Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme0.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme0.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme0.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme0.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme0.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme0.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. HL ) |
8 |
7
|
hllatd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. Lat ) |
9 |
|
simp2ll |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A ) |
10 |
|
simp2rl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A ) |
11 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
12 |
11 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
13 |
7 9 10 12
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
14 |
|
simp1r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> W e. H ) |
15 |
11 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
16 |
14 15
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> W e. ( Base ` K ) ) |
17 |
11 1 3
|
latmle2 |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W ) |
18 |
8 13 16 17
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( ( P .\/ Q ) ./\ W ) .<_ W ) |
19 |
6 18
|
eqbrtrid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> U .<_ W ) |
20 |
|
simp2lr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W ) |
21 |
|
nbrne2 |
|- ( ( U .<_ W /\ -. P .<_ W ) -> U =/= P ) |
22 |
19 20 21
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> U =/= P ) |
23 |
|
simp2rr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. Q .<_ W ) |
24 |
|
nbrne2 |
|- ( ( U .<_ W /\ -. Q .<_ W ) -> U =/= Q ) |
25 |
19 23 24
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> U =/= Q ) |
26 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) ) |
27 |
1 2 3 4 5 6
|
cdlemeulpq |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) ) |
28 |
26 9 10 27
|
syl12anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> U .<_ ( P .\/ Q ) ) |
29 |
22 25 28
|
3jca |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( U =/= P /\ U =/= Q /\ U .<_ ( P .\/ Q ) ) ) |
30 |
29 19
|
jca |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( ( U =/= P /\ U =/= Q /\ U .<_ ( P .\/ Q ) ) /\ U .<_ W ) ) |