Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme22.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme22.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme22.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme22.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme22.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme22f2.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
cdleme22f2.f |
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) |
8 |
|
cdleme22f2.n |
|- N = ( ( P .\/ Q ) ./\ ( F .\/ ( ( T .\/ S ) ./\ W ) ) ) |
9 |
|
simp11 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( K e. HL /\ W e. H ) ) |
10 |
|
simp2l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
11 |
|
simp2r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
12 |
9 10 11
|
3jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) |
13 |
|
simp12 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( T e. A /\ -. T .<_ W ) ) |
14 |
|
simp31l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> S e. A ) |
15 |
|
simp33 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( V e. A /\ V .<_ W ) ) |
16 |
|
simp32l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> S =/= T ) |
17 |
16
|
necomd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> T =/= S ) |
18 |
|
simp32r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> S .<_ ( T .\/ V ) ) |
19 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> K e. HL ) |
20 |
|
hlcvl |
|- ( K e. HL -> K e. CvLat ) |
21 |
19 20
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> K e. CvLat ) |
22 |
|
simp12l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> T e. A ) |
23 |
|
simp33l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> V e. A ) |
24 |
|
simp33r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> V .<_ W ) |
25 |
|
simp31r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. S .<_ W ) |
26 |
|
nbrne2 |
|- ( ( V .<_ W /\ -. S .<_ W ) -> V =/= S ) |
27 |
26
|
necomd |
|- ( ( V .<_ W /\ -. S .<_ W ) -> S =/= V ) |
28 |
24 25 27
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> S =/= V ) |
29 |
1 2 4
|
cvlatexch2 |
|- ( ( K e. CvLat /\ ( S e. A /\ T e. A /\ V e. A ) /\ S =/= V ) -> ( S .<_ ( T .\/ V ) -> T .<_ ( S .\/ V ) ) ) |
30 |
21 14 22 23 28 29
|
syl131anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( S .<_ ( T .\/ V ) -> T .<_ ( S .\/ V ) ) ) |
31 |
18 30
|
mpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> T .<_ ( S .\/ V ) ) |
32 |
1 2 3 4 5 6 7 8
|
cdleme22f |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( T e. A /\ -. T .<_ W ) /\ S e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( T =/= S /\ T .<_ ( S .\/ V ) ) ) -> N .<_ ( F .\/ V ) ) |
33 |
12 13 14 15 17 31 32
|
syl132anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> N .<_ ( F .\/ V ) ) |
34 |
|
simp31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( S e. A /\ -. S .<_ W ) ) |
35 |
|
simp133 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> P =/= Q ) |
36 |
|
simp132 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> T .<_ ( P .\/ Q ) ) |
37 |
|
simp131 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. S .<_ ( P .\/ Q ) ) |
38 |
1 2 3 4 5 6 7 8
|
cdleme7ga |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( T e. A /\ -. T .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> N e. A ) |
39 |
12 13 34 35 36 37 38
|
syl123anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> N e. A ) |
40 |
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 ) |
41 |
9 10 11 34 35 37 40
|
syl132anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> F e. A ) |
42 |
1 2 3 4 5 6 7 8
|
cdleme7 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( T e. A /\ -. T .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. N .<_ W ) |
43 |
12 13 34 35 36 37 42
|
syl123anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. N .<_ W ) |
44 |
|
nbrne2 |
|- ( ( V .<_ W /\ -. N .<_ W ) -> V =/= N ) |
45 |
44
|
necomd |
|- ( ( V .<_ W /\ -. N .<_ W ) -> N =/= V ) |
46 |
24 43 45
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> N =/= V ) |
47 |
1 2 4
|
cvlatexch2 |
|- ( ( K e. CvLat /\ ( N e. A /\ F e. A /\ V e. A ) /\ N =/= V ) -> ( N .<_ ( F .\/ V ) -> F .<_ ( N .\/ V ) ) ) |
48 |
21 39 41 23 46 47
|
syl131anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( N .<_ ( F .\/ V ) -> F .<_ ( N .\/ V ) ) ) |
49 |
33 48
|
mpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> F .<_ ( N .\/ V ) ) |