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 |
|
dalem15.m |
|- ./\ = ( meet ` K ) |
6 |
|
dalem15.n |
|- N = ( LLines ` K ) |
7 |
|
dalem15.o |
|- O = ( LPlanes ` K ) |
8 |
|
dalem15.y |
|- Y = ( ( P .\/ Q ) .\/ R ) |
9 |
|
dalem15.z |
|- Z = ( ( S .\/ T ) .\/ U ) |
10 |
|
dalem15.x |
|- X = ( Y ./\ Z ) |
11 |
|
eqid |
|- ( LVols ` K ) = ( LVols ` K ) |
12 |
|
eqid |
|- ( Y .\/ C ) = ( Y .\/ C ) |
13 |
1 2 3 4 7 11 8 9 12
|
dalem14 |
|- ( ( ph /\ Y =/= Z ) -> ( Y .\/ Z ) e. ( LVols ` K ) ) |
14 |
1
|
dalemkehl |
|- ( ph -> K e. HL ) |
15 |
1
|
dalemyeo |
|- ( ph -> Y e. O ) |
16 |
1
|
dalemzeo |
|- ( ph -> Z e. O ) |
17 |
3 5 6 7 11
|
2lplnmj |
|- ( ( K e. HL /\ Y e. O /\ Z e. O ) -> ( ( Y ./\ Z ) e. N <-> ( Y .\/ Z ) e. ( LVols ` K ) ) ) |
18 |
14 15 16 17
|
syl3anc |
|- ( ph -> ( ( Y ./\ Z ) e. N <-> ( Y .\/ Z ) e. ( LVols ` K ) ) ) |
19 |
18
|
adantr |
|- ( ( ph /\ Y =/= Z ) -> ( ( Y ./\ Z ) e. N <-> ( Y .\/ Z ) e. ( LVols ` K ) ) ) |
20 |
13 19
|
mpbird |
|- ( ( ph /\ Y =/= Z ) -> ( Y ./\ Z ) e. N ) |
21 |
10 20
|
eqeltrid |
|- ( ( ph /\ Y =/= Z ) -> X e. N ) |