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 |
|
4thatlem0.d |
|- D = ( ( R .\/ T ) ./\ ( P .\/ S ) ) |
11 |
1 2 3 4 5 6 7 8
|
4atexlemtlw |
|- ( ph -> T .<_ W ) |
12 |
1 2 3 4 5 6 7 8 9
|
4atexlemnclw |
|- ( ph -> -. C .<_ W ) |
13 |
|
nbrne2 |
|- ( ( T .<_ W /\ -. C .<_ W ) -> T =/= C ) |
14 |
11 12 13
|
syl2anc |
|- ( ph -> T =/= C ) |
15 |
1
|
4atexlemk |
|- ( ph -> K e. HL ) |
16 |
1
|
4atexlemq |
|- ( ph -> Q e. A ) |
17 |
1
|
4atexlemt |
|- ( ph -> T e. A ) |
18 |
3 5
|
hlatjcom |
|- ( ( K e. HL /\ Q e. A /\ T e. A ) -> ( Q .\/ T ) = ( T .\/ Q ) ) |
19 |
15 16 17 18
|
syl3anc |
|- ( ph -> ( Q .\/ T ) = ( T .\/ Q ) ) |
20 |
|
simp221 |
|- ( ( ( ( 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 ) ) ) -> R e. A ) |
21 |
1 20
|
sylbi |
|- ( ph -> R e. A ) |
22 |
3 5
|
hlatjcom |
|- ( ( K e. HL /\ R e. A /\ T e. A ) -> ( R .\/ T ) = ( T .\/ R ) ) |
23 |
15 21 17 22
|
syl3anc |
|- ( ph -> ( R .\/ T ) = ( T .\/ R ) ) |
24 |
19 23
|
oveq12d |
|- ( ph -> ( ( Q .\/ T ) ./\ ( R .\/ T ) ) = ( ( T .\/ Q ) ./\ ( T .\/ R ) ) ) |
25 |
1
|
4atexlemkc |
|- ( ph -> K e. CvLat ) |
26 |
1
|
4atexlemp |
|- ( ph -> P e. A ) |
27 |
1
|
4atexlempnq |
|- ( ph -> P =/= Q ) |
28 |
|
simp223 |
|- ( ( ( ( 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 ) ) ) -> ( P .\/ R ) = ( Q .\/ R ) ) |
29 |
1 28
|
sylbi |
|- ( ph -> ( P .\/ R ) = ( Q .\/ R ) ) |
30 |
5 3
|
cvlsupr6 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> R =/= Q ) |
31 |
30
|
necomd |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> Q =/= R ) |
32 |
25 26 16 21 27 29 31
|
syl132anc |
|- ( ph -> Q =/= R ) |
33 |
1 2 3 4 5 6 7 8
|
4atexlemntlpq |
|- ( ph -> -. T .<_ ( P .\/ Q ) ) |
34 |
5 3
|
cvlsupr7 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> ( P .\/ Q ) = ( R .\/ Q ) ) |
35 |
25 26 16 21 27 29 34
|
syl132anc |
|- ( ph -> ( P .\/ Q ) = ( R .\/ Q ) ) |
36 |
3 5
|
hlatjcom |
|- ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) = ( R .\/ Q ) ) |
37 |
15 16 21 36
|
syl3anc |
|- ( ph -> ( Q .\/ R ) = ( R .\/ Q ) ) |
38 |
35 37
|
eqtr4d |
|- ( ph -> ( P .\/ Q ) = ( Q .\/ R ) ) |
39 |
38
|
breq2d |
|- ( ph -> ( T .<_ ( P .\/ Q ) <-> T .<_ ( Q .\/ R ) ) ) |
40 |
33 39
|
mtbid |
|- ( ph -> -. T .<_ ( Q .\/ R ) ) |
41 |
2 3 4 5
|
2llnma2 |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ T e. A ) /\ ( Q =/= R /\ -. T .<_ ( Q .\/ R ) ) ) -> ( ( T .\/ Q ) ./\ ( T .\/ R ) ) = T ) |
42 |
15 16 21 17 32 40 41
|
syl132anc |
|- ( ph -> ( ( T .\/ Q ) ./\ ( T .\/ R ) ) = T ) |
43 |
24 42
|
eqtr2d |
|- ( ph -> T = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) |
44 |
43
|
adantr |
|- ( ( ph /\ C = D ) -> T = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) |
45 |
1
|
4atexlemkl |
|- ( ph -> K e. Lat ) |
46 |
1 3 5
|
4atexlemqtb |
|- ( ph -> ( Q .\/ T ) e. ( Base ` K ) ) |
47 |
1 3 5
|
4atexlempsb |
|- ( ph -> ( P .\/ S ) e. ( Base ` K ) ) |
48 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
