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 |
|
dalem1.o |
|- O = ( LPlanes ` K ) |
6 |
|
dalem1.y |
|- Y = ( ( P .\/ Q ) .\/ R ) |
7 |
1
|
dalemkehl |
|- ( ph -> K e. HL ) |
8 |
1
|
dalempea |
|- ( ph -> P e. A ) |
9 |
1
|
dalemqea |
|- ( ph -> Q e. A ) |
10 |
1
|
dalemsea |
|- ( ph -> S e. A ) |
11 |
1
|
dalemtea |
|- ( ph -> T e. A ) |
12 |
3 4
|
hlatj4 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) = ( ( P .\/ S ) .\/ ( Q .\/ T ) ) ) |
13 |
7 8 9 10 11 12
|
syl122anc |
|- ( ph -> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) = ( ( P .\/ S ) .\/ ( Q .\/ T ) ) ) |
14 |
1 2 3 4 5 6
|
dalempjsen |
|- ( ph -> ( P .\/ S ) e. ( LLines ` K ) ) |
15 |
1 2 3 4 5 6
|
dalemqnet |
|- ( ph -> Q =/= T ) |
16 |
|
eqid |
|- ( LLines ` K ) = ( LLines ` K ) |
17 |
3 4 16
|
llni2 |
|- ( ( ( K e. HL /\ Q e. A /\ T e. A ) /\ Q =/= T ) -> ( Q .\/ T ) e. ( LLines ` K ) ) |
18 |
7 9 11 15 17
|
syl31anc |
|- ( ph -> ( Q .\/ T ) e. ( LLines ` K ) ) |
19 |
1 2 3 4 5 6
|
dalem1 |
|- ( ph -> ( P .\/ S ) =/= ( Q .\/ T ) ) |
20 |
1 2 3 4 5 6
|
dalemcea |
|- ( ph -> C e. A ) |
21 |
1
|
dalemclpjs |
|- ( ph -> C .<_ ( P .\/ S ) ) |
22 |
1
|
dalemclqjt |
|- ( ph -> C .<_ ( Q .\/ T ) ) |
23 |
|
eqid |
|- ( meet ` K ) = ( meet ` K ) |
24 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
25 |
2 23 24 4 16
|
2llnm4 |
|- ( ( K e. HL /\ ( C e. A /\ ( P .\/ S ) e. ( LLines ` K ) /\ ( Q .\/ T ) e. ( LLines ` K ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) ) ) -> ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) =/= ( 0. ` K ) ) |
26 |
7 20 14 18 21 22 25
|
syl132anc |
|- ( ph -> ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) =/= ( 0. ` K ) ) |
27 |
23 24 4 16
|
2llnmat |
|- ( ( ( K e. HL /\ ( P .\/ S ) e. ( LLines ` K ) /\ ( Q .\/ T ) e. ( LLines ` K ) ) /\ ( ( P .\/ S ) =/= ( Q .\/ T ) /\ ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) =/= ( 0. ` K ) ) ) -> ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) e. A ) |
28 |
7 14 18 19 26 27
|
syl32anc |
|- ( ph -> ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) e. A ) |
29 |
3 23 4 16 5
|
2llnmj |
|- ( ( K e. HL /\ ( P .\/ S ) e. ( LLines ` K ) /\ ( Q .\/ T ) e. ( LLines ` K ) ) -> ( ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) e. A <-> ( ( P .\/ S ) .\/ ( Q .\/ T ) ) e. O ) ) |
30 |
7 14 18 29
|
syl3anc |
|- ( ph -> ( ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) e. A <-> ( ( P .\/ S ) .\/ ( Q .\/ T ) ) e. O ) ) |
31 |
28 30
|
mpbid |
|- ( ph -> ( ( P .\/ S ) .\/ ( Q .\/ T ) ) e. O ) |
32 |
13 31
|
eqeltrd |
|- ( ph -> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) e. O ) |