Step |
Hyp |
Ref |
Expression |
1 |
|
2llnja.l |
|- .<_ = ( le ` K ) |
2 |
|
2llnja.j |
|- .\/ = ( join ` K ) |
3 |
|
2llnja.a |
|- A = ( Atoms ` K ) |
4 |
|
2llnja.n |
|- N = ( LLines ` K ) |
5 |
|
2llnja.p |
|- P = ( LPlanes ` K ) |
6 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
7 |
|
simpl1l |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> K e. HL ) |
8 |
7
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> K e. Lat ) |
9 |
|
simpl21 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> Q e. A ) |
10 |
|
simpl22 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> R e. A ) |
11 |
6 2 3
|
hlatjcl |
|- ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) ) |
12 |
7 9 10 11
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( Q .\/ R ) e. ( Base ` K ) ) |
13 |
|
simpl31 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> S e. A ) |
14 |
|
simpl32 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> T e. A ) |
15 |
6 2 3
|
hlatjcl |
|- ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) ) |
16 |
7 13 14 15
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( S .\/ T ) e. ( Base ` K ) ) |
17 |
6 2
|
latjcl |
|- ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) -> ( ( Q .\/ R ) .\/ ( S .\/ T ) ) e. ( Base ` K ) ) |
18 |
8 12 16 17
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( Q .\/ R ) .\/ ( S .\/ T ) ) e. ( Base ` K ) ) |
19 |
|
simpl1r |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> W e. P ) |
20 |
6 5
|
lplnbase |
|- ( W e. P -> W e. ( Base ` K ) ) |
21 |
19 20
|
syl |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> W e. ( Base ` K ) ) |
22 |
|
simpr1 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( Q .\/ R ) .<_ W ) |
23 |
|
simpr2 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( S .\/ T ) .<_ W ) |
24 |
6 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( ( Q .\/ R ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W ) <-> ( ( Q .\/ R ) .\/ ( S .\/ T ) ) .<_ W ) ) |
25 |
8 12 16 21 24
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W ) <-> ( ( Q .\/ R ) .\/ ( S .\/ T ) ) .<_ W ) ) |
26 |
22 23 25
|
mpbi2and |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( Q .\/ R ) .\/ ( S .\/ T ) ) .<_ W ) |
27 |
6 3
|
atbase |
|- ( T e. A -> T e. ( Base ` K ) ) |
28 |
14 27
|
syl |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> T e. ( Base ` K ) ) |
29 |
6 2
|
latjcl |
|- ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> ( ( Q .\/ R ) .\/ T ) e. ( Base ` K ) ) |
30 |
8 12 28 29
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( Q .\/ R ) .\/ T ) e. ( Base ` K ) ) |
31 |
6 3
|
atbase |
|- ( S e. A -> S e. ( Base ` K ) ) |
32 |
13 31
|
syl |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> S e. ( Base ` K ) ) |
33 |
6 1 2
|
latlej2 |
|- ( ( K e. Lat /\ S e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> T .<_ ( S .\/ T ) ) |
34 |
8 32 28 33
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> T .<_ ( S .\/ T ) ) |
35 |
6 1 2
|
latjlej2 |
|- ( ( K e. Lat /\ ( T e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) ) -> ( T .<_ ( S .\/ T ) -> ( ( Q .\/ R ) .\/ T ) .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) ) |
36 |
8 28 16 12 35
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( T .<_ ( S .\/ T ) -> ( ( Q .\/ R ) .\/ T ) .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) ) |
37 |
34 36
|
mpd |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( Q .\/ R ) .\/ T ) .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) |
38 |
6 1 8 30 18 21 37 26
|
lattrd |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( Q .\/ R ) .\/ T ) .<_ W ) |
39 |
38
|
3adant3 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> ( ( Q .\/ R ) .\/ T ) .<_ W ) |
