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 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) ) |
9 |
|
simp21 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> F e. T ) |
10 |
|
simp22 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
11 |
|
simp23 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
12 |
|
simp3r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> -. ( R ` F ) .<_ ( P .\/ Q ) ) |
13 |
|
simpl1l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> K e. HL ) |
14 |
|
simp23l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> Q e. A ) |
15 |
14
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> Q e. A ) |
16 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( K e. HL /\ W e. H ) ) |
17 |
|
simpl21 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> F e. T ) |
18 |
1 4 5 6
|
ltrnat |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ Q e. A ) -> ( F ` Q ) e. A ) |
19 |
16 17 15 18
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( F ` Q ) e. A ) |
20 |
1 2 4
|
hlatlej1 |
|- ( ( K e. HL /\ Q e. A /\ ( F ` Q ) e. A ) -> Q .<_ ( Q .\/ ( F ` Q ) ) ) |
21 |
13 15 19 20
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> Q .<_ ( Q .\/ ( F ` Q ) ) ) |
22 |
|
simpl22 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
23 |
1 2 4 5 6 7
|
trljat1 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` F ) ) = ( P .\/ ( F ` P ) ) ) |
24 |
16 17 22 23
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( P .\/ ( R ` F ) ) = ( P .\/ ( F ` P ) ) ) |
25 |
|
simpr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) |
26 |
24 25
|
eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( P .\/ ( R ` F ) ) = ( Q .\/ ( F ` Q ) ) ) |
27 |
21 26
|
breqtrrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> Q .<_ ( P .\/ ( R ` F ) ) ) |
28 |
|
simpl3r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> -. ( R ` F ) .<_ ( P .\/ Q ) ) |
29 |
|
simpll1 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( K e. HL /\ W e. H ) ) |
30 |
22
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( P e. A /\ -. P .<_ W ) ) |
31 |
17
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> F e. T ) |
32 |
|
simpr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( F ` P ) = P ) |
33 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
34 |
1 33 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 ) ) |
35 |
29 30 31 32 34
|
syl112anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( R ` F ) = ( 0. ` K ) ) |
36 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
37 |
13 36
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> K e. AtLat ) |
38 |
|
simp22l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> P e. A ) |
39 |
38
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> P e. A ) |
40 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
41 |
40 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
42 |
13 39 15 41
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
43 |
40 1 33
|
atl0le |
|- ( ( K e. AtLat /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( 0. ` K ) .<_ ( P .\/ Q ) ) |
44 |
37 42 43
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( 0. ` K ) .<_ ( P .\/ Q ) ) |
45 |
44
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( 0. ` K ) .<_ ( P .\/ Q ) ) |
46 |
35 45
|
eqbrtrd |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( R ` F ) .<_ ( P .\/ Q ) ) |
47 |
46
|
ex |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( ( F ` P ) = P -> ( R ` F ) .<_ ( P .\/ Q ) ) ) |
48 |
47
|
necon3bd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( -. ( R ` F ) .<_ ( P .\/ Q ) -> ( F ` P ) =/= P ) ) |
49 |
28 48
|
mpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( F ` P ) =/= P ) |
50 |
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 ) |
51 |
16 22 17 49 50
|
syl112anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( R ` F ) e. A ) |
52 |
|
simpl3l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> P =/= Q ) |
53 |
52
|
necomd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> Q =/= P ) |
54 |
1 2 4
|
hlatexch1 |
|- ( ( K e. HL /\ ( Q e. A /\ ( R ` F ) e. A /\ P e. A ) /\ Q =/= P ) -> ( Q .<_ ( P .\/ ( R ` F ) ) -> ( R ` F ) .<_ ( P .\/ Q ) ) ) |
55 |
13 15 51 39 53 54
|
syl131anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( Q .<_ ( P .\/ ( R ` F ) ) -> ( R ` F ) .<_ ( P .\/ Q ) ) ) |
56 |
27 55
|
mpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( R ` F ) .<_ ( P .\/ Q ) ) |
57 |
56
|
ex |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) -> ( R ` F ) .<_ ( P .\/ Q ) ) ) |
58 |
57
|
necon3bd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( -. ( R ` F ) .<_ ( P .\/ Q ) -> ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) |
59 |
12 58
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) |
60 |
1 2 3 4 5 6 7
|
trlval3 |
|- ( ( ( 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 ) ) ) ) |
61 |
8 9 10 11 59 60
|
syl113anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) |