| 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 |
|
dalempnes.o |
|- O = ( LPlanes ` K ) |
| 6 |
|
dalempnes.y |
|- Y = ( ( P .\/ Q ) .\/ R ) |
| 7 |
1
|
dalemkelat |
|- ( ph -> K e. Lat ) |
| 8 |
1 4
|
dalempeb |
|- ( ph -> P e. ( Base ` K ) ) |
| 9 |
1
|
dalemkehl |
|- ( ph -> K e. HL ) |
| 10 |
1
|
dalemqea |
|- ( ph -> Q e. A ) |
| 11 |
1
|
dalemrea |
|- ( ph -> R e. A ) |
| 12 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
| 13 |
12 3 4
|
hlatjcl |
|- ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) ) |
| 14 |
9 10 11 13
|
syl3anc |
|- ( ph -> ( Q .\/ R ) e. ( Base ` K ) ) |
| 15 |
12 2 3
|
latlej1 |
|- ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) -> P .<_ ( P .\/ ( Q .\/ R ) ) ) |
| 16 |
7 8 14 15
|
syl3anc |
|- ( ph -> P .<_ ( P .\/ ( Q .\/ R ) ) ) |
| 17 |
1
|
dalempea |
|- ( ph -> P e. A ) |
| 18 |
3 4
|
hlatjass |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) ) |
| 19 |
9 17 10 11 18
|
syl13anc |
|- ( ph -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) ) |
| 20 |
16 19
|
breqtrrd |
|- ( ph -> P .<_ ( ( P .\/ Q ) .\/ R ) ) |
| 21 |
20 6
|
breqtrrdi |
|- ( ph -> P .<_ Y ) |