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 2 3 4 5 6 7 8 9
|
4atexlemc |
|- ( ph -> C e. A ) |
11 |
10
|
adantr |
|- ( ( ph /\ C =/= S ) -> C e. A ) |
12 |
1 2 3 4 5 6 7 8 9
|
4atexlemnclw |
|- ( ph -> -. C .<_ W ) |
13 |
12
|
adantr |
|- ( ( ph /\ C =/= S ) -> -. C .<_ W ) |
14 |
1 2 3 4 5 6 7 8
|
4atexlemntlpq |
|- ( ph -> -. T .<_ ( P .\/ Q ) ) |
15 |
|
id |
|- ( C = P -> C = P ) |
16 |
9 15
|
eqtr3id |
|- ( C = P -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = P ) |
17 |
16
|
adantl |
|- ( ( ph /\ C = P ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = P ) |
18 |
1
|
4atexlemkl |
|- ( ph -> K e. Lat ) |
19 |
1 3 5
|
4atexlemqtb |
|- ( ph -> ( Q .\/ T ) e. ( Base ` K ) ) |
20 |
1 3 5
|
4atexlempsb |
|- ( ph -> ( P .\/ S ) e. ( Base ` K ) ) |
21 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
22 |
21 2 4
|
latmle1 |
|- ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) ) |
23 |
18 19 20 22
|
syl3anc |
|- ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) ) |
24 |
1
|
4atexlemk |
|- ( ph -> K e. HL ) |
25 |
1
|
4atexlemq |
|- ( ph -> Q e. A ) |
26 |
1
|
4atexlemt |
|- ( ph -> T e. A ) |
27 |
3 5
|
hlatjcom |
|- ( ( K e. HL /\ Q e. A /\ T e. A ) -> ( Q .\/ T ) = ( T .\/ Q ) ) |
28 |
24 25 26 27
|
syl3anc |
|- ( ph -> ( Q .\/ T ) = ( T .\/ Q ) ) |
29 |
23 28
|
breqtrd |
|- ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( T .\/ Q ) ) |
30 |
29
|
adantr |
|- ( ( ph /\ C = P ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( T .\/ Q ) ) |
31 |
17 30
|
eqbrtrrd |
|- ( ( ph /\ C = P ) -> P .<_ ( T .\/ Q ) ) |
32 |
1
|
4atexlemkc |
|- ( ph -> K e. CvLat ) |
33 |
1
|
4atexlemp |
|- ( ph -> P e. A ) |
34 |
1
|
4atexlempnq |
|- ( ph -> P =/= Q ) |
35 |
2 3 5
|
cvlatexch2 |
|- ( ( K e. CvLat /\ ( P e. A /\ T e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .<_ ( T .\/ Q ) -> T .<_ ( P .\/ Q ) ) ) |
36 |
32 33 26 25 34 35
|
syl131anc |
|- ( ph -> ( P .<_ ( T .\/ Q ) -> T .<_ ( P .\/ Q ) ) ) |
37 |
36
|
adantr |
|- ( ( ph /\ C = P ) -> ( P .<_ ( T .\/ Q ) -> T .<_ ( P .\/ Q ) ) ) |
38 |
31 37
|
mpd |
|- ( ( ph /\ C = P ) -> T .<_ ( P .\/ Q ) ) |
39 |
38
|
ex |
|- ( ph -> ( C = P -> T .<_ ( P .\/ Q ) ) ) |
40 |
39
|
necon3bd |
|- ( ph -> ( -. T .<_ ( P .\/ Q ) -> C =/= P ) ) |
41 |
14 40
|
mpd |
|- ( ph -> C =/= P ) |
42 |
41
|
adantr |
|- ( ( ph /\ C =/= S ) -> C =/= P ) |
43 |
|
simpr |
|- ( ( ph /\ C =/= S ) -> C =/= S ) |
44 |
21 2 4
|
latmle2 |
|- ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( P .\/ S ) ) |
45 |
18 19 20 44
|
syl3anc |
|- ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( P .\/ S ) ) |
46 |
9 45
|
eqbrtrid |
|- ( ph -> C .<_ ( P .\/ S ) ) |
47 |
46
|
adantr |
|- ( ( ph /\ C =/= S ) -> C .<_ ( P .\/ S ) ) |
48 |
1
|
4atexlems |
|- ( ph -> S e. A ) |
49 |
1 2 3 5
|
4atexlempns |
|- ( ph -> P =/= S ) |
50 |
5 2 3
|
cvlsupr2 |
|- ( ( K e. CvLat /\ ( P e. A /\ S e. A /\ C e. A ) /\ P =/= S ) -> ( ( P .\/ C ) = ( S .\/ C ) <-> ( C =/= P /\ C =/= S /\ C .<_ ( P .\/ S ) ) ) ) |
51 |
32 33 48 10 49 50
|
syl131anc |
|- ( ph -> ( ( P .\/ C ) = ( S .\/ C ) <-> ( C =/= P /\ C =/= S /\ C .<_ ( P .\/ S ) ) ) ) |
52 |
51
|
adantr |
|- ( ( ph /\ C =/= S ) -> ( ( P .\/ C ) = ( S .\/ C ) <-> ( C =/= P /\ C =/= S /\ C .<_ ( P .\/ S ) ) ) ) |
53 |
42 43 47 52
|
mpbir3and |
|- ( ( ph /\ C =/= S ) -> ( P .\/ C ) = ( S .\/ C ) ) |
54 |
|
breq1 |
|- ( z = C -> ( z .<_ W <-> C .<_ W ) ) |
55 |
54
|
notbid |
|- ( z = C -> ( -. z .<_ W <-> -. C .<_ W ) ) |
56 |
|
oveq2 |
|- ( z = C -> ( P .\/ z ) = ( P .\/ C ) ) |
57 |
|
oveq2 |
|- ( z = C -> ( S .\/ z ) = ( S .\/ C ) ) |
58 |
56 57
|
eqeq12d |
|- ( z = C -> ( ( P .\/ z ) = ( S .\/ z ) <-> ( P .\/ C ) = ( S .\/ C ) ) ) |
59 |
55 58
|
anbi12d |
|- ( z = C -> ( ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) <-> ( -. C .<_ W /\ ( P .\/ C ) = ( S .\/ C ) ) ) ) |
60 |
59
|
rspcev |
|- ( ( C e. A /\ ( -. C .<_ W /\ ( P .\/ C ) = ( S .\/ C ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |
61 |
11 13 53 60
|
syl12anc |
|- ( ( ph /\ C =/= S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |