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 |
5
|
dalemddea |
|- ( ps -> d e. A ) |
7 |
5
|
dalemccea |
|- ( ps -> c e. A ) |
8 |
6 7
|
jca |
|- ( ps -> ( d e. A /\ c e. A ) ) |
9 |
8
|
3ad2ant3 |
|- ( ( ph /\ Y = Z /\ ps ) -> ( d e. A /\ c e. A ) ) |
10 |
5
|
dalem-ddly |
|- ( ps -> -. d .<_ Y ) |
11 |
10
|
3ad2ant3 |
|- ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Y ) |
12 |
|
simp2 |
|- ( ( ph /\ Y = Z /\ ps ) -> Y = Z ) |
13 |
12
|
breq2d |
|- ( ( ph /\ Y = Z /\ ps ) -> ( d .<_ Y <-> d .<_ Z ) ) |
14 |
11 13
|
mtbid |
|- ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Z ) |
15 |
5
|
dalemccnedd |
|- ( ps -> c =/= d ) |
16 |
15
|
3ad2ant3 |
|- ( ( ph /\ Y = Z /\ ps ) -> c =/= d ) |
17 |
5
|
dalem-ccly |
|- ( ps -> -. c .<_ Y ) |
18 |
17
|
3ad2ant3 |
|- ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Y ) |
19 |
12
|
breq2d |
|- ( ( ph /\ Y = Z /\ ps ) -> ( c .<_ Y <-> c .<_ Z ) ) |
20 |
18 19
|
mtbid |
|- ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Z ) |
21 |
5
|
dalemclccjdd |
|- ( ps -> C .<_ ( c .\/ d ) ) |
22 |
21
|
3ad2ant3 |
|- ( ( ph /\ Y = Z /\ ps ) -> C .<_ ( c .\/ d ) ) |
23 |
1
|
dalemkehl |
|- ( ph -> K e. HL ) |
24 |
23
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> K e. HL ) |
25 |
7
|
3ad2ant3 |
|- ( ( ph /\ Y = Z /\ ps ) -> c e. A ) |
26 |
6
|
3ad2ant3 |
|- ( ( ph /\ Y = Z /\ ps ) -> d e. A ) |
27 |
3 4
|
hlatjcom |
|- ( ( K e. HL /\ c e. A /\ d e. A ) -> ( c .\/ d ) = ( d .\/ c ) ) |
28 |
24 25 26 27
|
syl3anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ d ) = ( d .\/ c ) ) |
29 |
22 28
|
breqtrd |
|- ( ( ph /\ Y = Z /\ ps ) -> C .<_ ( d .\/ c ) ) |
30 |
16 20 29
|
3jca |
|- ( ( ph /\ Y = Z /\ ps ) -> ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) ) |
31 |
9 14 30
|
3jca |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( d e. A /\ c e. A ) /\ -. d .<_ Z /\ ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) ) ) |