Step |
Hyp |
Ref |
Expression |
1 |
|
dalema.ph |
|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) ) |
2 |
|
dalemc.l |
|- .<_ = ( le ` K ) |
3 |
|
dalemc.j |
|- .\/ = ( join ` K ) |
4 |
|
dalemc.a |
|- A = ( Atoms ` K ) |
5 |
|
dalem11.m |
|- ./\ = ( meet ` K ) |
6 |
|
dalem11.o |
|- O = ( LPlanes ` K ) |
7 |
|
dalem11.y |
|- Y = ( ( P .\/ Q ) .\/ R ) |
8 |
|
dalem11.z |
|- Z = ( ( S .\/ T ) .\/ U ) |
9 |
|
dalem11.x |
|- X = ( Y ./\ Z ) |
10 |
|
dalem11.e |
|- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) ) |
11 |
1 2 3 4 7 8
|
dalemrot |
|- ( ph -> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) ) |
12 |
|
biid |
|- ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) ) |
13 |
|
eqid |
|- ( ( Q .\/ R ) .\/ P ) = ( ( Q .\/ R ) .\/ P ) |
14 |
|
eqid |
|- ( ( T .\/ U ) .\/ S ) = ( ( T .\/ U ) .\/ S ) |
15 |
|
eqid |
|- ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) ) = ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) ) |
16 |
12 2 3 4 5 6 13 14 15 10
|
dalem10 |
|- ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) -> E .<_ ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) ) ) |
17 |
11 16
|
syl |
|- ( ph -> E .<_ ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) ) ) |
18 |
1 3 4
|
dalemqrprot |
|- ( ph -> ( ( Q .\/ R ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) ) |
19 |
7 18
|
eqtr4id |
|- ( ph -> Y = ( ( Q .\/ R ) .\/ P ) ) |
20 |
1
|
dalemkehl |
|- ( ph -> K e. HL ) |
21 |
1
|
dalemtea |
|- ( ph -> T e. A ) |
22 |
1
|
dalemuea |
|- ( ph -> U e. A ) |
23 |
1
|
dalemsea |
|- ( ph -> S e. A ) |
24 |
3 4
|
hlatjrot |
|- ( ( K e. HL /\ ( T e. A /\ U e. A /\ S e. A ) ) -> ( ( T .\/ U ) .\/ S ) = ( ( S .\/ T ) .\/ U ) ) |
25 |
20 21 22 23 24
|
syl13anc |
|- ( ph -> ( ( T .\/ U ) .\/ S ) = ( ( S .\/ T ) .\/ U ) ) |
26 |
8 25
|
eqtr4id |
|- ( ph -> Z = ( ( T .\/ U ) .\/ S ) ) |
27 |
19 26
|
oveq12d |
|- ( ph -> ( Y ./\ Z ) = ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) ) ) |
28 |
9 27
|
syl5eq |
|- ( ph -> X = ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) ) ) |
29 |
17 28
|
breqtrrd |
|- ( ph -> E .<_ X ) |