Step |
Hyp |
Ref |
Expression |
1 |
|
cvlexch.b |
|- B = ( Base ` K ) |
2 |
|
cvlexch.l |
|- .<_ = ( le ` K ) |
3 |
|
cvlexch.j |
|- .\/ = ( join ` K ) |
4 |
|
cvlexch.a |
|- A = ( Atoms ` K ) |
5 |
|
cvllat |
|- ( K e. CvLat -> K e. Lat ) |
6 |
5
|
adantr |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> K e. Lat ) |
7 |
|
simpr3 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> X e. B ) |
8 |
|
simpr2 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> Q e. A ) |
9 |
1 4
|
atbase |
|- ( Q e. A -> Q e. B ) |
10 |
8 9
|
syl |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> Q e. B ) |
11 |
1 2 3
|
latlej1 |
|- ( ( K e. Lat /\ X e. B /\ Q e. B ) -> X .<_ ( X .\/ Q ) ) |
12 |
6 7 10 11
|
syl3anc |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> X .<_ ( X .\/ Q ) ) |
13 |
12
|
3adant3 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> X .<_ ( X .\/ Q ) ) |
14 |
13
|
adantr |
|- ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) /\ P .<_ ( X .\/ Q ) ) -> X .<_ ( X .\/ Q ) ) |
15 |
|
simpr |
|- ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) /\ P .<_ ( X .\/ Q ) ) -> P .<_ ( X .\/ Q ) ) |
16 |
|
simpr1 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> P e. A ) |
17 |
1 4
|
atbase |
|- ( P e. A -> P e. B ) |
18 |
16 17
|
syl |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> P e. B ) |
19 |
1 3
|
latjcl |
|- ( ( K e. Lat /\ X e. B /\ Q e. B ) -> ( X .\/ Q ) e. B ) |
20 |
6 7 10 19
|
syl3anc |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> ( X .\/ Q ) e. B ) |
21 |
1 2 3
|
latjle12 |
|- ( ( K e. Lat /\ ( X e. B /\ P e. B /\ ( X .\/ Q ) e. B ) ) -> ( ( X .<_ ( X .\/ Q ) /\ P .<_ ( X .\/ Q ) ) <-> ( X .\/ P ) .<_ ( X .\/ Q ) ) ) |
22 |
6 7 18 20 21
|
syl13anc |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> ( ( X .<_ ( X .\/ Q ) /\ P .<_ ( X .\/ Q ) ) <-> ( X .\/ P ) .<_ ( X .\/ Q ) ) ) |
23 |
22
|
3adant3 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( ( X .<_ ( X .\/ Q ) /\ P .<_ ( X .\/ Q ) ) <-> ( X .\/ P ) .<_ ( X .\/ Q ) ) ) |
24 |
23
|
adantr |
|- ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) /\ P .<_ ( X .\/ Q ) ) -> ( ( X .<_ ( X .\/ Q ) /\ P .<_ ( X .\/ Q ) ) <-> ( X .\/ P ) .<_ ( X .\/ Q ) ) ) |
25 |
14 15 24
|
mpbi2and |
|- ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) /\ P .<_ ( X .\/ Q ) ) -> ( X .\/ P ) .<_ ( X .\/ Q ) ) |
26 |
1 2 3
|
latlej1 |
|- ( ( K e. Lat /\ X e. B /\ P e. B ) -> X .<_ ( X .\/ P ) ) |
27 |
6 7 18 26
|
syl3anc |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> X .<_ ( X .\/ P ) ) |
28 |
27
|
3adant3 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> X .<_ ( X .\/ P ) ) |
29 |
28
|
adantr |
|- ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) /\ P .<_ ( X .\/ Q ) ) -> X .<_ ( X .\/ P ) ) |
30 |
1 2 3 4
|
cvlexch1 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( X .\/ Q ) -> Q .<_ ( X .\/ P ) ) ) |
