Step |
Hyp |
Ref |
Expression |
1 |
|
dih2dimb.l |
|- .<_ = ( le ` K ) |
2 |
|
dih2dimb.j |
|- .\/ = ( join ` K ) |
3 |
|
dih2dimb.a |
|- A = ( Atoms ` K ) |
4 |
|
dih2dimb.h |
|- H = ( LHyp ` K ) |
5 |
|
dih2dimb.u |
|- U = ( ( DVecH ` K ) ` W ) |
6 |
|
dih2dimb.s |
|- .(+) = ( LSSum ` U ) |
7 |
|
dih2dimb.i |
|- I = ( ( DIsoH ` K ) ` W ) |
8 |
|
dih2dimb.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
9 |
|
dih2dimb.p |
|- ( ph -> ( P e. A /\ P .<_ W ) ) |
10 |
|
dih2dimb.q |
|- ( ph -> ( Q e. A /\ Q .<_ W ) ) |
11 |
|
eqid |
|- ( ( DIsoB ` K ) ` W ) = ( ( DIsoB ` K ) ` W ) |
12 |
1 2 3 4 5 6 11 8 9 10
|
dib2dim |
|- ( ph -> ( ( ( DIsoB ` K ) ` W ) ` ( P .\/ Q ) ) C_ ( ( ( ( DIsoB ` K ) ` W ) ` P ) .(+) ( ( ( DIsoB ` K ) ` W ) ` Q ) ) ) |
13 |
8
|
simpld |
|- ( ph -> K e. HL ) |
14 |
9
|
simpld |
|- ( ph -> P e. A ) |
15 |
10
|
simpld |
|- ( ph -> Q e. A ) |
16 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
17 |
16 2 3
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
18 |
13 14 15 17
|
syl3anc |
|- ( ph -> ( P .\/ Q ) e. ( Base ` K ) ) |
19 |
9
|
simprd |
|- ( ph -> P .<_ W ) |
20 |
10
|
simprd |
|- ( ph -> Q .<_ W ) |
21 |
13
|
hllatd |
|- ( ph -> K e. Lat ) |
22 |
16 3
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
23 |
14 22
|
syl |
|- ( ph -> P e. ( Base ` K ) ) |
24 |
16 3
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
25 |
15 24
|
syl |
|- ( ph -> Q e. ( Base ` K ) ) |
26 |
8
|
simprd |
|- ( ph -> W e. H ) |
27 |
16 4
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
28 |
26 27
|
syl |
|- ( ph -> W e. ( Base ` K ) ) |
29 |
16 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( P .<_ W /\ Q .<_ W ) <-> ( P .\/ Q ) .<_ W ) ) |
30 |
21 23 25 28 29
|
syl13anc |
|- ( ph -> ( ( P .<_ W /\ Q .<_ W ) <-> ( P .\/ Q ) .<_ W ) ) |
31 |
19 20 30
|
mpbi2and |
|- ( ph -> ( P .\/ Q ) .<_ W ) |
32 |
16 1 4 7 11
|
dihvalb |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ ( P .\/ Q ) .<_ W ) ) -> ( I ` ( P .\/ Q ) ) = ( ( ( DIsoB ` K ) ` W ) ` ( P .\/ Q ) ) ) |
33 |
8 18 31 32
|
syl12anc |
|- ( ph -> ( I ` ( P .\/ Q ) ) = ( ( ( DIsoB ` K ) ` W ) ` ( P .\/ Q ) ) ) |
34 |
16 1 4 7 11
|
dihvalb |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. ( Base ` K ) /\ P .<_ W ) ) -> ( I ` P ) = ( ( ( DIsoB ` K ) ` W ) ` P ) ) |
35 |
8 23 19 34
|
syl12anc |
|- ( ph -> ( I ` P ) = ( ( ( DIsoB ` K ) ` W ) ` P ) ) |
36 |
16 1 4 7 11
|
dihvalb |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. ( Base ` K ) /\ Q .<_ W ) ) -> ( I ` Q ) = ( ( ( DIsoB ` K ) ` W ) ` Q ) ) |
37 |
8 25 20 36
|
syl12anc |
|- ( ph -> ( I ` Q ) = ( ( ( DIsoB ` K ) ` W ) ` Q ) ) |
38 |
35 37
|
oveq12d |
|- ( ph -> ( ( I ` P ) .(+) ( I ` Q ) ) = ( ( ( ( DIsoB ` K ) ` W ) ` P ) .(+) ( ( ( DIsoB ` K ) ` W ) ` Q ) ) ) |
39 |
12 33 38
|
3sstr4d |
|- ( ph -> ( I ` ( P .\/ Q ) ) C_ ( ( I ` P ) .(+) ( I ` Q ) ) ) |