49 |
48 2 4
|
latmle1 |
|- ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) ) |
50 |
45 46 47 49
|
syl3anc |
|- ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) ) |
51 |
9 50
|
eqbrtrid |
|- ( ph -> C .<_ ( Q .\/ T ) ) |
52 |
51
|
adantr |
|- ( ( ph /\ C = D ) -> C .<_ ( Q .\/ T ) ) |
53 |
|
simpr |
|- ( ( ph /\ C = D ) -> C = D ) |
54 |
48 3 5
|
hlatjcl |
|- ( ( K e. HL /\ R e. A /\ T e. A ) -> ( R .\/ T ) e. ( Base ` K ) ) |
55 |
15 21 17 54
|
syl3anc |
|- ( ph -> ( R .\/ T ) e. ( Base ` K ) ) |
56 |
48 2 4
|
latmle1 |
|- ( ( K e. Lat /\ ( R .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( R .\/ T ) ./\ ( P .\/ S ) ) .<_ ( R .\/ T ) ) |
57 |
45 55 47 56
|
syl3anc |
|- ( ph -> ( ( R .\/ T ) ./\ ( P .\/ S ) ) .<_ ( R .\/ T ) ) |
58 |
10 57
|
eqbrtrid |
|- ( ph -> D .<_ ( R .\/ T ) ) |
59 |
58
|
adantr |
|- ( ( ph /\ C = D ) -> D .<_ ( R .\/ T ) ) |
60 |
53 59
|
eqbrtrd |
|- ( ( ph /\ C = D ) -> C .<_ ( R .\/ T ) ) |
61 |
1 2 3 4 5 6 7 8 9
|
4atexlemc |
|- ( ph -> C e. A ) |
62 |
48 5
|
atbase |
|- ( C e. A -> C e. ( Base ` K ) ) |
63 |
61 62
|
syl |
|- ( ph -> C e. ( Base ` K ) ) |
64 |
48 2 4
|
latlem12 |
|- ( ( K e. Lat /\ ( C e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( R .\/ T ) e. ( Base ` K ) ) ) -> ( ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ T ) ) <-> C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) ) |
65 |
45 63 46 55 64
|
syl13anc |
|- ( ph -> ( ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ T ) ) <-> C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) ) |
66 |
65
|
adantr |
|- ( ( ph /\ C = D ) -> ( ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ T ) ) <-> C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) ) |
67 |
52 60 66
|
mpbi2and |
|- ( ( ph /\ C = D ) -> C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) |
68 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
69 |
15 68
|
syl |
|- ( ph -> K e. AtLat ) |
70 |
43 17
|
eqeltrrd |
|- ( ph -> ( ( Q .\/ T ) ./\ ( R .\/ T ) ) e. A ) |
71 |
2 5
|
atcmp |
|- ( ( K e. AtLat /\ C e. A /\ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) e. A ) -> ( C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) <-> C = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) ) |
72 |
69 61 70 71
|
syl3anc |
|- ( ph -> ( C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) <-> C = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) ) |
73 |
72
|
adantr |
|- ( ( ph /\ C = D ) -> ( C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) <-> C = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) ) |
74 |
67 73
|
mpbid |
|- ( ( ph /\ C = D ) -> C = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) |
75 |
44 74
|
eqtr4d |
|- ( ( ph /\ C = D ) -> T = C ) |
76 |
75
|
ex |
|- ( ph -> ( C = D -> T = C ) ) |
77 |
76
|
necon3d |
|- ( ph -> ( T =/= C -> C =/= D ) ) |
78 |
14 77
|
mpd |
|- ( ph -> C =/= D ) |