40 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> K e. HL ) |
41 |
|
simp121 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> Q e. A ) |
42 |
|
simp122 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> R e. A ) |
43 |
|
simp132 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> T e. A ) |
44 |
|
simp123 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> Q =/= R ) |
45 |
|
simp23 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> ( Q .\/ R ) =/= ( S .\/ T ) ) |
46 |
|
simpl3 |
|- ( ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) /\ T .<_ ( Q .\/ R ) ) -> S .<_ ( Q .\/ R ) ) |
47 |
|
simpr |
|- ( ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) /\ T .<_ ( Q .\/ R ) ) -> T .<_ ( Q .\/ R ) ) |
48 |
6 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ T e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) ) -> ( ( S .<_ ( Q .\/ R ) /\ T .<_ ( Q .\/ R ) ) <-> ( S .\/ T ) .<_ ( Q .\/ R ) ) ) |
49 |
8 32 28 12 48
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( S .<_ ( Q .\/ R ) /\ T .<_ ( Q .\/ R ) ) <-> ( S .\/ T ) .<_ ( Q .\/ R ) ) ) |
50 |
49
|
3adant3 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> ( ( S .<_ ( Q .\/ R ) /\ T .<_ ( Q .\/ R ) ) <-> ( S .\/ T ) .<_ ( Q .\/ R ) ) ) |
51 |
50
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) /\ T .<_ ( Q .\/ R ) ) -> ( ( S .<_ ( Q .\/ R ) /\ T .<_ ( Q .\/ R ) ) <-> ( S .\/ T ) .<_ ( Q .\/ R ) ) ) |
52 |
46 47 51
|
mpbi2and |
|- ( ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) /\ T .<_ ( Q .\/ R ) ) -> ( S .\/ T ) .<_ ( Q .\/ R ) ) |
53 |
|
simpl3 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( S e. A /\ T e. A /\ S =/= T ) ) |
54 |
1 2 3
|
ps-1 |
|- ( ( K e. HL /\ ( S e. A /\ T e. A /\ S =/= T ) /\ ( Q e. A /\ R e. A ) ) -> ( ( S .\/ T ) .<_ ( Q .\/ R ) <-> ( S .\/ T ) = ( Q .\/ R ) ) ) |
55 |
7 53 9 10 54
|
syl112anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( S .\/ T ) .<_ ( Q .\/ R ) <-> ( S .\/ T ) = ( Q .\/ R ) ) ) |
56 |
55
|
3adant3 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> ( ( S .\/ T ) .<_ ( Q .\/ R ) <-> ( S .\/ T ) = ( Q .\/ R ) ) ) |
57 |
56
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) /\ T .<_ ( Q .\/ R ) ) -> ( ( S .\/ T ) .<_ ( Q .\/ R ) <-> ( S .\/ T ) = ( Q .\/ R ) ) ) |
58 |
52 57
|
mpbid |
|- ( ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) /\ T .<_ ( Q .\/ R ) ) -> ( S .\/ T ) = ( Q .\/ R ) ) |
59 |
58
|
eqcomd |
|- ( ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) /\ T .<_ ( Q .\/ R ) ) -> ( Q .\/ R ) = ( S .\/ T ) ) |
60 |
59
|
ex |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> ( T .<_ ( Q .\/ R ) -> ( Q .\/ R ) = ( S .\/ T ) ) ) |
61 |
60
|
necon3ad |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> ( ( Q .\/ R ) =/= ( S .\/ T ) -> -. T .<_ ( Q .\/ R ) ) ) |
62 |
45 61
|
mpd |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> -. T .<_ ( Q .\/ R ) ) |
63 |
1 2 3 5
|
lplni2 |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ T e. A ) /\ ( Q =/= R /\ -. T .<_ ( Q .\/ R ) ) ) -> ( ( Q .\/ R ) .\/ T ) e. P ) |
64 |
40 41 42 43 44 62 63
|
syl132anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> ( ( Q .\/ R ) .\/ T ) e. P ) |
65 |
|
simp11r |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> W e. P ) |
66 |
1 5
|
lplncmp |
|- ( ( K e. HL /\ ( ( Q .\/ R ) .\/ T ) e. P /\ W e. P ) -> ( ( ( Q .\/ R ) .\/ T ) .<_ W <-> ( ( Q .\/ R ) .\/ T ) = W ) ) |
67 |
40 64 65 66
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> ( ( ( Q .\/ R ) .\/ T ) .<_ W <-> ( ( Q .\/ R ) .\/ T ) = W ) ) |
68 |
39 67
|
mpbid |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> ( ( Q .\/ R ) .\/ T ) = W ) |
