Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme4.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme4.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme4.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme4.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme4.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme4.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
cdleme4.f |
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) |
8 |
|
cdleme4.g |
|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) |
9 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL ) |
10 |
|
simp12l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. A ) |
11 |
|
simp13l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. A ) |
12 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
13 |
12 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
14 |
9 10 11 13
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
15 |
|
simp11 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) ) |
16 |
|
simp12 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
17 |
|
simp13 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
18 |
|
simp2r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( S e. A /\ -. S .<_ W ) ) |
19 |
|
simp31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= Q ) |
20 |
|
simp33 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) ) |
21 |
1 2 3 4 5 6 7
|
cdleme3fa |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F e. A ) |
22 |
15 16 17 18 19 20 21
|
syl132anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> F e. A ) |
23 |
|
simp2l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R e. A /\ -. R .<_ W ) ) |
24 |
|
simp2rl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A ) |
25 |
|
simp32 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) ) |
26 |
|
eqid |
|- ( ( R .\/ S ) ./\ W ) = ( ( R .\/ S ) ./\ W ) |
27 |
1 2 3 4 5 6 7 8 26
|
cdleme7b |
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) e. A ) |
28 |
15 23 24 20 25 27
|
syl113anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) e. A ) |
29 |
12 2 4
|
hlatjcl |
|- ( ( K e. HL /\ F e. A /\ ( ( R .\/ S ) ./\ W ) e. A ) -> ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) ) |
30 |
9 22 28 29
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) ) |
31 |
9
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. Lat ) |
32 |
|
eqid |
|- ( Lines ` K ) = ( Lines ` K ) |
33 |
|
eqid |
|- ( pmap ` K ) = ( pmap ` K ) |
34 |
2 4 32 33
|
linepmap |
|- ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( ( pmap ` K ) ` ( P .\/ Q ) ) e. ( Lines ` K ) ) |
35 |
31 10 11 19 34
|
syl31anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( pmap ` K ) ` ( P .\/ Q ) ) e. ( Lines ` K ) ) |
36 |
|
simp2ll |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. A ) |
37 |
12 2 4
|
hlatjcl |
|- ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) ) |
38 |
9 36 24 37
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R .\/ S ) e. ( Base ` K ) ) |
39 |
|
simp11r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. H ) |
40 |
12 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
41 |
39 40
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. ( Base ` K ) ) |
42 |
12 1 3
|
latmle2 |
|- ( ( K e. Lat /\ ( R .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( R .\/ S ) ./\ W ) .<_ W ) |
43 |
31 38 41 42
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) .<_ W ) |
44 |
1 2 3 4 5 6 7
|
cdleme3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. F .<_ W ) |
45 |
15 16 17 18 19 20 44
|
syl132anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. F .<_ W ) |
46 |
|
nbrne2 |
|- ( ( ( ( R .\/ S ) ./\ W ) .<_ W /\ -. F .<_ W ) -> ( ( R .\/ S ) ./\ W ) =/= F ) |
47 |
46
|
necomd |
|- ( ( ( ( R .\/ S ) ./\ W ) .<_ W /\ -. F .<_ W ) -> F =/= ( ( R .\/ S ) ./\ W ) ) |
48 |
43 45 47
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> F =/= ( ( R .\/ S ) ./\ W ) ) |
49 |
2 4 32 33
|
linepmap |
|- ( ( ( K e. Lat /\ F e. A /\ ( ( R .\/ S ) ./\ W ) e. A ) /\ F =/= ( ( R .\/ S ) ./\ W ) ) -> ( ( pmap ` K ) ` ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) e. ( Lines ` K ) ) |
50 |
31 22 28 48 49
|
syl31anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( pmap ` K ) ` ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) e. ( Lines ` K ) ) |
