| 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 |
|
dalem9.o |
|- O = ( LPlanes ` K ) |
| 6 |
|
dalem9.v |
|- V = ( LVols ` K ) |
| 7 |
|
dalem9.y |
|- Y = ( ( P .\/ Q ) .\/ R ) |
| 8 |
|
dalem9.z |
|- Z = ( ( S .\/ T ) .\/ U ) |
| 9 |
|
dalem9.w |
|- W = ( Y .\/ C ) |
| 10 |
1
|
dalemkehl |
|- ( ph -> K e. HL ) |
| 11 |
10
|
adantr |
|- ( ( ph /\ Y =/= Z ) -> K e. HL ) |
| 12 |
1
|
dalemyeo |
|- ( ph -> Y e. O ) |
| 13 |
12
|
adantr |
|- ( ( ph /\ Y =/= Z ) -> Y e. O ) |
| 14 |
1 2 3 4 5 7
|
dalemcea |
|- ( ph -> C e. A ) |
| 15 |
14
|
adantr |
|- ( ( ph /\ Y =/= Z ) -> C e. A ) |
| 16 |
1 2 3 4 5 7 8
|
dalem-cly |
|- ( ( ph /\ Y =/= Z ) -> -. C .<_ Y ) |
| 17 |
2 3 4 5 6
|
lvoli3 |
|- ( ( ( K e. HL /\ Y e. O /\ C e. A ) /\ -. C .<_ Y ) -> ( Y .\/ C ) e. V ) |
| 18 |
11 13 15 16 17
|
syl31anc |
|- ( ( ph /\ Y =/= Z ) -> ( Y .\/ C ) e. V ) |
| 19 |
9 18
|
eqeltrid |
|- ( ( ph /\ Y =/= Z ) -> W e. V ) |