Step |
Hyp |
Ref |
Expression |
1 |
|
lvoli3.l |
|- .<_ = ( le ` K ) |
2 |
|
lvoli3.j |
|- .\/ = ( join ` K ) |
3 |
|
lvoli3.a |
|- A = ( Atoms ` K ) |
4 |
|
lvoli3.p |
|- P = ( LPlanes ` K ) |
5 |
|
lvoli3.v |
|- V = ( LVols ` K ) |
6 |
|
simpl2 |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> X e. P ) |
7 |
|
simpl3 |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> Q e. A ) |
8 |
|
simpr |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> -. Q .<_ X ) |
9 |
|
eqidd |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> ( X .\/ Q ) = ( X .\/ Q ) ) |
10 |
|
breq2 |
|- ( y = X -> ( r .<_ y <-> r .<_ X ) ) |
11 |
10
|
notbid |
|- ( y = X -> ( -. r .<_ y <-> -. r .<_ X ) ) |
12 |
|
oveq1 |
|- ( y = X -> ( y .\/ r ) = ( X .\/ r ) ) |
13 |
12
|
eqeq2d |
|- ( y = X -> ( ( X .\/ Q ) = ( y .\/ r ) <-> ( X .\/ Q ) = ( X .\/ r ) ) ) |
14 |
11 13
|
anbi12d |
|- ( y = X -> ( ( -. r .<_ y /\ ( X .\/ Q ) = ( y .\/ r ) ) <-> ( -. r .<_ X /\ ( X .\/ Q ) = ( X .\/ r ) ) ) ) |
15 |
|
breq1 |
|- ( r = Q -> ( r .<_ X <-> Q .<_ X ) ) |
16 |
15
|
notbid |
|- ( r = Q -> ( -. r .<_ X <-> -. Q .<_ X ) ) |
17 |
|
oveq2 |
|- ( r = Q -> ( X .\/ r ) = ( X .\/ Q ) ) |
18 |
17
|
eqeq2d |
|- ( r = Q -> ( ( X .\/ Q ) = ( X .\/ r ) <-> ( X .\/ Q ) = ( X .\/ Q ) ) ) |
19 |
16 18
|
anbi12d |
|- ( r = Q -> ( ( -. r .<_ X /\ ( X .\/ Q ) = ( X .\/ r ) ) <-> ( -. Q .<_ X /\ ( X .\/ Q ) = ( X .\/ Q ) ) ) ) |
20 |
14 19
|
rspc2ev |
|- ( ( X e. P /\ Q e. A /\ ( -. Q .<_ X /\ ( X .\/ Q ) = ( X .\/ Q ) ) ) -> E. y e. P E. r e. A ( -. r .<_ y /\ ( X .\/ Q ) = ( y .\/ r ) ) ) |
21 |
6 7 8 9 20
|
syl112anc |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> E. y e. P E. r e. A ( -. r .<_ y /\ ( X .\/ Q ) = ( y .\/ r ) ) ) |
22 |
|
simpl1 |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> K e. HL ) |
23 |
22
|
hllatd |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> K e. Lat ) |
24 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
25 |
24 4
|
lplnbase |
|- ( X e. P -> X e. ( Base ` K ) ) |
26 |
6 25
|
syl |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> X e. ( Base ` K ) ) |
27 |
24 3
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
28 |
7 27
|
syl |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> Q e. ( Base ` K ) ) |
29 |
24 2
|
latjcl |
|- ( ( K e. Lat /\ X e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( X .\/ Q ) e. ( Base ` K ) ) |
30 |
23 26 28 29
|
syl3anc |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> ( X .\/ Q ) e. ( Base ` K ) ) |
31 |
24 1 2 3 4 5
|
islvol3 |
|- ( ( K e. HL /\ ( X .\/ Q ) e. ( Base ` K ) ) -> ( ( X .\/ Q ) e. V <-> E. y e. P E. r e. A ( -. r .<_ y /\ ( X .\/ Q ) = ( y .\/ r ) ) ) ) |
32 |
22 30 31
|
syl2anc |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> ( ( X .\/ Q ) e. V <-> E. y e. P E. r e. A ( -. r .<_ y /\ ( X .\/ Q ) = ( y .\/ r ) ) ) ) |
33 |
21 32
|
mpbird |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> ( X .\/ Q ) e. V ) |