69 |
37
|
3adant3 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> ( ( Q .\/ R ) .\/ T ) .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) |
70 |
68 69
|
eqbrtrrd |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ S .<_ ( Q .\/ R ) ) -> W .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) |
71 |
70
|
3expia |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( S .<_ ( Q .\/ R ) -> W .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) ) |
72 |
6 2
|
latjcl |
|- ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( Q .\/ R ) .\/ S ) e. ( Base ` K ) ) |
73 |
8 12 32 72
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( Q .\/ R ) .\/ S ) e. ( Base ` K ) ) |
74 |
6 1 2
|
latlej1 |
|- ( ( K e. Lat /\ S e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> S .<_ ( S .\/ T ) ) |
75 |
8 32 28 74
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> S .<_ ( S .\/ T ) ) |
76 |
6 1 2
|
latjlej2 |
|- ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) ) -> ( S .<_ ( S .\/ T ) -> ( ( Q .\/ R ) .\/ S ) .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) ) |
77 |
8 32 16 12 76
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( S .<_ ( S .\/ T ) -> ( ( Q .\/ R ) .\/ S ) .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) ) |
78 |
75 77
|
mpd |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( Q .\/ R ) .\/ S ) .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) |
79 |
6 1 8 73 18 21 78 26
|
lattrd |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( Q .\/ R ) .\/ S ) .<_ W ) |
80 |
79
|
3adant3 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> ( ( Q .\/ R ) .\/ S ) .<_ W ) |
81 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> K e. HL ) |
82 |
|
simp121 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> Q e. A ) |
83 |
|
simp122 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> R e. A ) |
84 |
|
simp131 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> S e. A ) |
85 |
|
simp123 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> Q =/= R ) |
86 |
|
simp3 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> -. S .<_ ( Q .\/ R ) ) |
87 |
1 2 3 5
|
lplni2 |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( ( Q .\/ R ) .\/ S ) e. P ) |
88 |
81 82 83 84 85 86 87
|
syl132anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> ( ( Q .\/ R ) .\/ S ) e. P ) |
89 |
|
simp11r |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> W e. P ) |
90 |
1 5
|
lplncmp |
|- ( ( K e. HL /\ ( ( Q .\/ R ) .\/ S ) e. P /\ W e. P ) -> ( ( ( Q .\/ R ) .\/ S ) .<_ W <-> ( ( Q .\/ R ) .\/ S ) = W ) ) |
91 |
81 88 89 90
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> ( ( ( Q .\/ R ) .\/ S ) .<_ W <-> ( ( Q .\/ R ) .\/ S ) = W ) ) |
92 |
80 91
|
mpbid |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> ( ( Q .\/ R ) .\/ S ) = W ) |
93 |
78
|
3adant3 |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> ( ( Q .\/ R ) .\/ S ) .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) |
94 |
92 93
|
eqbrtrrd |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) /\ -. S .<_ ( Q .\/ R ) ) -> W .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) |
95 |
94
|
3expia |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( -. S .<_ ( Q .\/ R ) -> W .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) ) |
96 |
71 95
|
pm2.61d |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> W .<_ ( ( Q .\/ R ) .\/ ( S .\/ T ) ) ) |
97 |
6 1 8 18 21 26 96
|
latasymd |
|- ( ( ( ( K e. HL /\ W e. P ) /\ ( Q e. A /\ R e. A /\ Q =/= R ) /\ ( S e. A /\ T e. A /\ S =/= T ) ) /\ ( ( Q .\/ R ) .<_ W /\ ( S .\/ T ) .<_ W /\ ( Q .\/ R ) =/= ( S .\/ T ) ) ) -> ( ( Q .\/ R ) .\/ ( S .\/ T ) ) = W ) |