Step |
Hyp |
Ref |
Expression |
1 |
|
trlval3.l |
|- .<_ = ( le ` K ) |
2 |
|
trlval3.j |
|- .\/ = ( join ` K ) |
3 |
|
trlval3.m |
|- ./\ = ( meet ` K ) |
4 |
|
trlval3.a |
|- A = ( Atoms ` K ) |
5 |
|
trlval3.h |
|- H = ( LHyp ` K ) |
6 |
|
trlval3.t |
|- T = ( ( LTrn ` K ) ` W ) |
7 |
|
trlval3.r |
|- R = ( ( trL ` K ) ` W ) |
8 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( K e. HL /\ W e. H ) ) |
9 |
|
simpl31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( P e. A /\ -. P .<_ W ) ) |
10 |
|
simpl2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> F e. T ) |
11 |
|
simpr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( F ` P ) = P ) |
12 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
13 |
1 12 4 5 6 7
|
trl0 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = ( 0. ` K ) ) |
14 |
8 9 10 11 13
|
syl112anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( R ` F ) = ( 0. ` K ) ) |
15 |
|
simpl33 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) |
16 |
|
simpl1l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> K e. HL ) |
17 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
18 |
16 17
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> K e. AtLat ) |
19 |
11
|
oveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( P .\/ ( F ` P ) ) = ( P .\/ P ) ) |
20 |
|
simp31l |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) -> P e. A ) |
21 |
20
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> P e. A ) |
22 |
2 4
|
hlatjidm |
|- ( ( K e. HL /\ P e. A ) -> ( P .\/ P ) = P ) |
23 |
16 21 22
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( P .\/ P ) = P ) |
24 |
19 23
|
eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( P .\/ ( F ` P ) ) = P ) |
25 |
24 21
|
eqeltrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( P .\/ ( F ` P ) ) e. A ) |
26 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
27 |
|
simp2 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) -> F e. T ) |
28 |
|
simp31 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
29 |
|
simp32 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
30 |
1 4 5 6
|
ltrn2ateq |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( ( F ` P ) = P <-> ( F ` Q ) = Q ) ) |
31 |
26 27 28 29 30
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) -> ( ( F ` P ) = P <-> ( F ` Q ) = Q ) ) |
32 |
31
|
biimpa |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( F ` Q ) = Q ) |
33 |
32
|
oveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( F ` Q ) ) = ( Q .\/ Q ) ) |
34 |
|
simp32l |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) -> Q e. A ) |
35 |
34
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> Q e. A ) |
36 |
2 4
|
hlatjidm |
|- ( ( K e. HL /\ Q e. A ) -> ( Q .\/ Q ) = Q ) |
37 |
16 35 36
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( Q .\/ Q ) = Q ) |
38 |
33 37
|
eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( F ` Q ) ) = Q ) |
39 |
38 35
|
eqeltrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( F ` Q ) ) e. A ) |
40 |
3 12 4
|
atnem0 |
|- ( ( K e. AtLat /\ ( P .\/ ( F ` P ) ) e. A /\ ( Q .\/ ( F ` Q ) ) e. A ) -> ( ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) <-> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) ) |
41 |
18 25 39 40
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) <-> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) ) |
42 |
15 41
|
mpbid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) |
43 |
14 42
|
eqtr4d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) = P ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) |
44 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( K e. HL /\ W e. H ) ) |
45 |
|
simpl2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> F e. T ) |
46 |
|
simpl31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( P e. A /\ -. P .<_ W ) ) |
47 |
1 2 3 4 5 6 7
|
trlval2 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ W ) ) |
48 |
44 45 46 47
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ W ) ) |
49 |
|
simpl1l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> K e. HL ) |
50 |
49
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> K e. Lat ) |
51 |
20
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> P e. A ) |
52 |
1 4 5 6
|
ltrnat |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A ) |
53 |
44 45 51 52
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( F ` P ) e. A ) |
54 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
55 |
54 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ ( F ` P ) e. A ) -> ( P .\/ ( F ` P ) ) e. ( Base ` K ) ) |
56 |
49 51 53 55
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( P .\/ ( F ` P ) ) e. ( Base ` K ) ) |
57 |
|
simpl1r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> W e. H ) |
58 |
