Step |
Hyp |
Ref |
Expression |
1 |
|
dalem.ph |
|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) ) |
2 |
|
dalem.l |
|- .<_ = ( le ` K ) |
3 |
|
dalem.j |
|- .\/ = ( join ` K ) |
4 |
|
dalem.a |
|- A = ( Atoms ` K ) |
5 |
|
dalem.ps |
|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) ) |
6 |
|
dalem21.m |
|- ./\ = ( meet ` K ) |
7 |
|
dalem21.o |
|- O = ( LPlanes ` K ) |
8 |
|
dalem21.y |
|- Y = ( ( P .\/ Q ) .\/ R ) |
9 |
|
dalem21.z |
|- Z = ( ( S .\/ T ) .\/ U ) |
10 |
1
|
dalemkehl |
|- ( ph -> K e. HL ) |
11 |
10
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> K e. HL ) |
12 |
1 2 3 4 5
|
dalemcjden |
|- ( ( ph /\ ps ) -> ( c .\/ d ) e. ( LLines ` K ) ) |
13 |
12
|
3adant2 |
|- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ d ) e. ( LLines ` K ) ) |
14 |
1 2 3 4 7 8
|
dalempjsen |
|- ( ph -> ( P .\/ S ) e. ( LLines ` K ) ) |
15 |
14
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ S ) e. ( LLines ` K ) ) |
16 |
1 2 3 4 7 8
|
dalemply |
|- ( ph -> P .<_ Y ) |
17 |
16
|
adantr |
|- ( ( ph /\ Y = Z ) -> P .<_ Y ) |
18 |
1 2 3 4 9
|
dalemsly |
|- ( ( ph /\ Y = Z ) -> S .<_ Y ) |
19 |
1
|
dalemkelat |
|- ( ph -> K e. Lat ) |
20 |
1 4
|
dalempeb |
|- ( ph -> P e. ( Base ` K ) ) |
21 |
1 4
|
dalemseb |
|- ( ph -> S e. ( Base ` K ) ) |
22 |
1 7
|
dalemyeb |
|- ( ph -> Y e. ( Base ` K ) ) |
23 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
24 |
23 2 3
|
latjle12 |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ S e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( P .<_ Y /\ S .<_ Y ) <-> ( P .\/ S ) .<_ Y ) ) |
25 |
19 20 21 22 24
|
syl13anc |
|- ( ph -> ( ( P .<_ Y /\ S .<_ Y ) <-> ( P .\/ S ) .<_ Y ) ) |
26 |
25
|
adantr |
|- ( ( ph /\ Y = Z ) -> ( ( P .<_ Y /\ S .<_ Y ) <-> ( P .\/ S ) .<_ Y ) ) |
27 |
17 18 26
|
mpbi2and |
|- ( ( ph /\ Y = Z ) -> ( P .\/ S ) .<_ Y ) |
28 |
27
|
3adant3 |
|- ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ S ) .<_ Y ) |
29 |
5
|
dalem-ccly |
|- ( ps -> -. c .<_ Y ) |
30 |
29
|
adantl |
|- ( ( ph /\ ps ) -> -. c .<_ Y ) |
31 |
19
|
adantr |
|- ( ( ph /\ ps ) -> K e. Lat ) |
32 |
5 4
|
dalemcceb |
|- ( ps -> c e. ( Base ` K ) ) |
33 |
32
|
adantl |
|- ( ( ph /\ ps ) -> c e. ( Base ` K ) ) |
34 |
5
|
dalemddea |
|- ( ps -> d e. A ) |
35 |
23 4
|
atbase |
|- ( d e. A -> d e. ( Base ` K ) ) |
36 |
34 35
|
syl |
|- ( ps -> d e. ( Base ` K ) ) |
37 |
36
|
adantl |
|- ( ( ph /\ ps ) -> d e. ( Base ` K ) ) |
38 |
23 2 3
|
latlej1 |
|- ( ( K e. Lat /\ c e. ( Base ` K ) /\ d e. ( Base ` K ) ) -> c .<_ ( c .\/ d ) ) |
39 |
31 33 37 38
|
syl3anc |
|- ( ( ph /\ ps ) -> c .<_ ( c .\/ d ) ) |
40 |
|
eqid |
|- ( LLines ` K ) = ( LLines ` K ) |
41 |
23 40
|
llnbase |
|- ( ( c .\/ d ) e. ( LLines ` K ) -> ( c .\/ d ) e. ( Base ` K ) ) |
42 |
12 41
|
syl |
|- ( ( ph /\ ps ) -> ( c .\/ d ) e. ( Base ` K ) ) |
43 |
22
|
adantr |
|- ( ( ph /\ ps ) -> Y e. ( Base ` K ) ) |
44 |
23 2
|
lattr |
|- ( ( K e. Lat /\ ( c e. ( Base ` K ) /\ ( c .\/ d ) e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( c .<_ ( c .\/ d ) /\ ( c .\/ d ) .<_ Y ) -> c .<_ Y ) ) |
45 |
31 33 42 43 44
