) ) |
17 |
12 13 14 16
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( P .\/ S ) e. ( Base ` K ) ) |
18 |
|
simp11r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> W e. H ) |
19 |
15 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
20 |
18 19
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> W e. ( Base ` K ) ) |
21 |
1 2 4
|
hlatlej1 |
|- ( ( K e. HL /\ P e. A /\ S e. A ) -> P .<_ ( P .\/ S ) ) |
22 |
12 13 14 21
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> P .<_ ( P .\/ S ) ) |
23 |
15 1 2 3 4
|
atmod3i1 |
|- ( ( K e. HL /\ ( P e. A /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ P .<_ ( P .\/ S ) ) -> ( P .\/ ( ( P .\/ S ) ./\ W ) ) = ( ( P .\/ S ) ./\ ( P .\/ W ) ) ) |
24 |
12 13 17 20 22 23
|
syl131anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( P .\/ ( ( P .\/ S ) ./\ W ) ) = ( ( P .\/ S ) ./\ ( P .\/ W ) ) ) |
25 |
11 24
|
syl5eq |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( P .\/ C ) = ( ( P .\/ S ) ./\ ( P .\/ W ) ) ) |
26 |
25
|
oveq1d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( P .\/ C ) ./\ Q ) = ( ( ( P .\/ S ) ./\ ( P .\/ W ) ) ./\ Q ) ) |
27 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
28 |
12 27
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> K e. OL ) |
29 |
12
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> K e. Lat ) |
30 |
15 4
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
31 |
13 30
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> P e. ( Base ` K ) ) |
32 |
15 2
|
latjcl |
|- ( ( K e. Lat /\ P e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( P .\/ W ) e. ( Base ` K ) ) |
33 |
29 31 20 32
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( P .\/ W ) e. ( Base ` K ) ) |
34 |
|
simp13l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> Q e. A ) |
35 |
15 4
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
36 |
34 35
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> Q e. ( Base ` K ) ) |
37 |
15 3
|
latmrot |
|- ( ( K e. OL /\ ( ( P .\/ S ) e. ( Base ` K ) /\ ( P .\/ W ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) ) -> ( ( ( P .\/ S ) ./\ ( P .\/ W ) ) ./\ Q ) = ( ( Q ./\ ( P .\/ S ) ) ./\ ( P .\/ W ) ) ) |
38 |
28 17 33 36 37
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( ( P .\/ S ) ./\ ( P .\/ W ) ) ./\ Q ) = ( ( Q ./\ ( P .\/ S ) ) ./\ ( P .\/ W ) ) ) |
39 |
|
simp31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> -. S .<_ ( P .\/ Q ) ) |
40 |
|
simp23l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> P =/= Q ) |
41 |
40
|
necomd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> Q =/= P ) |
42 |
1 2 4
|
hlatexch1 |
|- ( ( K e. HL /\ ( Q e. A /\ S e. A /\ P e. A ) /\ Q =/= P ) -> ( Q .<_ ( P .\/ S ) -> S .<_ ( P .\/ Q ) ) ) |
43 |
12 34 14 13 41 42
|
syl131anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( Q .<_ ( P .\/ S ) -> S .<_ ( P .\/ Q ) ) ) |
44 |
39 43
|
mtod |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> -. Q .<_ ( P .\/ S ) ) |
45 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
46 |
12 45
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> K e. AtLat ) |
47 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
48 |
15 1 3 47 4
|
atnle |
|- ( ( K e. AtLat /\ Q e. A /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( -. Q .<_ ( P .\/ S ) <-> ( Q ./\ ( P .\/ S ) ) = ( 0. ` K ) ) ) |
49 |
46 34 17 48
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( -. Q .<_ ( P .\/ S ) <-> ( Q ./\ ( P .\/ S ) ) = ( 0. ` K ) ) ) |
50 |
44 49
|
mpbid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( Q ./\ ( P .\/ S ) ) = ( 0. ` K ) ) |
51 |
50
|
oveq1d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( Q ./\ ( P .\/ S ) ) ./\ ( P .\/ W ) ) = ( ( 0. ` K ) ./\ ( P .\/ W ) ) ) |
52 |
15 3 47
|
olm02 |
|- ( ( K e. OL /\ ( P .\/ W ) e. ( Base ` K ) ) -> ( ( 0. ` K ) ./\ ( P .\/ W ) ) = ( 0. ` K ) ) |
53 |
28 33 52
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( 0. ` K ) ./\ ( P .\/ W ) ) = ( 0. ` K ) ) |
54 |
38 51 53
|
3eqtrrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( 0. ` K ) = ( ( ( P .\/ S ) ./\ ( P .\/ W ) ) ./\ Q ) ) |
55 |
26 54
|
eqtr4d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( P .\/ C ) ./\ Q ) = ( 0. ` K ) ) |
56 |
55
|
oveq1d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( ( P .\/ C ) ./\ Q ) .\/ C ) = ( ( 0. ` K ) .\/ C ) ) |
57 |
15 2 3 4 5 9
|
cdleme9b |
|- ( ( K e. HL /\ ( P e. A /\ S e. A /\ W e. H ) ) -> C e. ( Base ` K ) ) |
58 |
12 13 14 18 57
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> C e. ( Base ` K ) ) |
59 |
15 2
|
latjcl |
|- ( ( K e. Lat /\ P e. ( Base ` K ) /\ C e. ( Base ` K ) ) -> ( P .\/ C ) e. ( Base ` K ) ) |
60 |
29 31 58 59
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( P .\/ C ) e. ( Base ` K ) ) |
61 |
15 1 2
|
latlej2 |
|- ( ( K e. Lat /\ P e. ( Base ` K ) /\ C e. ( Base ` K ) ) -> C .<_ ( P .\/ C ) ) |
62 |
29 31 58 61
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> C .<_ ( P .\/ C ) ) |
63 |
15 1 2 3 4
|
atmod2i2 |
|- ( ( K e. HL /\ ( Q e. A /\ ( P .\/ C ) e. ( Base ` K ) /\ C e. ( Base ` K ) ) /\ C .<_ ( P .\/ C ) ) -> ( ( ( P .\/ C ) ./\ Q ) .\/ C ) = ( ( P .\/ C ) ./\ ( Q .\/ C ) ) ) |
64 |
12 34 60 58 62 63
|
syl131anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( ( P .\/ C ) ./\ Q ) .\/ C ) = ( ( P .\/ C ) ./\ ( Q .\/ C ) ) ) |
65 |
15 2 47
|
olj02 |
|- ( ( K e. OL /\ C e. ( Base ` K ) ) -> ( ( 0. ` K ) .\/ C ) = C ) |
66 |
28 58 65
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( 0. ` K ) .\/ C ) = C ) |
67 |
56 64 66
|
3eqtr3d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( P .\/ C ) ./\ ( Q .\/ C ) ) = C ) |