Step |
Hyp |
Ref |
Expression |
1 |
|
lplnn0.z |
|- .0. = ( 0. ` K ) |
2 |
|
lplnn0.p |
|- P = ( LPlanes ` K ) |
3 |
|
eqid |
|- ( Atoms ` K ) = ( Atoms ` K ) |
4 |
3
|
atex |
|- ( K e. HL -> ( Atoms ` K ) =/= (/) ) |
5 |
|
n0 |
|- ( ( Atoms ` K ) =/= (/) <-> E. p p e. ( Atoms ` K ) ) |
6 |
4 5
|
sylib |
|- ( K e. HL -> E. p p e. ( Atoms ` K ) ) |
7 |
6
|
adantr |
|- ( ( K e. HL /\ X e. P ) -> E. p p e. ( Atoms ` K ) ) |
8 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
9 |
8 3 2
|
lplnnleat |
|- ( ( K e. HL /\ X e. P /\ p e. ( Atoms ` K ) ) -> -. X ( le ` K ) p ) |
10 |
9
|
3expa |
|- ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> -. X ( le ` K ) p ) |
11 |
|
hlop |
|- ( K e. HL -> K e. OP ) |
12 |
11
|
ad2antrr |
|- ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> K e. OP ) |
13 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
14 |
13 3
|
atbase |
|- ( p e. ( Atoms ` K ) -> p e. ( Base ` K ) ) |
15 |
14
|
adantl |
|- ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> p e. ( Base ` K ) ) |
16 |
13 8 1
|
op0le |
|- ( ( K e. OP /\ p e. ( Base ` K ) ) -> .0. ( le ` K ) p ) |
17 |
12 15 16
|
syl2anc |
|- ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> .0. ( le ` K ) p ) |
18 |
|
breq1 |
|- ( X = .0. -> ( X ( le ` K ) p <-> .0. ( le ` K ) p ) ) |
19 |
17 18
|
syl5ibrcom |
|- ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> ( X = .0. -> X ( le ` K ) p ) ) |
20 |
19
|
necon3bd |
|- ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> ( -. X ( le ` K ) p -> X =/= .0. ) ) |
21 |
10 20
|
mpd |
|- ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> X =/= .0. ) |
22 |
7 21
|
exlimddv |
|- ( ( K e. HL /\ X e. P ) -> X =/= .0. ) |