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 |
|
cdleme7.v |
|- V = ( ( R .\/ S ) ./\ W ) |
10 |
|
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 ) |
11 |
10
|
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 ) |
12 |
|
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 ) |
13 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
14 |
13 4
|
atbase |
|- ( R e. A -> R e. ( Base ` K ) ) |
15 |
12 14
|
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 ) ) ) -> R e. ( Base ` K ) ) |
16 |
|
hlop |
|- ( K e. HL -> K e. OP ) |
17 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
18 |
13 17
|
op0cl |
|- ( K e. OP -> ( 0. ` K ) e. ( Base ` K ) ) |
19 |
10 16 18
|
3syl |
|- ( ( ( ( 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 ) ) ) -> ( 0. ` K ) e. ( Base ` K ) ) |
20 |
13 2
|
latjcl |
|- ( ( K e. Lat /\ R e. ( Base ` K ) /\ ( 0. ` K ) e. ( Base ` K ) ) -> ( R .\/ ( 0. ` K ) ) e. ( Base ` K ) ) |
21 |
11 15 19 20
|
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 .\/ ( 0. ` K ) ) e. ( Base ` K ) ) |
22 |
|
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 ) |
23 |
|
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 ) |
24 |
13 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
25 |
10 22 23 24
|
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 ) ) |
26 |
|
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 ) ) |
27 |
|
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 ) |
28 |
1 2 3 4 5 6 7 13
|
cdleme1b |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> F e. ( Base ` K ) ) |
29 |
26 22 23 27 28
|
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 ) ) ) -> F e. ( Base ` K ) ) |
30 |
13 2 4
|
hlatjcl |
|- ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) ) |
31 |
10 12 27 30
|
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 ) ) |
32 |
|
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 ) |
33 |
13 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
34 |
32 33
|
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 ) ) |
35 |
13 3
|
latmcl |
|- ( ( K e. Lat /\ ( R .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) ) |
36 |
11 31 34 35
|
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 ) ) |
37 |
13 2
|
latjcl |
|- ( ( K e. Lat /\ F e. ( Base ` K ) /\ ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) ) -> ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) ) |
38 |
11 29 36 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 ) ) ) -> ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) ) |
39 |
13 3
|
latmcl |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) e. ( Base ` K ) ) |
40 |
11 25 38 39
|
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 ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) e. ( Base ` K ) ) |
41 |
8 40
|
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. ( Base ` K ) ) |
42 |
13 2
|
latjcl |
|- ( ( K e. Lat /\ R e. ( Base ` K ) /\ G e. ( Base ` K ) ) -> ( R .\/ G ) e. ( Base ` K ) ) |
43 |
11 15 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 .\/ G ) e. ( Base ` K ) ) |
44 |
13 1 3
|
latmle2 |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W ) |
45 |
11 25 34 44
|
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 ) ./\ W ) .<_ W ) |
46 |
6 45
|
eqbrtrid |
|- ( ( ( ( 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 .<_ W ) |
47 |
|
simp2lr |
|- ( ( ( ( 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 .<_ W ) |
48 |
|
nbrne2 |
|- ( ( U .<_ W /\ -. R .<_ W ) -> U =/= R ) |
49 |
48
|
necomd |
|- ( ( U .<_ W /\ -. R .<_ W ) -> R =/= U ) |
50 |
46 47 49
|
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 ) ) ) -> R =/= U ) |
51 |
|
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 ) ) |
52 |
|
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 ) |
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 |
26 51 23 52 53
|
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 ) |
55 |
|
eqid |
|- ( |
56 |
2 55 4
|
atcvr1 |
|- ( ( K e. HL /\ R e. A /\ U e. A ) -> ( R =/= U <-> R ( |
57 |
10 12 54 56
|
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 =/= U <-> R ( |
58 |
50 57
|
mpbid |
|- ( ( ( ( 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 ( |
59 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
60 |
10 59
|
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. OL ) |
61 |
13 2 17
|
olj01 |
|- ( ( K e. OL /\ R e. ( Base ` K ) ) -> ( R .\/ ( 0. ` K ) ) = R ) |
62 |
60 15 61
|
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 ) ) ) -> ( R .\/ ( 0. ` K ) ) = R ) |
63 |
|
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 ) ) |
64 |
|
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 ) ) |
65 |
|
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 ) ) |
66 |
1 2 3 4 5 6 7 8
|
cdleme5 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R .\/ G ) = ( P .\/ Q ) ) |
67 |
26 22 23 63 64 65 66
|
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 ) ) ) -> ( R .\/ G ) = ( P .\/ Q ) ) |
68 |
1 2 3 4 5 6
|
cdleme4 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( P .\/ Q ) = ( R .\/ U ) ) |
69 |
26 22 23 63 65 68
|
syl131anc |
|- ( ( ( ( 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 ) = ( R .\/ U ) ) |
70 |
67 69
|
eqtrd |
|- ( ( ( ( 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 .\/ G ) = ( R .\/ U ) ) |
71 |
58 62 70
|
3brtr4d |
|- ( ( ( ( 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 .\/ ( 0. ` K ) ) ( |
72 |
13 55
|
cvrne |
|- ( ( ( K e. HL /\ ( R .\/ ( 0. ` K ) ) e. ( Base ` K ) /\ ( R .\/ G ) e. ( Base ` K ) ) /\ ( R .\/ ( 0. ` K ) ) ( ( R .\/ ( 0. ` K ) ) =/= ( R .\/ G ) ) |
73 |
10 21 43 71 72
|
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 ) ) ) -> ( R .\/ ( 0. ` K ) ) =/= ( R .\/ G ) ) |
74 |
|
oveq2 |
|- ( ( 0. ` K ) = G -> ( R .\/ ( 0. ` K ) ) = ( R .\/ G ) ) |
75 |
74
|
necon3i |
|- ( ( R .\/ ( 0. ` K ) ) =/= ( R .\/ G ) -> ( 0. ` K ) =/= G ) |
76 |
73 75
|
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 ) ) ) -> ( 0. ` K ) =/= G ) |
77 |
76
|
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 ) ) ) -> G =/= ( 0. ` K ) ) |