Step |
Hyp |
Ref |
Expression |
1 |
|
dihjatc1.b |
|- B = ( Base ` K ) |
2 |
|
dihjatc1.l |
|- .<_ = ( le ` K ) |
3 |
|
dihjatc1.h |
|- H = ( LHyp ` K ) |
4 |
|
dihjatc1.j |
|- .\/ = ( join ` K ) |
5 |
|
dihjatc1.m |
|- ./\ = ( meet ` K ) |
6 |
|
dihjatc1.a |
|- A = ( Atoms ` K ) |
7 |
|
dihjatc1.u |
|- U = ( ( DVecH ` K ) ` W ) |
8 |
|
dihjatc1.s |
|- .(+) = ( LSSum ` U ) |
9 |
|
dihjatc1.i |
|- I = ( ( DIsoH ` K ) ` W ) |
10 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> K e. HL ) |
11 |
10
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> K e. Lat ) |
12 |
|
simp2l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> Q e. A ) |
13 |
1 6
|
atbase |
|- ( Q e. A -> Q e. B ) |
14 |
12 13
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> Q e. B ) |
15 |
|
simp12 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> X e. B ) |
16 |
|
simp13 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> Y e. B ) |
17 |
1 5
|
latmcl |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B ) |
18 |
11 15 16 17
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( X ./\ Y ) e. B ) |
19 |
1 4
|
latjcom |
|- ( ( K e. Lat /\ Q e. B /\ ( X ./\ Y ) e. B ) -> ( Q .\/ ( X ./\ Y ) ) = ( ( X ./\ Y ) .\/ Q ) ) |
20 |
11 14 18 19
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( Q .\/ ( X ./\ Y ) ) = ( ( X ./\ Y ) .\/ Q ) ) |
21 |
20
|
fveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( I ` ( Q .\/ ( X ./\ Y ) ) ) = ( I ` ( ( X ./\ Y ) .\/ Q ) ) ) |
22 |
1 2 3 4 5 6 7 8 9
|
dihjatc1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( I ` ( ( X ./\ Y ) .\/ Q ) ) = ( ( I ` Q ) .(+) ( I ` ( X ./\ Y ) ) ) ) |
23 |
21 22
|
eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( I ` ( Q .\/ ( X ./\ Y ) ) ) = ( ( I ` Q ) .(+) ( I ` ( X ./\ Y ) ) ) ) |