Step |
Hyp |
Ref |
Expression |
1 |
|
4thatlem.ph |
|- ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) ) |
2 |
|
4thatlem0.l |
|- .<_ = ( le ` K ) |
3 |
|
4thatlem0.j |
|- .\/ = ( join ` K ) |
4 |
|
4thatlem0.m |
|- ./\ = ( meet ` K ) |
5 |
|
4thatlem0.a |
|- A = ( Atoms ` K ) |
6 |
|
4thatlem0.h |
|- H = ( LHyp ` K ) |
7 |
|
4thatlem0.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
4thatlem0.v |
|- V = ( ( P .\/ S ) ./\ W ) |
9 |
|
4thatlem0.c |
|- C = ( ( Q .\/ T ) ./\ ( P .\/ S ) ) |
10 |
1
|
4atexlemkl |
|- ( ph -> K e. Lat ) |
11 |
1 3 5
|
4atexlemqtb |
|- ( ph -> ( Q .\/ T ) e. ( Base ` K ) ) |
12 |
1 3 5
|
4atexlempsb |
|- ( ph -> ( P .\/ S ) e. ( Base ` K ) ) |
13 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
14 |
13 4
|
latmcom |
|- ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) |
15 |
10 11 12 14
|
syl3anc |
|- ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) |
16 |
9 15
|
syl5eq |
|- ( ph -> C = ( ( P .\/ S ) ./\ ( Q .\/ T ) ) ) |
17 |
1
|
4atexlemk |
|- ( ph -> K e. HL ) |
18 |
1
|
4atexlemp |
|- ( ph -> P e. A ) |
19 |
1
|
4atexlems |
|- ( ph -> S e. A ) |
20 |
1
|
4atexlemq |
|- ( ph -> Q e. A ) |
21 |
1
|
4atexlemt |
|- ( ph -> T e. A ) |
22 |
1 2 3 5
|
4atexlempns |
|- ( ph -> P =/= S ) |
23 |
1 2 3 4 5 6 7 8
|
4atexlemntlpq |
|- ( ph -> -. T .<_ ( P .\/ Q ) ) |
24 |
2 3 5
|
atnlej2 |
|- ( ( K e. HL /\ ( T e. A /\ P e. A /\ Q e. A ) /\ -. T .<_ ( P .\/ Q ) ) -> T =/= Q ) |
25 |
24
|
necomd |
|- ( ( K e. HL /\ ( T e. A /\ P e. A /\ Q e. A ) /\ -. T .<_ ( P .\/ Q ) ) -> Q =/= T ) |
26 |
17 21 18 20 23 25
|
syl131anc |
|- ( ph -> Q =/= T ) |
27 |
1
|
4atexlempnq |
|- ( ph -> P =/= Q ) |
28 |
1
|
4atexlemnslpq |
|- ( ph -> -. S .<_ ( P .\/ Q ) ) |
29 |
2 3 5
|
4atlem0ae |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. Q .<_ ( P .\/ S ) ) |
30 |
17 18 20 19 27 28 29
|
syl132anc |
|- ( ph -> -. Q .<_ ( P .\/ S ) ) |
31 |
13 5
|
atbase |
|- ( T e. A -> T e. ( Base ` K ) ) |
32 |
21 31
|
syl |
|- ( ph -> T e. ( Base ` K ) ) |
33 |
1 2 3 4 5 6 7
|
4atexlemu |
|- ( ph -> U e. A ) |
34 |
1 2 3 4 5 6 7 8
|
4atexlemv |
|- ( ph -> V e. A ) |
35 |
13 3 5
|
hlatjcl |
|- ( ( K e. HL /\ U e. A /\ V e. A ) -> ( U .\/ V ) e. ( Base ` K ) ) |
36 |
17 33 34 35
|
syl3anc |
|- ( ph -> ( U .\/ V ) e. ( Base ` K ) ) |
37 |
13 5
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
38 |
20 37
|
syl |
|- ( ph -> Q e. ( Base ` K ) ) |
39 |
13 3
|
latjcl |
|- ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( ( P .\/ S ) .\/ Q ) e. ( Base ` K ) ) |
40 |
10 12 38 39
|
syl3anc |
|- ( ph -> ( ( P .\/ S ) .\/ Q ) e. ( Base ` K ) ) |
41 |
1
|
4atexlemkc |
|- ( ph -> K e. CvLat ) |
42 |
1 2 3 4 5 6 7 8
|
4atexlemunv |
|- ( ph -> U =/= V ) |
43 |
1
|
4atexlemutvt |
|- ( ph -> ( U .\/ T ) = ( V .\/ T ) ) |
44 |
5 2 3
|
cvlsupr4 |
|- ( ( K e. CvLat /\ ( U e. A /\ V e. A /\ T e. A ) /\ ( U =/= V /\ ( U .\/ T ) = ( V .\/ T ) ) ) -> T .<_ ( U .\/ V ) ) |
45 |
41 33 34 21 42 43 44
|
syl132anc |
