Step |
Hyp |
Ref |
Expression |
1 |
|
atltcvr.s |
|- .< = ( lt ` K ) |
2 |
|
atltcvr.j |
|- .\/ = ( join ` K ) |
3 |
|
atltcvr.a |
|- A = ( Atoms ` K ) |
4 |
|
atltcvr.c |
|- C = ( |
5 |
|
oveq1 |
|- ( Q = R -> ( Q .\/ R ) = ( R .\/ R ) ) |
6 |
|
simpr3 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. A ) |
7 |
2 3
|
hlatjidm |
|- ( ( K e. HL /\ R e. A ) -> ( R .\/ R ) = R ) |
8 |
6 7
|
syldan |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( R .\/ R ) = R ) |
9 |
5 8
|
sylan9eqr |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q = R ) -> ( Q .\/ R ) = R ) |
10 |
9
|
breq2d |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q = R ) -> ( P .< ( Q .\/ R ) <-> P .< R ) ) |
11 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
12 |
11
|
adantr |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. AtLat ) |
13 |
|
simpr1 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. A ) |
14 |
1 3
|
atnlt |
|- ( ( K e. AtLat /\ P e. A /\ R e. A ) -> -. P .< R ) |
15 |
12 13 6 14
|
syl3anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> -. P .< R ) |
16 |
15
|
pm2.21d |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .< R -> P C ( Q .\/ R ) ) ) |
17 |
16
|
adantr |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q = R ) -> ( P .< R -> P C ( Q .\/ R ) ) ) |
18 |
10 17
|
sylbid |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q = R ) -> ( P .< ( Q .\/ R ) -> P C ( Q .\/ R ) ) ) |
19 |
|
simpl |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. HL ) |
20 |
|
hllat |
|- ( K e. HL -> K e. Lat ) |
21 |
20
|
adantr |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. Lat ) |
22 |
|
simpr2 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. A ) |
23 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
24 |
23 3
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
25 |
22 24
|
syl |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. ( Base ` K ) ) |
26 |
23 3
|
atbase |
|- ( R e. A -> R e. ( Base ` K ) ) |
27 |
6 26
|
syl |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. ( Base ` K ) ) |
28 |
23 2
|
latjcl |
|- ( ( K e. Lat /\ Q e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( Q .\/ R ) e. ( Base ` K ) ) |
29 |
21 25 27 28
|
syl3anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( Q .\/ R ) e. ( Base ` K ) ) |
30 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
31 |
30 1
|
pltle |
|- ( ( K e. HL /\ P e. A /\ ( Q .\/ R ) e. ( Base ` K ) ) -> ( P .< ( Q .\/ R ) -> P ( le ` K ) ( Q .\/ R ) ) ) |
32 |
19 13 29 31
|
syl3anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .< ( Q .\/ R ) -> P ( le ` K ) ( Q .\/ R ) ) ) |
33 |
32
|
adantr |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q =/= R ) -> ( P .< ( Q .\/ R ) -> P ( le ` K ) ( Q .\/ R ) ) ) |
34 |
|
simpll |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) -> K e. HL ) |
35 |
|
simplr |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) -> ( P e. A /\ Q e. A /\ R e. A ) ) |
36 |
|
simpr |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) -> ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) |
37 |
34 35 36
|
3jca |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) -> ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) ) |
38 |
37
|
anassrs |
|- ( ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q =/= R ) /\ P ( le ` K ) ( Q .\/ R ) ) -> ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) ) |
39 |
30 2 4 3
|
atcvrj2 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) -> P C ( Q .\/ R ) ) |
40 |
38 39
|
syl |
|- ( ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q =/= R ) /\ P ( le ` K ) ( Q .\/ R ) ) -> P C ( Q .\/ R ) ) |
41 |
40
|
ex |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q =/= R ) -> ( P ( le ` K ) ( Q .\/ R ) -> P C ( Q .\/ R ) ) ) |
42 |
33 41
|
syld |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q =/= R ) -> ( P .< ( Q .\/ R ) -> P C ( Q .\/ R ) ) ) |
43 |
18 42
|
pm2.61dane |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .< ( Q .\/ R ) -> P C ( Q .\/ R ) ) ) |
44 |
23 3
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
45 |
13 44
|
syl |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. ( Base ` K ) ) |
46 |
23 1 4
|
cvrlt |
|- ( ( ( K e. HL /\ P e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) /\ P C ( Q .\/ R ) ) -> P .< ( Q .\/ R ) ) |
47 |
46
|
ex |
|- ( ( K e. HL /\ P e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) -> ( P C ( Q .\/ R ) -> P .< ( Q .\/ R ) ) ) |
48 |
19 45 29 47
|
syl3anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P C ( Q .\/ R ) -> P .< ( Q .\/ R ) ) ) |
49 |
43 48
|
impbid |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .< ( Q .\/ R ) <-> P C ( Q .\/ R ) ) ) |