Step |
Hyp |
Ref |
Expression |
1 |
|
dalem.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 |
|
dalem.l |
|- .<_ = ( le ` K ) |
3 |
|
dalem.j |
|- .\/ = ( join ` K ) |
4 |
|
dalem.a |
|- A = ( Atoms ` K ) |
5 |
|
dalem.ps |
|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) ) |
6 |
1
|
dalemkehl |
|- ( ph -> K e. HL ) |
7 |
6
|
adantr |
|- ( ( ph /\ ps ) -> K e. HL ) |
8 |
5
|
dalemccea |
|- ( ps -> c e. A ) |
9 |
8
|
adantl |
|- ( ( ph /\ ps ) -> c e. A ) |
10 |
5
|
dalemddea |
|- ( ps -> d e. A ) |
11 |
10
|
adantl |
|- ( ( ph /\ ps ) -> d e. A ) |
12 |
5
|
dalemccnedd |
|- ( ps -> c =/= d ) |
13 |
12
|
adantl |
|- ( ( ph /\ ps ) -> c =/= d ) |
14 |
|
eqid |
|- ( LLines ` K ) = ( LLines ` K ) |
15 |
3 4 14
|
llni2 |
|- ( ( ( K e. HL /\ c e. A /\ d e. A ) /\ c =/= d ) -> ( c .\/ d ) e. ( LLines ` K ) ) |
16 |
7 9 11 13 15
|
syl31anc |
|- ( ( ph /\ ps ) -> ( c .\/ d ) e. ( LLines ` K ) ) |