|- ( ph -> T .<_ ( U .\/ V ) ) |
46 |
13 3 5
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
47 |
17 18 20 46
|
syl3anc |
|- ( ph -> ( P .\/ Q ) e. ( Base ` K ) ) |
48 |
1 6
|
4atexlemwb |
|- ( ph -> W e. ( Base ` K ) ) |
49 |
13 2 4
|
latmle1 |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) ) |
50 |
10 47 48 49
|
syl3anc |
|- ( ph -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) ) |
51 |
7 50
|
eqbrtrid |
|- ( ph -> U .<_ ( P .\/ Q ) ) |
52 |
13 2 4
|
latmle1 |
|- ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) ) |
53 |
10 12 48 52
|
syl3anc |
|- ( ph -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) ) |
54 |
8 53
|
eqbrtrid |
|- ( ph -> V .<_ ( P .\/ S ) ) |
55 |
13 5
|
atbase |
|- ( U e. A -> U e. ( Base ` K ) ) |
56 |
33 55
|
syl |
|- ( ph -> U e. ( Base ` K ) ) |
57 |
13 5
|
atbase |
|- ( V e. A -> V e. ( Base ` K ) ) |
58 |
34 57
|
syl |
|- ( ph -> V e. ( Base ` K ) ) |
59 |
13 2 3
|
latjlej12 |
|- ( ( K e. Lat /\ ( U e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) /\ ( V e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) ) -> ( ( U .<_ ( P .\/ Q ) /\ V .<_ ( P .\/ S ) ) -> ( U .\/ V ) .<_ ( ( P .\/ Q ) .\/ ( P .\/ S ) ) ) ) |
60 |
10 56 47 58 12 59
|
syl122anc |
|- ( ph -> ( ( U .<_ ( P .\/ Q ) /\ V .<_ ( P .\/ S ) ) -> ( U .\/ V ) .<_ ( ( P .\/ Q ) .\/ ( P .\/ S ) ) ) ) |
61 |
51 54 60
|
mp2and |
|- ( ph -> ( U .\/ V ) .<_ ( ( P .\/ Q ) .\/ ( P .\/ S ) ) ) |
62 |
3 5
|
hlatjass |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ S ) = ( P .\/ ( Q .\/ S ) ) ) |
63 |
17 18 20 19 62
|
syl13anc |
|- ( ph -> ( ( P .\/ Q ) .\/ S ) = ( P .\/ ( Q .\/ S ) ) ) |
64 |
13 5
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
65 |
18 64
|
syl |
|- ( ph -> P e. ( Base ` K ) ) |
66 |
13 5
|
atbase |
|- ( S e. A -> S e. ( Base ` K ) ) |
67 |
19 66
|
syl |
|- ( ph -> S e. ( Base ` K ) ) |
68 |
13 3
|
latj32 |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( P .\/ Q ) .\/ S ) = ( ( P .\/ S ) .\/ Q ) ) |
69 |
10 65 38 67 68
|
syl13anc |
|- ( ph -> ( ( P .\/ Q ) .\/ S ) = ( ( P .\/ S ) .\/ Q ) ) |
70 |
13 3
|
latjjdi |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( P .\/ ( Q .\/ S ) ) = ( ( P .\/ Q ) .\/ ( P .\/ S ) ) ) |
71 |
10 65 38 67 70
|
syl13anc |
|- ( ph -> ( P .\/ ( Q .\/ S ) ) = ( ( P .\/ Q ) .\/ ( P .\/ S ) ) ) |
72 |
63 69 71
|
3eqtr3rd |
|- ( ph -> ( ( P .\/ Q ) .\/ ( P .\/ S ) ) = ( ( P .\/ S ) .\/ Q ) ) |
73 |
61 72
|
breqtrd |
|- ( ph -> ( U .\/ V ) .<_ ( ( P .\/ S ) .\/ Q ) ) |
74 |
13 2 10 32 36 40 45 73
|
lattrd |
|- ( ph -> T .<_ ( ( P .\/ S ) .\/ Q ) ) |
75 |
2 3 4 5
|
2atmat |
|- ( ( ( K e. HL /\ P e. A /\ S e. A ) /\ ( Q e. A /\ T e. A /\ P =/= S ) /\ ( Q =/= T /\ -. Q .<_ ( P .\/ S ) /\ T .<_ ( ( P .\/ S ) .\/ Q ) ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. A ) |
76 |
17 18 19 20 21 22 26 30 74 75
|
syl333anc |
|- ( ph -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. A ) |
77 |
16 76
|
eqeltrd |
|- ( ph -> C e. A ) |