| 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 |
|
dalemrot.y |
|- Y = ( ( P .\/ Q ) .\/ R ) |
| 6 |
|
dalemrot.z |
|- Z = ( ( S .\/ T ) .\/ U ) |
| 7 |
1
|
dalemkehl |
|- ( ph -> K e. HL ) |
| 8 |
1 4
|
dalemceb |
|- ( ph -> C e. ( Base ` K ) ) |
| 9 |
7 8
|
jca |
|- ( ph -> ( K e. HL /\ C e. ( Base ` K ) ) ) |
| 10 |
1
|
dalemqea |
|- ( ph -> Q e. A ) |
| 11 |
1
|
dalemrea |
|- ( ph -> R e. A ) |
| 12 |
1
|
dalempea |
|- ( ph -> P e. A ) |
| 13 |
10 11 12
|
3jca |
|- ( ph -> ( Q e. A /\ R e. A /\ P e. A ) ) |
| 14 |
1
|
dalemtea |
|- ( ph -> T e. A ) |
| 15 |
1
|
dalemuea |
|- ( ph -> U e. A ) |
| 16 |
1
|
dalemsea |
|- ( ph -> S e. A ) |
| 17 |
14 15 16
|
3jca |
|- ( ph -> ( T e. A /\ U e. A /\ S e. A ) ) |
| 18 |
9 13 17
|
3jca |
|- ( ph -> ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) ) |
| 19 |
1 3 4
|
dalemqrprot |
|- ( ph -> ( ( Q .\/ R ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) ) |
| 20 |
1
|
dalemyeo |
|- ( ph -> Y e. O ) |
| 21 |
5 20
|
eqeltrrid |
|- ( ph -> ( ( P .\/ Q ) .\/ R ) e. O ) |
| 22 |
19 21
|
eqeltrd |
|- ( ph -> ( ( Q .\/ R ) .\/ P ) e. O ) |
| 23 |
3 4
|
hlatjrot |
|- ( ( K e. HL /\ ( T e. A /\ U e. A /\ S e. A ) ) -> ( ( T .\/ U ) .\/ S ) = ( ( S .\/ T ) .\/ U ) ) |
| 24 |
7 14 15 16 23
|
syl13anc |
|- ( ph -> ( ( T .\/ U ) .\/ S ) = ( ( S .\/ T ) .\/ U ) ) |
| 25 |
1
|
dalemzeo |
|- ( ph -> Z e. O ) |
| 26 |
6 25
|
eqeltrrid |
|- ( ph -> ( ( S .\/ T ) .\/ U ) e. O ) |
| 27 |
24 26
|
eqeltrd |
|- ( ph -> ( ( T .\/ U ) .\/ S ) e. O ) |
| 28 |
22 27
|
jca |
|- ( ph -> ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) ) |
| 29 |
|
simp312 |
|- ( ( ( ( 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 ) ) ) ) -> -. C .<_ ( Q .\/ R ) ) |
| 30 |
1 29
|
sylbi |
|- ( ph -> -. C .<_ ( Q .\/ R ) ) |
| 31 |
|
simp313 |
|- ( ( ( ( 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 ) ) ) ) -> -. C .<_ ( R .\/ P ) ) |
| 32 |
1 31
|
sylbi |
|- ( ph -> -. C .<_ ( R .\/ P ) ) |
| 33 |
1
|
dalem-clpjq |
|- ( ph -> -. C .<_ ( P .\/ Q ) ) |
| 34 |
30 32 33
|
3jca |
|- ( ph -> ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) ) |
| 35 |
|
simp322 |
|- ( ( ( ( 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 ) ) ) ) -> -. C .<_ ( T .\/ U ) ) |
| 36 |
1 35
|
sylbi |
|- ( ph -> -. C .<_ ( T .\/ U ) ) |
| 37 |
|
simp323 |
|- ( ( ( ( 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 ) ) ) ) -> -. C .<_ ( U .\/ S ) ) |
| 38 |
1 37
|
sylbi |
|- ( ph -> -. C .<_ ( U .\/ S ) ) |
| 39 |
|
simp321 |
|- ( ( ( ( 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 ) ) ) ) -> -. C .<_ ( S .\/ T ) ) |
| 40 |
1 39
|
sylbi |
|- ( ph -> -. C .<_ ( S .\/ T ) ) |
| 41 |
36 38 40
|
3jca |
|- ( ph -> ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) ) |
| 42 |
1
|
dalemclqjt |
|- ( ph -> C .<_ ( Q .\/ T ) ) |
| 43 |
1
|
dalemclrju |
|- ( ph -> C .<_ ( R .\/ U ) ) |
| 44 |
1
|
dalemclpjs |
|- ( ph -> C .<_ ( P .\/ S ) ) |
| 45 |
42 43 44
|
3jca |
|- ( ph -> ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) |
| 46 |
34 41 45
|
3jca |
|- ( ph -> ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) |
| 47 |
18 28 46
|
3jca |
|- ( ph -> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) ) |