31 |
30
|
imp |
|- ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) /\ P .<_ ( X .\/ Q ) ) -> Q .<_ ( X .\/ P ) ) |
32 |
1 3
|
latjcl |
|- ( ( K e. Lat /\ X e. B /\ P e. B ) -> ( X .\/ P ) e. B ) |
33 |
6 7 18 32
|
syl3anc |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> ( X .\/ P ) e. B ) |
34 |
1 2 3
|
latjle12 |
|- ( ( K e. Lat /\ ( X e. B /\ Q e. B /\ ( X .\/ P ) e. B ) ) -> ( ( X .<_ ( X .\/ P ) /\ Q .<_ ( X .\/ P ) ) <-> ( X .\/ Q ) .<_ ( X .\/ P ) ) ) |
35 |
6 7 10 33 34
|
syl13anc |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> ( ( X .<_ ( X .\/ P ) /\ Q .<_ ( X .\/ P ) ) <-> ( X .\/ Q ) .<_ ( X .\/ P ) ) ) |
36 |
35
|
3adant3 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( ( X .<_ ( X .\/ P ) /\ Q .<_ ( X .\/ P ) ) <-> ( X .\/ Q ) .<_ ( X .\/ P ) ) ) |
37 |
36
|
adantr |
|- ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) /\ P .<_ ( X .\/ Q ) ) -> ( ( X .<_ ( X .\/ P ) /\ Q .<_ ( X .\/ P ) ) <-> ( X .\/ Q ) .<_ ( X .\/ P ) ) ) |
38 |
29 31 37
|
mpbi2and |
|- ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) /\ P .<_ ( X .\/ Q ) ) -> ( X .\/ Q ) .<_ ( X .\/ P ) ) |
39 |
1 2
|
latasymb |
|- ( ( K e. Lat /\ ( X .\/ P ) e. B /\ ( X .\/ Q ) e. B ) -> ( ( ( X .\/ P ) .<_ ( X .\/ Q ) /\ ( X .\/ Q ) .<_ ( X .\/ P ) ) <-> ( X .\/ P ) = ( X .\/ Q ) ) ) |
40 |
6 33 20 39
|
syl3anc |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> ( ( ( X .\/ P ) .<_ ( X .\/ Q ) /\ ( X .\/ Q ) .<_ ( X .\/ P ) ) <-> ( X .\/ P ) = ( X .\/ Q ) ) ) |
41 |
40
|
3adant3 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( ( ( X .\/ P ) .<_ ( X .\/ Q ) /\ ( X .\/ Q ) .<_ ( X .\/ P ) ) <-> ( X .\/ P ) = ( X .\/ Q ) ) ) |
42 |
41
|
adantr |
|- ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) /\ P .<_ ( X .\/ Q ) ) -> ( ( ( X .\/ P ) .<_ ( X .\/ Q ) /\ ( X .\/ Q ) .<_ ( X .\/ P ) ) <-> ( X .\/ P ) = ( X .\/ Q ) ) ) |
43 |
25 38 42
|
mpbi2and |
|- ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) /\ P .<_ ( X .\/ Q ) ) -> ( X .\/ P ) = ( X .\/ Q ) ) |
44 |
43
|
ex |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( X .\/ Q ) -> ( X .\/ P ) = ( X .\/ Q ) ) ) |
45 |
1 2 3
|
latlej2 |
|- ( ( K e. Lat /\ X e. B /\ P e. B ) -> P .<_ ( X .\/ P ) ) |
46 |
6 7 18 45
|
syl3anc |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> P .<_ ( X .\/ P ) ) |
47 |
|
breq2 |
|- ( ( X .\/ P ) = ( X .\/ Q ) -> ( P .<_ ( X .\/ P ) <-> P .<_ ( X .\/ Q ) ) ) |
48 |
46 47
|
syl5ibcom |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> ( ( X .\/ P ) = ( X .\/ Q ) -> P .<_ ( X .\/ Q ) ) ) |
49 |
48
|
3adant3 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( ( X .\/ P ) = ( X .\/ Q ) -> P .<_ ( X .\/ Q ) ) ) |
50 |
44 49
|
impbid |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( X .\/ Q ) <-> ( X .\/ P ) = ( X .\/ Q ) ) ) |