Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme9.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme9.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme9.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme9.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme9.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme9.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
cdleme9.f |
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) |
8 |
|
cdleme9.c |
|- C = ( ( P .\/ S ) ./\ W ) |
9 |
1 2 3 4 5 6 7 8
|
cdleme3d |
|- F = ( ( S .\/ U ) ./\ ( Q .\/ C ) ) |
10 |
9
|
oveq1i |
|- ( F .\/ C ) = ( ( ( S .\/ U ) ./\ ( Q .\/ C ) ) .\/ C ) |
11 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> K e. HL ) |
12 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( K e. HL /\ W e. H ) ) |
13 |
|
simp21 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( P e. A /\ -. P .<_ W ) ) |
14 |
|
simp23l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> S e. A ) |
15 |
11
|
hllatd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> K e. Lat ) |
16 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
17 |
16 4
|
atbase |
|- ( S e. A -> S e. ( Base ` K ) ) |
18 |
14 17
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> S e. ( Base ` K ) ) |
19 |
|
simp21l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> P e. A ) |
20 |
16 4
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
21 |
19 20
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> P e. ( Base ` K ) ) |
22 |
|
simp22 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> Q e. A ) |
23 |
16 4
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
24 |
22 23
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> Q e. ( Base ` K ) ) |
25 |
|
simp3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> -. S .<_ ( P .\/ Q ) ) |
26 |
16 1 2
|
latnlej1l |
|- ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. S .<_ ( P .\/ Q ) ) -> S =/= P ) |
27 |
26
|
necomd |
|- ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. S .<_ ( P .\/ Q ) ) -> P =/= S ) |
28 |
15 18 21 24 25 27
|
syl131anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> P =/= S ) |
29 |
1 2 3 4 5 8
|
cdleme9a |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( S e. A /\ P =/= S ) ) -> C e. A ) |
30 |
12 13 14 28 29
|
syl112anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> C e. A ) |
31 |
1 2 3 4 5 6 16
|
cdleme0aa |
|- ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> U e. ( Base ` K ) ) |
32 |
12 19 22 31
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> U e. ( Base ` K ) ) |
33 |
16 2
|
latjcl |
|- ( ( K e. Lat /\ S e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> ( S .\/ U ) e. ( Base ` K ) ) |
34 |
15 18 32 33
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( S .\/ U ) e. ( Base ` K ) ) |
35 |
16 2 4
|
hlatjcl |
|- ( ( K e. HL /\ Q e. A /\ C e. A ) -> ( Q .\/ C ) e. ( Base ` K ) ) |
36 |
11 22 30 35
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( Q .\/ C ) e. ( Base ` K ) ) |
37 |
1 2 4
|
hlatlej2 |
|- ( ( K e. HL /\ Q e. A /\ C e. A ) -> C .<_ ( Q .\/ C ) ) |
38 |
11 22 30 37
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> C .<_ ( Q .\/ C ) ) |
39 |
16 1 2 3 4
|
atmod4i1 |
|- ( ( K e. HL /\ ( C e. A /\ ( S .\/ U ) e. ( Base ` K ) /\ ( Q .\/ C ) e. ( Base ` K ) ) /\ C .<_ ( Q .\/ C ) ) -> ( ( ( S .\/ U ) ./\ ( Q .\/ C ) ) .\/ C ) = ( ( ( S .\/ U ) .\/ C ) ./\ ( Q .\/ C ) ) ) |
40 |
11 30 34 36 38 39
|
syl131anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( ( S .\/ U ) ./\ ( Q .\/ C ) ) .\/ C ) = ( ( ( S .\/ U ) .\/ C ) ./\ ( Q .\/ C ) ) ) |