|
syl13anc |
|- ( ( ph /\ ps ) -> ( ( c .<_ ( c .\/ d ) /\ ( c .\/ d ) .<_ Y ) -> c .<_ Y ) ) |
46 |
39 45
|
mpand |
|- ( ( ph /\ ps ) -> ( ( c .\/ d ) .<_ Y -> c .<_ Y ) ) |
47 |
30 46
|
mtod |
|- ( ( ph /\ ps ) -> -. ( c .\/ d ) .<_ Y ) |
48 |
47
|
3adant2 |
|- ( ( ph /\ Y = Z /\ ps ) -> -. ( c .\/ d ) .<_ Y ) |
49 |
|
nbrne2 |
|- ( ( ( P .\/ S ) .<_ Y /\ -. ( c .\/ d ) .<_ Y ) -> ( P .\/ S ) =/= ( c .\/ d ) ) |
50 |
28 48 49
|
syl2anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ S ) =/= ( c .\/ d ) ) |
51 |
50
|
necomd |
|- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ d ) =/= ( P .\/ S ) ) |
52 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
53 |
10 52
|
syl |
|- ( ph -> K e. AtLat ) |
54 |
53
|
adantr |
|- ( ( ph /\ ps ) -> K e. AtLat ) |
55 |
1
|
dalempea |
|- ( ph -> P e. A ) |
56 |
1
|
dalemsea |
|- ( ph -> S e. A ) |
57 |
23 3 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) ) |
58 |
10 55 56 57
|
syl3anc |
|- ( ph -> ( P .\/ S ) e. ( Base ` K ) ) |
59 |
58
|
adantr |
|- ( ( ph /\ ps ) -> ( P .\/ S ) e. ( Base ` K ) ) |
60 |
23 6
|
latmcl |
|- ( ( K e. Lat /\ ( c .\/ d ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) e. ( Base ` K ) ) |
61 |
31 42 59 60
|
syl3anc |
|- ( ( ph /\ ps ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) e. ( Base ` K ) ) |
62 |
1 2 3 4 7 8
|
dalemcea |
|- ( ph -> C e. A ) |
63 |
62
|
adantr |
|- ( ( ph /\ ps ) -> C e. A ) |
64 |
5
|
dalemclccjdd |
|- ( ps -> C .<_ ( c .\/ d ) ) |
65 |
64
|
adantl |
|- ( ( ph /\ ps ) -> C .<_ ( c .\/ d ) ) |
66 |
1
|
dalemclpjs |
|- ( ph -> C .<_ ( P .\/ S ) ) |
67 |
66
|
adantr |
|- ( ( ph /\ ps ) -> C .<_ ( P .\/ S ) ) |
68 |
1 4
|
dalemceb |
|- ( ph -> C e. ( Base ` K ) ) |
69 |
68
|
adantr |
|- ( ( ph /\ ps ) -> C e. ( Base ` K ) ) |
70 |
23 2 6
|
latlem12 |
|- ( ( K e. Lat /\ ( C e. ( Base ` K ) /\ ( c .\/ d ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) ) -> ( ( C .<_ ( c .\/ d ) /\ C .<_ ( P .\/ S ) ) <-> C .<_ ( ( c .\/ d ) ./\ ( P .\/ S ) ) ) ) |
71 |
31 69 42 59 70
|
syl13anc |
|- ( ( ph /\ ps ) -> ( ( C .<_ ( c .\/ d ) /\ C .<_ ( P .\/ S ) ) <-> C .<_ ( ( c .\/ d ) ./\ ( P .\/ S ) ) ) ) |
72 |
65 67 71
|
mpbi2and |
|- ( ( ph /\ ps ) -> C .<_ ( ( c .\/ d ) ./\ ( P .\/ S ) ) ) |
73 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
74 |
23 2 73 4
|
atlen0 |
|- ( ( ( K e. AtLat /\ ( ( c .\/ d ) ./\ ( P .\/ S ) ) e. ( Base ` K ) /\ C e. A ) /\ C .<_ ( ( c .\/ d ) ./\ ( P .\/ S ) ) ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) =/= ( 0. ` K ) ) |
75 |
54 61 63 72 74
|
syl31anc |
|- ( ( ph /\ ps ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) =/= ( 0. ` K ) ) |
76 |
75
|
3adant2 |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) =/= ( 0. ` K ) ) |
77 |
6 73 4 40
|
2llnmat |
|- ( ( ( K e. HL /\ ( c .\/ d ) e. ( LLines ` K ) /\ ( P .\/ S ) e. ( LLines ` K ) ) /\ ( ( c .\/ d ) =/= ( P .\/ S ) /\ ( ( c .\/ d ) ./\ ( P .\/ S ) ) =/= ( 0. ` K ) ) ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) e. A ) |
78 |
11 13 15 51 76 77
|
syl32anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) e. A ) |