51 |
12 4
|
atbase |
|- ( F e. A -> F e. ( Base ` K ) ) |
52 |
22 51
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> F e. ( Base ` K ) ) |
53 |
12 3
|
latmcl |
|- ( ( K e. Lat /\ ( R .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) ) |
54 |
31 38 41 53
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) ) |
55 |
12 1 2
|
latlej2 |
|- ( ( K e. Lat /\ F e. ( Base ` K ) /\ ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) ) -> ( ( R .\/ S ) ./\ W ) .<_ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) |
56 |
31 52 54 55
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) .<_ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) |
57 |
1 2 3 4 5 6 7 8 26
|
cdleme7c |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> U =/= ( ( R .\/ S ) ./\ W ) ) |
58 |
15 16 11 23 18 19 25 20 57
|
syl323anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> U =/= ( ( R .\/ S ) ./\ W ) ) |
59 |
58
|
necomd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) =/= U ) |
60 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
61 |
9 60
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. AtLat ) |
62 |
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 ) |
63 |
15 16 11 19 62
|
syl112anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> U e. A ) |
64 |
1 4
|
atncmp |
|- ( ( K e. AtLat /\ ( ( R .\/ S ) ./\ W ) e. A /\ U e. A ) -> ( -. ( ( R .\/ S ) ./\ W ) .<_ U <-> ( ( R .\/ S ) ./\ W ) =/= U ) ) |
65 |
61 28 63 64
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( -. ( ( R .\/ S ) ./\ W ) .<_ U <-> ( ( R .\/ S ) ./\ W ) =/= U ) ) |
66 |
59 65
|
mpbird |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. ( ( R .\/ S ) ./\ W ) .<_ U ) |
67 |
12 1 3
|
latlem12 |
|- ( ( K e. Lat /\ ( ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( ( R .\/ S ) ./\ W ) .<_ ( P .\/ Q ) /\ ( ( R .\/ S ) ./\ W ) .<_ W ) <-> ( ( R .\/ S ) ./\ W ) .<_ ( ( P .\/ Q ) ./\ W ) ) ) |
68 |
31 54 14 41 67
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( ( ( R .\/ S ) ./\ W ) .<_ ( P .\/ Q ) /\ ( ( R .\/ S ) ./\ W ) .<_ W ) <-> ( ( R .\/ S ) ./\ W ) .<_ ( ( P .\/ Q ) ./\ W ) ) ) |
69 |
68
|
biimpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( ( ( R .\/ S ) ./\ W ) .<_ ( P .\/ Q ) /\ ( ( R .\/ S ) ./\ W ) .<_ W ) -> ( ( R .\/ S ) ./\ W ) .<_ ( ( P .\/ Q ) ./\ W ) ) ) |
70 |
43 69
|
mpan2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( ( R .\/ S ) ./\ W ) .<_ ( P .\/ Q ) -> ( ( R .\/ S ) ./\ W ) .<_ ( ( P .\/ Q ) ./\ W ) ) ) |
71 |
6
|
breq2i |
|- ( ( ( R .\/ S ) ./\ W ) .<_ U <-> ( ( R .\/ S ) ./\ W ) .<_ ( ( P .\/ Q ) ./\ W ) ) |
72 |
70 71
|
syl6ibr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( ( R .\/ S ) ./\ W ) .<_ ( P .\/ Q ) -> ( ( R .\/ S ) ./\ W ) .<_ U ) ) |
73 |
66 72
|
mtod |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. ( ( R .\/ S ) ./\ W ) .<_ ( P .\/ Q ) ) |
74 |
|
nbrne1 |
|- ( ( ( ( R .\/ S ) ./\ W ) .<_ ( F .\/ ( ( R .\/ S ) ./\ W ) ) /\ -. ( ( R .\/ S ) ./\ W ) .<_ ( P .\/ Q ) ) -> ( F .\/ ( ( R .\/ S ) ./\ W ) ) =/= ( P .\/ Q ) ) |
75 |
74
|
necomd |
|- ( ( ( ( R .\/ S ) ./\ W ) .<_ ( F .\/ ( ( R .\/ S ) ./\ W ) ) /\ -. ( ( R .\/ S ) ./\ W ) .<_ ( P .\/ Q ) ) -> ( P .\/ Q ) =/= ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) |
76 |
56 73 75
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) =/= ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) |
77 |
1 2 3 4 5 6 7 8 26
|
cdleme7e |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> G =/= ( 0. ` K ) ) |
78 |
8 77
|
eqnetrrid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) =/= ( 0. ` K ) ) |
79 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
80 |
12 3 79 4 32 33
|
2lnat |
|- ( ( ( K e. HL /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) ) /\ ( ( ( pmap ` K ) ` ( P .\/ Q ) ) e. ( Lines ` K ) /\ ( ( pmap ` K ) ` ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) e. ( Lines ` K ) ) /\ ( ( P .\/ Q ) =/= ( F .\/ ( ( R .\/ S ) ./\ W ) ) /\ ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) =/= ( 0. ` K ) ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) e. A ) |
81 |
9 14 30 35 50 76 78 80
|
syl322anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) e. A ) |
82 |
8 81
|
eqeltrid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> G e. A ) |