54 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
59 |
57 58
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> W e. ( Base ` K ) ) |
60 |
54 1 3
|
latmle1 |
|- ( ( K e. Lat /\ ( P .\/ ( F ` P ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) .<_ ( P .\/ ( F ` P ) ) ) |
61 |
50 56 59 60
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) .<_ ( P .\/ ( F ` P ) ) ) |
62 |
48 61
|
eqbrtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) .<_ ( P .\/ ( F ` P ) ) ) |
63 |
|
simpl32 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( Q e. A /\ -. Q .<_ W ) ) |
64 |
1 2 3 4 5 6 7
|
trlval2 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( R ` F ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) ) |
65 |
44 45 63 64
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) ) |
66 |
34
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> Q e. A ) |
67 |
1 4 5 6
|
ltrnat |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ Q e. A ) -> ( F ` Q ) e. A ) |
68 |
44 45 66 67
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( F ` Q ) e. A ) |
69 |
54 2 4
|
hlatjcl |
|- ( ( K e. HL /\ Q e. A /\ ( F ` Q ) e. A ) -> ( Q .\/ ( F ` Q ) ) e. ( Base ` K ) ) |
70 |
49 66 68 69
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( Q .\/ ( F ` Q ) ) e. ( Base ` K ) ) |
71 |
54 1 3
|
latmle1 |
|- ( ( K e. Lat /\ ( Q .\/ ( F ` Q ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( Q .\/ ( F ` Q ) ) ./\ W ) .<_ ( Q .\/ ( F ` Q ) ) ) |
72 |
50 70 59 71
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( Q .\/ ( F ` Q ) ) ./\ W ) .<_ ( Q .\/ ( F ` Q ) ) ) |
73 |
65 72
|
eqbrtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) .<_ ( Q .\/ ( F ` Q ) ) ) |
74 |
54 5 6 7
|
trlcl |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) ) |
75 |
44 45 74
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) e. ( Base ` K ) ) |
76 |
54 1 3
|
latlem12 |
|- ( ( K e. Lat /\ ( ( R ` F ) e. ( Base ` K ) /\ ( P .\/ ( F ` P ) ) e. ( Base ` K ) /\ ( Q .\/ ( F ` Q ) ) e. ( Base ` K ) ) ) -> ( ( ( R ` F ) .<_ ( P .\/ ( F ` P ) ) /\ ( R ` F ) .<_ ( Q .\/ ( F ` Q ) ) ) <-> ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) ) |
77 |
50 75 56 70 76
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( ( R ` F ) .<_ ( P .\/ ( F ` P ) ) /\ ( R ` F ) .<_ ( Q .\/ ( F ` Q ) ) ) <-> ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) ) |
78 |
62 73 77
|
mpbi2and |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) |
79 |
49 17
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> K e. AtLat ) |
80 |
|
simpr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( F ` P ) =/= P ) |
81 |
1 4 5 6 7
|
trlat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A ) |
82 |
44 46 45 80 81
|
syl112anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) e. A ) |
83 |
54 3
|
latmcl |
|- ( ( K e. Lat /\ ( P .\/ ( F ` P ) ) e. ( Base ` K ) /\ ( Q .\/ ( F ` Q ) ) e. ( Base ` K ) ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. ( Base ` K ) ) |
84 |
50 56 70 83
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. ( Base ` K ) ) |
85 |
54 1 12 4
|
atlen0 |
|- ( ( ( K e. AtLat /\ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. ( Base ` K ) /\ ( R ` F ) e. A ) /\ ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) =/= ( 0. ` K ) ) |
86 |
79 84 82 78 85
|
syl31anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) =/= ( 0. ` K ) ) |
87 |
86
|
neneqd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> -. ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) |
88 |
|
simpl33 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) |
89 |
2 3 12 4
|
2atmat0 |
|- ( ( ( K e. HL /\ P e. A /\ ( F ` P ) e. A ) /\ ( Q e. A /\ ( F ` Q ) e. A /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) -> ( ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. A \/ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) ) |
90 |
49 51 53 66 68 88 89
|
syl33anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. A \/ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) ) |
91 |
90
|
ord |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( -. ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. A -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) ) |
92 |
87 91
|
mt3d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. A ) |
93 |
1 4
|
atcmp |
|- ( ( K e. AtLat /\ ( R ` F ) e. A /\ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. A ) -> ( ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) <-> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) ) |
94 |
79 82 92 93
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) <-> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) ) |
95 |
78 94
|
mpbid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) |
96 |
43 95
|
pm2.61dane |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) |