41 |
8
|
oveq2i |
|- ( S .\/ C ) = ( S .\/ ( ( P .\/ S ) ./\ W ) ) |
42 |
16 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) ) |
43 |
11 19 14 42
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( P .\/ S ) e. ( Base ` K ) ) |
44 |
|
simp1r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> W e. H ) |
45 |
16 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
46 |
44 45
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> W e. ( Base ` K ) ) |
47 |
1 2 4
|
hlatlej2 |
|- ( ( K e. HL /\ P e. A /\ S e. A ) -> S .<_ ( P .\/ S ) ) |
48 |
11 19 14 47
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> S .<_ ( P .\/ S ) ) |
49 |
16 1 2 3 4
|
atmod3i1 |
|- ( ( K e. HL /\ ( S e. A /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ S .<_ ( P .\/ S ) ) -> ( S .\/ ( ( P .\/ S ) ./\ W ) ) = ( ( P .\/ S ) ./\ ( S .\/ W ) ) ) |
50 |
11 14 43 46 48 49
|
syl131anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( S .\/ ( ( P .\/ S ) ./\ W ) ) = ( ( P .\/ S ) ./\ ( S .\/ W ) ) ) |
51 |
|
simp23r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> -. S .<_ W ) |
52 |
|
eqid |
|- ( 1. ` K ) = ( 1. ` K ) |
53 |
1 2 52 4 5
|
lhpjat2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( S .\/ W ) = ( 1. ` K ) ) |
54 |
12 14 51 53
|
syl12anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( S .\/ W ) = ( 1. ` K ) ) |
55 |
54
|
oveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ S ) ./\ ( S .\/ W ) ) = ( ( P .\/ S ) ./\ ( 1. ` K ) ) ) |
56 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
57 |
11 56
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> K e. OL ) |
58 |
16 3 52
|
olm11 |
|- ( ( K e. OL /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( 1. ` K ) ) = ( P .\/ S ) ) |
59 |
57 43 58
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ S ) ./\ ( 1. ` K ) ) = ( P .\/ S ) ) |
60 |
50 55 59
|
3eqtrrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( P .\/ S ) = ( S .\/ ( ( P .\/ S ) ./\ W ) ) ) |
61 |
41 60
|
eqtr4id |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( S .\/ C ) = ( P .\/ S ) ) |
62 |
61
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( S .\/ C ) .\/ U ) = ( ( P .\/ S ) .\/ U ) ) |
63 |
16 4
|
atbase |
|- ( C e. A -> C e. ( Base ` K ) ) |
64 |
30 63
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> C e. ( Base ` K ) ) |
65 |
16 2
|
latj32 |
|- ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ U e. ( Base ` K ) /\ C e. ( Base ` K ) ) ) -> ( ( S .\/ U ) .\/ C ) = ( ( S .\/ C ) .\/ U ) ) |
66 |
15 18 32 64 65
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( S .\/ U ) .\/ C ) = ( ( S .\/ C ) .\/ U ) ) |
67 |
2 4
|
hlatj32 |
|- ( ( K e. HL /\ ( P e. A /\ S e. A /\ Q e. A ) ) -> ( ( P .\/ S ) .\/ Q ) = ( ( P .\/ Q ) .\/ S ) ) |
68 |
11 19 14 22 67
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ S ) .\/ Q ) = ( ( P .\/ Q ) .\/ S ) ) |
69 |
16 2
|
latjcom |
|- ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( Q .\/ ( P .\/ S ) ) = ( ( P .\/ S ) .\/ Q ) ) |
70 |
15 24 43 69
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( Q .\/ ( P .\/ S ) ) = ( ( P .\/ S ) .\/ Q ) ) |
71 |
6
|
oveq2i |
|- ( P .\/ U ) = ( P .\/ ( ( P .\/ Q ) ./\ W ) ) |
72 |
16 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
73 |
11 19 22 72
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
74 |
1 2 4
|
hlatlej1 |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> P .<_ ( P .\/ Q ) ) |
75 |
11 19 22 74
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> P .<_ ( P .\/ Q ) ) |
76 |
16 1 2 3 4
|
atmod3i1 |
