| 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 |
|
simpr |
|- ( ( ph /\ Y = Z ) -> Y = Z ) |
| 8 |
1 3 4
|
dalemqrprot |
|- ( ph -> ( ( Q .\/ R ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) ) |
| 9 |
5 8
|
eqtr4id |
|- ( ph -> Y = ( ( Q .\/ R ) .\/ P ) ) |
| 10 |
9
|
adantr |
|- ( ( ph /\ Y = Z ) -> Y = ( ( Q .\/ R ) .\/ P ) ) |
| 11 |
1
|
dalemkehl |
|- ( ph -> K e. HL ) |
| 12 |
1
|
dalemtea |
|- ( ph -> T e. A ) |
| 13 |
1
|
dalemuea |
|- ( ph -> U e. A ) |
| 14 |
1
|
dalemsea |
|- ( ph -> S e. A ) |
| 15 |
3 4
|
hlatjrot |
|- ( ( K e. HL /\ ( T e. A /\ U e. A /\ S e. A ) ) -> ( ( T .\/ U ) .\/ S ) = ( ( S .\/ T ) .\/ U ) ) |
| 16 |
11 12 13 14 15
|
syl13anc |
|- ( ph -> ( ( T .\/ U ) .\/ S ) = ( ( S .\/ T ) .\/ U ) ) |
| 17 |
6 16
|
eqtr4id |
|- ( ph -> Z = ( ( T .\/ U ) .\/ S ) ) |
| 18 |
17
|
adantr |
|- ( ( ph /\ Y = Z ) -> Z = ( ( T .\/ U ) .\/ S ) ) |
| 19 |
7 10 18
|
3eqtr3d |
|- ( ( ph /\ Y = Z ) -> ( ( Q .\/ R ) .\/ P ) = ( ( T .\/ U ) .\/ S ) ) |