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 |
|
dalemrotps.y |
|- Y = ( ( P .\/ Q ) .\/ R ) |
7 |
5
|
dalemccea |
|- ( ps -> c e. A ) |
8 |
5
|
dalemddea |
|- ( ps -> d e. A ) |
9 |
7 8
|
jca |
|- ( ps -> ( c e. A /\ d e. A ) ) |
10 |
9
|
adantl |
|- ( ( ph /\ ps ) -> ( c e. A /\ d e. A ) ) |
11 |
5
|
dalem-ccly |
|- ( ps -> -. c .<_ Y ) |
12 |
11
|
adantl |
|- ( ( ph /\ ps ) -> -. c .<_ Y ) |
13 |
1 3 4
|
dalemqrprot |
|- ( ph -> ( ( Q .\/ R ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) ) |
14 |
6 13
|
eqtr4id |
|- ( ph -> Y = ( ( Q .\/ R ) .\/ P ) ) |
15 |
14
|
breq2d |
|- ( ph -> ( c .<_ Y <-> c .<_ ( ( Q .\/ R ) .\/ P ) ) ) |
16 |
15
|
adantr |
|- ( ( ph /\ ps ) -> ( c .<_ Y <-> c .<_ ( ( Q .\/ R ) .\/ P ) ) ) |
17 |
12 16
|
mtbid |
|- ( ( ph /\ ps ) -> -. c .<_ ( ( Q .\/ R ) .\/ P ) ) |
18 |
5
|
dalemccnedd |
|- ( ps -> c =/= d ) |
19 |
18
|
necomd |
|- ( ps -> d =/= c ) |
20 |
19
|
adantl |
|- ( ( ph /\ ps ) -> d =/= c ) |
21 |
5
|
dalem-ddly |
|- ( ps -> -. d .<_ Y ) |
22 |
21
|
adantl |
|- ( ( ph /\ ps ) -> -. d .<_ Y ) |
23 |
14
|
breq2d |
|- ( ph -> ( d .<_ Y <-> d .<_ ( ( Q .\/ R ) .\/ P ) ) ) |
24 |
23
|
adantr |
|- ( ( ph /\ ps ) -> ( d .<_ Y <-> d .<_ ( ( Q .\/ R ) .\/ P ) ) ) |
25 |
22 24
|
mtbid |
|- ( ( ph /\ ps ) -> -. d .<_ ( ( Q .\/ R ) .\/ P ) ) |
26 |
5
|
dalemclccjdd |
|- ( ps -> C .<_ ( c .\/ d ) ) |
27 |
26
|
adantl |
|- ( ( ph /\ ps ) -> C .<_ ( c .\/ d ) ) |
28 |
20 25 27
|
3jca |
|- ( ( ph /\ ps ) -> ( d =/= c /\ -. d .<_ ( ( Q .\/ R ) .\/ P ) /\ C .<_ ( c .\/ d ) ) ) |
29 |
10 17 28
|
3jca |
|- ( ( ph /\ ps ) -> ( ( c e. A /\ d e. A ) /\ -. c .<_ ( ( Q .\/ R ) .\/ P ) /\ ( d =/= c /\ -. d .<_ ( ( Q .\/ R ) .\/ P ) /\ C .<_ ( c .\/ d ) ) ) ) |