|- ( ( K e. HL /\ ( P e. A /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ P .<_ ( P .\/ Q ) ) -> ( P .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ ( P .\/ W ) ) ) |
77 |
11 19 73 46 75 76
|
syl131anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( P .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ ( P .\/ W ) ) ) |
78 |
1 2 52 4 5
|
lhpjat2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ W ) = ( 1. ` K ) ) |
79 |
12 13 78
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( P .\/ W ) = ( 1. ` K ) ) |
80 |
79
|
oveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ Q ) ./\ ( P .\/ W ) ) = ( ( P .\/ Q ) ./\ ( 1. ` K ) ) ) |
81 |
16 3 52
|
olm11 |
|- ( ( K e. OL /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) ) |
82 |
57 73 81
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) ) |
83 |
77 80 82
|
3eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( P .\/ ( ( P .\/ Q ) ./\ W ) ) = ( P .\/ Q ) ) |
84 |
71 83
|
syl5eq |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( P .\/ U ) = ( P .\/ Q ) ) |
85 |
84
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ U ) .\/ S ) = ( ( P .\/ Q ) .\/ S ) ) |
86 |
68 70 85
|
3eqtr4d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( Q .\/ ( P .\/ S ) ) = ( ( P .\/ U ) .\/ S ) ) |
87 |
16 2
|
latj32 |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ U e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( P .\/ U ) .\/ S ) = ( ( P .\/ S ) .\/ U ) ) |
88 |
15 21 32 18 87
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ U ) .\/ S ) = ( ( P .\/ S ) .\/ U ) ) |
89 |
86 88
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( Q .\/ ( P .\/ S ) ) = ( ( P .\/ S ) .\/ U ) ) |
90 |
62 66 89
|
3eqtr4d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( S .\/ U ) .\/ C ) = ( Q .\/ ( P .\/ S ) ) ) |
91 |
90
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( ( S .\/ U ) .\/ C ) ./\ ( Q .\/ C ) ) = ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ C ) ) ) |
92 |
16 1 3
|
latmle1 |
|- ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) ) |
93 |
15 43 46 92
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) ) |
94 |
8 93
|
eqbrtrid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> C .<_ ( P .\/ S ) ) |
95 |
16 1 2
|
latjlej2 |
|- ( ( K e. Lat /\ ( C e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) ) -> ( C .<_ ( P .\/ S ) -> ( Q .\/ C ) .<_ ( Q .\/ ( P .\/ S ) ) ) ) |
96 |
15 64 43 24 95
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( C .<_ ( P .\/ S ) -> ( Q .\/ C ) .<_ ( Q .\/ ( P .\/ S ) ) ) ) |
97 |
94 96
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( Q .\/ C ) .<_ ( Q .\/ ( P .\/ S ) ) ) |
98 |
16 2
|
latjcl |
|- ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( Q .\/ ( P .\/ S ) ) e. ( Base ` K ) ) |
99 |
15 24 43 98
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( Q .\/ ( P .\/ S ) ) e. ( Base ` K ) ) |
100 |
16 1 3
|
latleeqm2 |
|- ( ( K e. Lat /\ ( Q .\/ C ) e. ( Base ` K ) /\ ( Q .\/ ( P .\/ S ) ) e. ( Base ` K ) ) -> ( ( Q .\/ C ) .<_ ( Q .\/ ( P .\/ S ) ) <-> ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ C ) ) = ( Q .\/ C ) ) ) |
101 |
15 36 99 100
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( Q .\/ C ) .<_ ( Q .\/ ( P .\/ S ) ) <-> ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ C ) ) = ( Q .\/ C ) ) ) |
102 |
97 101
|
mpbid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ C ) ) = ( Q .\/ C ) ) |
103 |
40 91 102
|
3eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( ( S .\/ U ) ./\ ( Q .\/ C ) ) .\/ C ) = ( Q .\/ C ) ) |
104 |
10 103
|
syl5eq |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( F .\/ C ) = ( Q .\/ C ) ) |