Step |
Hyp |
Ref |
Expression |
1 |
|
atcvreq0.b |
|- B = ( Base ` K ) |
2 |
|
atcvreq0.l |
|- .<_ = ( le ` K ) |
3 |
|
atcvreq0.z |
|- .0. = ( 0. ` K ) |
4 |
|
atcvreq0.c |
|- C = ( |
5 |
|
atcvreq0.a |
|- A = ( Atoms ` K ) |
6 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
7 |
1 6 3
|
atl0le |
|- ( ( K e. AtLat /\ X e. B ) -> .0. ( le ` K ) X ) |
8 |
7
|
3adant3 |
|- ( ( K e. AtLat /\ X e. B /\ P e. A ) -> .0. ( le ` K ) X ) |
9 |
8
|
adantr |
|- ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ X C P ) -> .0. ( le ` K ) X ) |
10 |
1 5
|
atbase |
|- ( P e. A -> P e. B ) |
11 |
|
eqid |
|- ( lt ` K ) = ( lt ` K ) |
12 |
1 11 4
|
cvrlt |
|- ( ( ( K e. AtLat /\ X e. B /\ P e. B ) /\ X C P ) -> X ( lt ` K ) P ) |
13 |
10 12
|
syl3anl3 |
|- ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ X C P ) -> X ( lt ` K ) P ) |
14 |
|
atlpos |
|- ( K e. AtLat -> K e. Poset ) |
15 |
14
|
3ad2ant1 |
|- ( ( K e. AtLat /\ X e. B /\ P e. A ) -> K e. Poset ) |
16 |
15
|
adantr |
|- ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ X C P ) -> K e. Poset ) |
17 |
1 3
|
atl0cl |
|- ( K e. AtLat -> .0. e. B ) |
18 |
17
|
3ad2ant1 |
|- ( ( K e. AtLat /\ X e. B /\ P e. A ) -> .0. e. B ) |
19 |
18
|
adantr |
|- ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ X C P ) -> .0. e. B ) |
20 |
10
|
3ad2ant3 |
|- ( ( K e. AtLat /\ X e. B /\ P e. A ) -> P e. B ) |
21 |
20
|
adantr |
|- ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ X C P ) -> P e. B ) |
22 |
|
simpl2 |
|- ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ X C P ) -> X e. B ) |
23 |
3 4 5
|
atcvr0 |
|- ( ( K e. AtLat /\ P e. A ) -> .0. C P ) |
24 |
23
|
3adant2 |
|- ( ( K e. AtLat /\ X e. B /\ P e. A ) -> .0. C P ) |
25 |
24
|
adantr |
|- ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ X C P ) -> .0. C P ) |
26 |
1 6 11 4
|
cvrnbtwn3 |
|- ( ( K e. Poset /\ ( .0. e. B /\ P e. B /\ X e. B ) /\ .0. C P ) -> ( ( .0. ( le ` K ) X /\ X ( lt ` K ) P ) <-> .0. = X ) ) |
27 |
16 19 21 22 25 26
|
syl131anc |
|- ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ X C P ) -> ( ( .0. ( le ` K ) X /\ X ( lt ` K ) P ) <-> .0. = X ) ) |
28 |
9 13 27
|
mpbi2and |
|- ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ X C P ) -> .0. = X ) |
29 |
28
|
eqcomd |
|- ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ X C P ) -> X = .0. ) |
30 |
29
|
ex |
|- ( ( K e. AtLat /\ X e. B /\ P e. A ) -> ( X C P -> X = .0. ) ) |
31 |
|
breq1 |
|- ( X = .0. -> ( X C P <-> .0. C P ) ) |
32 |
24 31
|
syl5ibrcom |
|- ( ( K e. AtLat /\ X e. B /\ P e. A ) -> ( X = .0. -> X C P ) ) |
33 |
30 32
|
impbid |
|- ( ( K e. AtLat /\ X e. B /\ P e. A ) -> ( X C P <-> X = .0. ) ) |